SPECIFIC HEAT MEASUREMENTS OF SOME SOLID … · Chapter 9 FURTHER DISCUSSION OF THE RESULTS 9-1 —...

SPECIFIC HEAT MEASUREMENTS OF SOME SOLID GASES

IN A He3 CRT OS TAT

by

JOHN C. BURFORD

© John C. Burford 1967

A Thesis

submi-t-ted in partial fulfillment of the requirements for the

degree of Doctor of Philosophy in the University of Toronto.

September 1967

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

V*--CONTENTS

Page

Abstract x

Chapter 1 INTRODUCTION1.1 — The Third Law of Thermodynamics-

The Residual Entropy. . ......11.2 — The Electronic Entropy of Oxygen............ .....20

Chapter 2 DESCRIPTION OF THE APPARATUS2.1 - Introduction................ 232.2 - The Main Features of the Cryostat.......... .....242 .3 - The Calorimeter............... .......... ......... 262.4 - The Heat Switch........................... ......282.5 - The Vacuum Systems....................... ..312.6 - The He 3 Systems....... 322.7 - The Gas Handling System...........................362.8 - The Electrical Systems............... 38

Chapter 3 PERFORMANCE OF THE APPARATUS ' - - —

3.1 — Preparation of the Samples....................... 463.2 — Preparations for the Specific Heat

Measurements............ . .51


3*3 - Procedure of the Measurements «........... .58

Chapter 4 THERMOMETER CALIBRATION AND TEMPERATURE .SCALES

4*1 — Introduction......... 61*>

4.2 — Considerations of the Choice ofCalorimeter Thermometer....6l

4*3 “ Thermometer Calibration Procedure. ..... .644.4 — Discussion of Errors in the

Temperature Scales........ 694*5 — The Thermometer Calibration

Interpolation Formulas.....76

Chapter 5 THE DATA HANDLING - THE CALORIMETER AND

.. COPPER MEASUREMENTS

5.1 - Reduction of the Raw Data ........ 835*2 - The Calorimeter Heat Capacity...................85

5-3 - The Copper Measurements............ ....90

Chapter 6 THE OXYGEN AND NITROGEN RESULTS

6.1 — The Purpose of the Measurements....... 966.2 — Presentation and Discussion of the Results 98

Chapter 7 THE "CARBON MONOXIDE AND~NITRIC OXIDE RESULTS

7.1 — Presentation and Discussion of the CO results.1037.2 — Presentation and Discussion of the NO results.104


Chapter 8 MEASUREMENTS ON THE IMPURE SAMPLES8.1 — Account of the Chronological Sequence

of the Experiments,8.2 — Discussion of the Results,..............

,106110

Chapter 9 FURTHER DISCUSSION OF THE RESULTS

9-1 — The Residual Entropy of CO and N O ..,

9-2 — The Effect of Oxygen as an Impurity.112

117

Acknowledgements

References

Appendices

1 .2.

Thermometer Calibration Data Specific Heat Results

2—A The Empty Calorimeter Results2—B The Copper Results

f

2-C The%

Oxygen Results2-D The Nitrogen Results2-E The Carbon Monoxide Results2-F The Nitric Oxide Results —2-G The CO-O2 Results2—H The ^2~^2 Results


ABSTRACT

The specific heats of solid CO, NO, 0 , N , and some dilute mixtures of 02 in CO and N2 in the temperature range 0.6° to 4°K are reported. The measurements were made in a mechanical heat switch calorimeter in a He" cryostat to which had been added a He stage. A commercial germanium resistance thermometer was used which was calibrated against the He^ and He^ vapor pressure scales.

The specific heats of CO and NO were measured in an attempt to settle the question as to the origin of the residual entropy of these two substances. For these cases, there exists a discrepancy between the entropy calculated from spectroscopic data (Sspec) and that calculated from specific heat data (Sca^); the difference Sgpec-Sca^ being called the residual entropy. For many years, the usually accepted explanation for the appearance of a positive, finite value of the residual entropy in the cases of CO and NO has been in terms of *frozen-inT non—equilibrium states of the crystal. For these cases, it was assumed that the orientation of the molecules becomes frozen—in at a high temperature because the forces tending to produce orientational order are insufficient to overcome the high potential barriers to molecular


rotation in the crystal. In this way, the disorder persists to

the absolute zero, resulting in the observed value of the residual entropy.

Recently, considerable doubt as to the validity of this

kind of argument has been raised, especially because molecular

rotation in solid CH^ has been demonstrated even at 1.8°K from a recent spin—lattice relaxation study. In an attempt to find a

specific heat anomaly which could remove the residual entropy,

this study of CO and NO was undertaken. No anomalous behavior in

either case was revealed down to 0.6°K. It is pointed out that

our present incomplete knowledge of molecular rotation in solids

at low temperatures needs to be improved by extensive infrared

absorption, spin—lattice relaxation, and other studies in order

to be in a better position to understand the origin of the resid

ual entropy in those few simple substances for which such an effect

persists.

During the course of this work, a sample of CO was found to

have been contaminated with CO2 and air. The specific heat

measurements on this sample revealed a rather broad anomaly, not accounted for by the Debye theory. A subsequent experiment in which more oxygen was deliberately added to the sample showed that the anomaly was caused by the oxygen impurity. A further exp

eriment in this series in which oxygen was added to a nitrogen


host was performed and the results allowed certain conclusions

to be made regarding the origin of the anomaly. The concentrations

of oxygen were very low, generally a few tenths percent.

The results may be interpreted in terms of a model consistent

with the low-lying rotational energy levels of the oxygen molecule.

The observed anomaly was in excellent agreement with the Schottky

anomaly for a system containing two levels with a degeneracy ratio,.j—

upper to lower, of 2:1, and an energy spacing of 5.14°,


CHAPTER 1

INTRODUCTION

1*1 — The Third Law of Thermodynamics, The Residual Entropy.

1. Historical sketch. The Third Law of Thermodynamics

has its origins in physical chemistry and it was mainly the

work of Nernst in the early 1900*s which laid the foundations

for the understanding of the Law as we know it today* Nernst

was originally interested in finding general rules of chemical

equilibria in gas reactions from the application of chemical

thermodynamics to the systems, and from these rather restricted

beginnings, the Law took shape. This section is devoted to a

brief description of the lines of thought which Nernst and

others used in developing the Law. Then the present status

of the Law will be made apparent, and finally, the case for a

re-examination of the simple gases, CO, ^ 0 and NO, which

apparently do not conform to the Law will be presented.

As a basis for predicting the conditions of chemical


equilibrium of a gas mixture, Nernst started with the Gibbs-

Helmholtz Equation for chemical reactions at constant volume,

where F is the change in Free Energy, and U is the change in

Internal Energy.

Now if F(T) is known, then U(T) is also known, but not

vice versa, for if there is a solution of Eq. 1.1, F(T), then

any other solution of the form (F(T) + constant x T) will

also be a solution. Of course, there is only one unique F(T)

for any given system under prescribed conditions, and Nernst

felt that the most suitable point of reference for finding the

form of F(T) was at the absolute zero of temperature, since the

term (constant x T) would vanish.

is a change in the total number of degrees of freedom during

a chemical reaction. Thus, 6 f/6 T becomes infinite at the

absolute zero, and as a point of reference for finding F(T),

the absolute zero has no significance. Nernst then turned to

solids, where classically, the specific heats on both sides of

the chemical equation are equal. If this remained true down to

the absolute zero then c> U/ c> T = 0 and c> F/ c> T can remain

Eq. 1.1

In the case of gases, <b U/1) T 0 in general, since there


finite there. Nernst then postulated that for condensed states,

both ^ F / ^ T and & U / & T become zero at the absolute zero,

I*t F/ <3 T — 6V/ T — 0 Nernst Heat TheoremT~> 0

Note that the first condition immediately gives the result that

the change in the entropy vanishes at the absolute zero, but

Nernst did not recognize this, and it was several years before

the attention of physicists was drawn to the theorem. Nernst

published this idea in 1906 and called it a Heat Theorem.

At first , the theorem was restricted to chemical reac

tions between condensed states and no account was taken of

quantum theory. Nernst assumed that the specific heats of the

condensed states remained at their classical values down to

the absolute zero. After the publication of the theorem, Nernst

set about testing it experimentally by measuring the specific

heats of various condensed substances at lower temperatures than

had been obtained previously. He developed the science of low

temperature calorimetry through his design of a hydrogen liqui-

fier and his adiabatic calorimeter. His work soon showed that

the specific heat of all substances tended to zero as the

absolute zero was approached. This was very satisfying to Nernst

because it was the simplest way of satisfying the condition


Lt(T->0) £>U/ <^T = 0, as well as being in agreement with the

trend predicted by Einstein some years earlier using quantum

.mechanics. With this success, Nernst extended his theorem to

include systems which did not undergo chemical reactions, that

is, purely physical systems. The theorem was used to predict

that, like the specific heat, the thermal expansion tended to

zero at the absolute zero, a prediction which was later verified

by Lindemann .

There are several corollaries of the theorem which have

since found general agreement, such as the unattainability of

the absolute zero, but the statement S = 0 at the absolute zero

is of most interest here and it will be discussed in the light

of its meaning in thermodynamics and statistical mechanics.

This statement of what is now known as the Third Law is very

useful in predicting properties of the solid state in terms of

inter-molecular forces, and a good demonstration of this is

found when the law encountered its first difficulty when it was

applied to a certain class of substances.

The result S = 0 at the absolute zero found an immediate explanation in quantum mechanics, namely, that at the absolute

zero, the substance is in a state of perfect order where all

systems are in the ground state, which is non-degenerate. Now

glasses and solutions are definitely not expected to be in this


state of perfect order at the absolute zero , and it was

interesting to Nernst to ask whether the Law could be applied

directly to such systems. Planck and Einstein stated that

such systems should have a non-zero entropy at the absolute

zero, but Nernst maintained that his Law could be applied

directly to any system, for he believed his Law to be a con

sequence of the Second Law and the vanishing specific heats,

and there was no doubt that the specific heats of all sub

stances vanished at the absolute zero, even for disordered

structures. This was one of the major difficulties with the

Law, and led Simon to propose a new formulation in 1927^ ^ stating:

At the absolute zero, the entropy differences

vanish between all those states of a system

between which reversible transitions are

possible, at least in principle.

In spite of these difficulties, it now seems certain

that the state of lowest energy at the absolute zero is one of

perfect order, in which the lattice vibrations have been frozen

out, the spins have been aligned, and so on. There has been

no general proof of this statement, except through the Third

Law, although intuitively it is satisfactory. The only


difficulty seems to be that fox* a certain class of substances,

the perfectly ordered state is not allowed to exist in the solid

and the. solid is then in a non-equilibrium state at the absolute

zero. Glasses and solutions are in this class, and it is

generally thought that crystalline CO, N^O, and NO may also be

included. This idea will be discussed later on.

2* Verification of the Law. The most fundamental way

of verifying the Third Law is to make a comparison between the

values of the entropy of a system evaluated first by the methods

of thermodynamics and second, by the methods of statistical

mechanics. It is noted that the two approaches are fundament

ally different. On the one hand, the thermodynamic approach

describes the macroscopic variables of the system and how they

behave when the system is subjected to changes of heat and work.

On the other hand, the statistical mechanical approach is con

cerned with the microscopic picture, where the behavior of the

particles comprising the system is analysed on an atomic scale.

Now the thermodynamic state of the system is known when a small

number of macroscopic variables is specified, such as the

pressure, the volume, the temperature, the chemical composition,

and so on. However, each thermodynamic state can be realized in

many ways on an atomic scale, and one refers to the number of


•ways of realizing the -thermodynamic state as the number of

micro—states corresponding to the given (thermodynamic) macro-

state. .If the number of different micro—states is g, then

the statistical definition of entropy is S = king, where k is

Boltzmannrs constant. Now the determination of the thermo

dynamic entropy depends on measurements of heat and temperature

(that is, on calorimetry), while the determination of the

statistical entropy depends on counting the number of micro-

states in the given macrostate. Also, the thermodynamic entropy

is obtained by accounting-for the quantities of heat which take

the system from an initial to a final thermodynamic state. Thus,

the value depends on the thermal behavior of the substance all

the, way from the initial to the final state. On the other hand,

the statistical entropy depends only on counting the number of

microstates in the macro-state at which the comparison is being

made, that is, the final thermodynamic state, and it requires

no knowledge of the existence of the solid and liquid phases if

the comparison is made in the gas phase, as is the usual practice

It is a major achievement of physics and chemistry that

the verification of the Third Law is shown by the fact that

the two approaches lead to the same result for the entropy of

most, but not all,substances. Because the Third Law is obeyed

in a great many cases, it was felt desirable to re-examine those


few simple cases which still do not appear to conform to the

requirement S — 0 at the absolute zerOj hut before this matter»

is discussed, it is appropriate to consider in more detail

what is meant by the notion of entropy, both in thermodynamics

and in statistical mechanics, and how the different values are arrived at in practice.

of a chemically pure substance in going reversably from the

where S(0) is the entropy of the system at the absolute zero,

which, according to the Third Law, is zero for stable and metastable equilibrium states.

Experimentally, we measure the amount of heat necessary to raise the temperature of a system which comprises more than one

A'. Entropy in thermodynamics. The entropy change

T*.

A ST,

If we start at the absolute zero (Tj =0), then

T*.

0


phase. Corrections for the higher temperature phase and the

latent heats must be made in order to obtain the specific heat

of the lower temperature phase under its saturated vapor, and

further corrections are necessary to obtain the specific heat

at constant pressure, which is required to determine the entropy

For an account of the procedure involved in making these correct

ions, one is referred to Fagerstroem*s thesis. Then the

entropy may be readily found by integrating Cp/T over the whole

experimental temperature range ^ to T2, which yields a value

of S(T2 ) «• S(Tj^). To obtain S(T^) — S(0), the specific heat is

extrapolated from T^ to the absolute zero. O f course, it is

advisable to extend the measurements to as low a ■ T^ value as

possible in order to be able to make a reliable estimate of

S(Tj^) — S(0). Then, having made our estimate of S(T^) — S(0),

we make use of the Third Law to put S(0) = 0 for all transitions

between states in stable or meta-stable equilibrium. Thus we

obtain S(T ) which may be compared with the value found from

statistical mechanics for the system in the same final thermo

dynamic state. The value of the thermodynamic entropy, S(T ),

found in this way is generally called the calorimetric entropy.

^ c a l •


10

B. Entropy in statistical mechanics. To calculate the entropy on this basis, we must first know the detailed microscopic properties of the systems which comprise the assembly

(the terms here taking their statistical mechanical sense)* It is possible to calculate approximate values of the entropy of

any molecule theoretically by idealizing the translational,

rotational, and vibrational motions. Thus, the translational

entropy is giveli. by the Sackur-Tetrode equation,

trans Nk r^lnT - InP + In fzJTm\ 3/?k5/2 +L 2 V ix2 ) 2 J

the rotational energy is given by the expression,

Ejpob = J(J+l). J = 0,1,2,,.* the rotational21

quantum number. I is the moment

of inertia of the molecule, and the vibrational energy is given by,

Evit) = w(v + ^), v = 0,1,2,... the vibrationalquantum number. w is the vibrational frequency.

The state sum (or partition function) of the assembly is given by,

^tot ^trans* ^rot* ^vib


where S R _3_(TlnZ)\ , where R is the gas constant.I dT /v

However, to obtain accurate values for the entropy, it

is necessary to take into account the facts that the molecule

is not rigid, and that the molecular vibrations are not harmonic

Exact entropy values may be obtained by using the exact spectro

scopically determined energy levels to find the state sum and

from that the entropy. In this way, no use is made of any

mechanical model of the molecule and the method should there

fore yield the most accurately obtainable entropy values,

provided that the degeneracies of the levels are known. However,

it has been shown that empirical formulae for expressing the

energy levels are sufficiently accurate for most purposes, a

fact which contributes a significant reduction in the work of

computation. Following the method developed by Giauque^^, the

centrifugal effects on the end-over-end rotational energies is

expressed by the following expansion formula,

Erot. = B . J (J+l) - D.J2(J+1)2 + ...

and the effect of anharmonicity on the molecular vibrational

energies is,

Evib = we-(v+2') “ xewe(v+2)2 + ...


where the constants B,D,we,xe, ... characteristic of each

molecule are determined from the band spectrum of the gas.

Also, if the molecule is in a higher electronic state, the

rotational and vibrational constants will be altered and

account must be taken of this.

With these equations representing the more accurate

energy levels, the entropy may now be calculated. This is a

very involved process and one is referred to the work of

Giauque^^ for the details. Let it be noted here that the

essence of the method involves the calculation of the rotational

and vibrational entropies together from the combined state sums,

and then the translational entropyj which is found from the

Sackur-Tetrode equation, is added to give the total entropy of

the gas, usually evaluated at the pressure of 1 atm and at the normal boiling point. Thus, we find the statistical entropy of

the gas in the hypothetical perfect gas state. This value of

the entropy (which excludes nuclear spin) is usually called the

spectroscopic entropy. SSpec.

C. Comparison between S5pec and Scaj. At this

point, we are not quite ready to compare SSpec with Sca^ for

the gas at the pressure of 1 atm and at its normal boiling point, because Sgpec here refers to the gas in the perfect gas state,


13

while Scal refers to the real gas. To correct for the non

ideality of the gas, it is usual practice to normalize Scal to

reduce it to the value of the gas in the perfect gas state

since this entropy value is the easier one to correct. First,

the real gas is imagined to have its pressure reduced from 1 to 0 atm where the real and perfect.gases are identical. Then,

the gas in the perfect state is imagined to be compressed back

to 1 atm. The entropy change may be readily calculated from a

knowledge of the equation of state of the real gas. With this

correction made, we are now in a position to compare Sca^ with

SSpec* Table 1.1 shows the entropy values of certain simple

molecular gases. The Sca]. values were obtained from specific

heat measurements usually down to 10 or. 15°K; the region below

these temperatures was covered by an extrapolation where the

specific heat was assumed to conform with the Debye theory.

3# The residual entropy. Reference to Table 1.1 shows

that for the last four entries, Sspec an<* ®cal no't agree. The difference between them is called either the zero—point, or the

residual entropy, Sres,

Sres = SSpec - Seal * the residual entropy.

Note that for these cases, SSpec is always greater than Sca^ .


TABLE 1,1 COMPARISON BETWEEN Sspec AND Scal FOR SOME SIMPLE

MOLECULES

ENTROPY (E.U.)MOLECULE TEMP Sspec Scal Sres REFERENCE

02 B.Pt. 40.68 40.70 - A

N 2 B.Pt. 36.42 36.53 - B

Cl B.Pt. 51.55 51.56 - C40

HC1 B.Pt. 41.45 41.3 - D

CH. B.Pt. 36.61 36.53 ~ E4

CO B.Pt. 37-8 37-O 0.8 F

NO B.Pt. 43.75 43.03 0.72 G

N20 B.Pt. 48.50 47.36 1.14 H

H20 298°K 45.10 44.29 0.81 I

See next page for References


REFERENCES:

A — W. F. Giauque and H. L. Johnston, J.Am.Chem.Soc. £1, 2300 (1929)

B - » and J. 0. Clayton, « » 4875 (1933)

C — ** and T. M. Powell, » n 61, 1970 (1939)

D — w and R. Wiebe, ** ** j[0, 101 (1928)

E *• A. Frank and K. Clusius, Zeit.Phys.Chem. B36. 291 (1937)F — E. K. Gill and J . A. Morrison, J.Chem.Phys. 45. 1585 (1966)

G - H. L. Johnston and W. F. Giauque, J.Am.Chem.Soc. J£l, 3194 (1929)

H - R. W. Blue and W. F. Giauque, J.Am.Chem.Soc. 991 (1935)

X - P. fic.T. <k A. T. A. JTlomSorij T,Cke*\*Phys. 33^175

NOTE: 1 E.U. (Entropy Unit) is equivalent to 1 cal/mole deg.


Therefore, the possibility always exists that the specific

heat measurements were not extended to low enough temperatures

and that an unforeseen specific heat contribution, not accounted

for by the Debye—type extrapolation, may have been overlooked.

For example, in the case of CH^, the first specific heat measure

ments, which extended only down to the liquid hydrogen temper

ature range ^), gave no hint whatever of the low temperature

anomaly which was found in later measurements to lower temper

atures^^. The anomaly which was found completely removed the

apparent disagreement between SSpec and Scai and in order to

reach agreement it was necessary to allow for the existence of

different nuclear spin species of the molecule. This was also true for hydrogen^).

A .second.type of possibility may arise which results in a

finite value of Sres and that is by the display of what is usually

called *frozen«in* disorder, where certain internal states of

the crystal are prevented from attaining equilibrium. A typical

case is a glass, which may be thought of as a rigid super-cooled

liquid which is prevented from crystallizing by the strong internal forces. Thus, even taking specific heat measurements down to the absolute zero on such a substance would yield a

value of Scai which would be too small by an amount equal to the

degree of frozen—in disorder, provided that the disorder


persisted down, to the absolute zero. If so^ then the substance

would not be expected to obey the Third Law which covers only

equilibrium states. This kind of explanation has been used

frequently to explain the residual entropy in CO, 3X141although there has been very little experimental evidence to

* confirm this picture in these cases. Because the types of

crystal disorder which are put forward to explain the residual

entropy in the cases of CO (and N2O) and of NO are different, they will now be discussed separately.

A. Carbon monoxide. The idea in this case is

that as the temperature of the crystal is lowered from the

freezing point, the molecules eventually cease their rotation

at a temperature where the thermal energy is of much the same

value as the height of the potential barriers which oppose

rotation. Now if the ends of the molecule are quite *similart,

then the energy of re«orientation of a molecule in the crystal

will be small and would probably be less than the height of the

potential barriers opposing rotation. In this case, since

rotation is no longer possible, the forces tending to produce

orientational order will be unsufficient to remove the disorder,

which was essentially frozen-in when the rotation died out. If

there is complete orientational disorder, then the residual entropy


16

would be Rln2 = I .38 E.U.. The fact that the observed residual

entropy is less than Rln2 for the case of CO is rationalized

by saying that a partial ordering takes place before re

orientation becomes impossible.

This explanation is commonly put forward for molecules

which have ’similar* (but not identical) ends (see, for example,

pg 63 of Wilks^®^), and it is thought that there is some support for the argument by the fact that molecules with very ’different*

ends do. not display a residual entropy. However, besides the

thorny question as to the meaning of the terms ’similar* and

’different* in this context, the whole acceptance of the argument

rests on the assumption that the CO molecules may not re—orient

themselves at low temperatures. If it can be shown that some

degree of rotation is possible, then the argument cannot hold

water.

Gill and Morrison have examined the various evidence

relating to the question of orientational disorder in crystalline

CO. Unfortunately, there have been no spin—lattice relaxation

measurements on CO which could tell us directly the state of

molecular rotation at low temperatures. However, Gill and

Morrison’s own capacitance data down to 6°K indicates that there is no change in the orientational motion of the molecules doivn

to this temperature. Because of the scant information regarding


the state of rotation in crystalline CO, it seemed worthwhile

to Gill and Morrison to extend the specific heat data of Clayton

and G i a u q u e t o lower temperatures especially because thermo

dynamic and spin-lattice relaxation measurements on crystalline

methane had shown that the usual explanation of the residual

entropy in this case, which was based on frozen—in disorder, was

incorrect. In addition, because methane was found to be able to

rotate fairly readily even down to 1.8°K, and because the

characteristic temperatures of methane and CO are comparable

(indicating a similarity in the effects of the inter-molecular

forces), then it was thought that a similar state of rotation

might appear in the two crystals.

The specific heat data of Gill and Morrison down to 2. 5°K

failed to show an anomalous behavior which would help explain

the residual entropy(9). for this reason, and also because a

third crystal structure modification was suspected to be dis

played at very low temperatures the present work, which

extended the specific heat data to still lower temperatures,

was undertaken. The results of this research are presented and

discussed in Sec. 7.1 and in Sec. 9.1.

B. Nitric oxide. One explanation of the residual

entropy of nitric oxide is as f o l l o w s ^ . Solid nitric oxide

may be considered as a solution of two polymerized isomers


(dimers) which are present in equal amounts. Now liquid nitric

oxide is known to be highly associated into N202 dimers and it is reasonable to suppose that the dimerization persists into

the solid state. If the structure of the dimers can take one

of two different forms,

N N or, N O

0 0 O N

then during solidification, since the energy difference between

the two isomers may be assumed small, the dimers will be present

in equal amounts in the two forms. Since the higher energy

dimer is not able to convert to the other form, then the disorder

persists to the absolute zero. For complete disorder, the

residual entropy would appear to be ^Rln2 per mole of NO (or

Rln2 per mole of N2O2). This value is in very close agreement(12 )with the observed value found by Johnston and Giauque .

However, this explanation may be criticized because it assumes

that all the dimers of each kind are completely oriented, for

the entropy of orientational disorder is neglected and only the

entropy of mixing is considered.

An alternative explanation which considers only the

orientational disorder is as follows (-*-3). The nitric oxide

dimer is assumed to have only one isomeric form,


N 00 N

where, in the crystal, the dimer can take one of two different

orientations. Complete orientational disorder is obtained when

there is no preferred orientation, and where the residual entropy

has the value ^Rln2 per mole of NO. Again, this kind of explan

ation rests on the assumption that the dimers are not able to

re-orient themselves in the lattice. As for the case of CO

(and N2O), this involves the question of molecular rotation in solids at low temperatures; a field which has been largely un

explored. However, some justification for the above viewpoint

has been provided by X-ray studies ^4)^ although this kind of

evidence may not be regarded as entirely convincing. The problem

of dimerization is one which has plagued the interpretation of

the physical behavior of solid oxygen for many years^^, and

since nitric oxide has received less attention .than oxygen in

this matter, we prefer to leave the question open and to investi

gate the behavior of the specific heat at temperatures below those

previously obtained in an attempt to discover a specific heat

anomaly which would remove the residual entropy. For this reason,

specific heat measurements on solid nitric oxide were made and

the results are presented and discussed in Sec. 7*2 and Sec. 9»1«


20

1.2 - The Electronic Entropy of Oxygen.

In the course of this work, some of the gas samples

were found to have been contaminated with air, which led to a

study of the effects of oxygen impurity. In order that the

results from the experiments on the dilute oxygen-CO and

systems may be interpreted, we must consider the molecular

structure of the oxygen molecule - in its electronic ground state.

It is believed that the specific heat results may be interpreted

in terms of a model which contains low-lying energy levels of

the molecule, the populations of which are altered by a change

in temperature, leading to the observed specific heat bumps.3 ~The electronic ground state of the molecule is /

c(see, for example, Hersberg Chapter S) and because the

nuclei have no spin, symmetry requirements of the total wave

function impose the condition that the molecular rotation quantum

number K may take only odd integer values (see, for example, Herzberg (-*-6) page ,i30). Each rotational level is split into

three levels by the interaction of the spin S with the rotation

of the molecule (Hund*s case (b)). While these levels were

originally called F levels by Mulliken, today they are sometimes

called the Kramers—Schiapp levels.

A theoretical treatment of the Kramers-Schlapp levels was

first given by Kramers^^ who considered only the spin-spin


interaction and showed it to be equivalent to a coupling between

the spin and the inter-molecular axis of the molecule. To improve agreement with experiment, S c h l a p p ( ! 8 ) included a term which took

into account the interaction between S and the magnetic field

produced by the molecular rotation. As a further refinement, Mizushima and Hill 9) considered the effect of centrifugal dis

tortion of the molecule and obtained still better agreement with

experiment. Thus, it is thought that the origin of the Kramers-

Schlapp levels is well understood.

The only rotational state of interest here is the ground

(K = l) state, where the total angular momentum quantum number

J ( = K + S) can take one of three values, 2, 1, or 0, correspond

ing to the three different orientations of S with respect to K.

Fig. 1.1 shows the ground state of the free O16O16 molecule.At very low temperatures, only the J = 0 state will be

occupied but as the temperature is raised the upper states will

become gradually filled. This process is accompanied by a Schottky

anomaly in the specific heat, the exact form of which may be

readily calculated from a knowledge of only the spacings of the energy levels and their degeneracies. The general expression for

the Schottky specific heat is,


6P» 5.704° K

£ « 2.702° K

J = 0K =1

FIG. 1.1 TRIPLET SPLITTING OF THE 0,60|g ROTATIONAL GROUND STATE (K = I) DUE TO THE INTERACTION BETWEEN K AND THE SPIN S = l


22

—^Schottky ~ iL_dT

fc=rvR S,E±Si exp(-Ei/kT)I" o ____________L*-n2 Si exp(- £±/kT)

where R is the gas constant, and gj is the degeneracy of the

i*th energy level, £ The entropy under a Schottky anomaly

may be written at once since it depends only on the degeneraciesof the levels. Following Rosenberg

for the Schottky entropy is,

(20), the general expression

SSchottky - R In 2t«0

where gc is the degeneracy of the ground state. Now the degeneracy

of the Kramers-Schlapp levels is given by (2J+1), since there are

(2J+1) distinct quantum states of the molecule corresponding to

the same J value. Thus, for the ground state triplet of the

molecule, the Schottky entropy is S = Rln9 per mole.

It must be remembered that in these discussions we have

referred to the molecule in the free state. However, when we

come to discuss the behavior of the energy levels when the molecule

is in close proximity to other molecules in the solid state, then

the energy level scheme (and hence the size and shape of the

Schottky anomaly) may have to be modified because of the influence

of the internal crystal fields. This matter is discussed in the

light of the results on the dilute oxygen mixtures in Sec. 9*2.


23

CHAPTER 2

DESCRIPTION OF THE APPARATUS

2.1 - Introduction .Essentially, the apparatus consisted of a conventional

low temperature He^ calorimeter cryostat to which was added a He^ stage which was included in order to obtain temperatures

below those normally available in He^ cryostats. Despite its

basic simplicity, the apparatus is rather unusual because it is

thought to be the first cryostat which was designed to measure,

in the He^ temperature range, the heat capacity of substances

which are gaseous at room temperature, other than the inert gas

solids. Most other solid gas calorimeter cryostats have been

designed for use in the interval from room temperature down to

liquid hydrogen temperatures, with occasional investigations in

the He4 temperature range. However, in view of the resurgence of

interest in solid gases which display a residual entropy, it

became desirable to investigate the thermal properties of solid

gases at much lower temperatures than were previously obtained with

them.


24

This chapter is devoted to a description of the apparatus.

First, the function of the main parts of the cryostat will be

described in relationship to each other, and then a more detailed

description of the individual components will be given.

2.2 — The main features of the cryostat.

Fig. 2.1 shows a diagram of the low temperature part of the

apparatus and Fig. 2.2 shows a photograph of the same part.

Referring to the Figures, the heart of the cryostat is the calor

imeter (see Sec. 2.3) which was suspended from its top and bottom

inside the copper shield by eight thin stranded terylene threads

(CoatsT Koban) which passed from the calorimeter to two brass

suspension rings which were mounted on four studded brass posts

fixed to the underside of the shield cap. The attitude of the

calorimeter could be adjusted by means of the pairs of nuts which

secured separately each of the four sides of the suspension rings

onto the posts.

The calorimeter was cooled through the heat switch (see Sec.

2.4) which provided a thermal link between the calorimeter and the He^ refrigerant chamber. The chamber was made of copper and was

an integral part of the shield system, since the chamber and the

shield cap were hard-soldered together. The use of hard solder

instead of the usual soft solder was made in order to reduce

temperature gradients in the shield which might have arisen if


He^pumping tube

Thermol shunt (copper) for gas filling tube

Terminol strip for electrical leads-----------Shield strutsGas filling tube vacuum jacket

Shield cop

Gas filling tube (A)Gas filling tube (B)

Shield con (copper)

Thread tension adjustment nuts

Colorimeter suspension rings (brass)

Interspace pumping tube

— Thermal shunt (copper) for He 3 vapor pressure tube

‘"—Shield pumping tube and vapor pressure tube

dd vacuum jacket

r f ' _ Outer con capHeat switch bellows

Fusite seals

He^vaporpressuresensingtubeHe3 vapor pressure cell(copper)

refrigerantchamber(copper)

Heat switchclampingplate

Outer can (brass)ColorimeterStudded posts (bross)Terylenesuspensionthreads

FIG. 2.1 THE CRYOSTAT


Reproduced with permission of the copyright owner. Further reproduction prohibited without permission

soft solder had been used. This is so because the thermal

conductivity of soft solder is very much reduced when it becomes

superconducting, a phenomenon which does not occur in the hard

solders. Provision was made for using the chamber for a magnetic

thermometer where the susceptibility of a paramagnetic salt

(which could be submerged in the He3 liquid inside the chamber)

could be measured with a set of coils mounted on the outside of

the chamber. The magnetic thermometer was not used in this work.

Mechanical support for the shield was provided by three l/8” thin- wall struts which were placed between the shield and outer can

caps. The struts helped to reduce the strain on the soldered

joints of the shield cap when the heat switch was in operation.

A manganin heater of about 500 ohms was wound non-inductively

around the shield can and a carbon radio resistor, which was used

as the shield thermometer, was placed on the shield cap with a

generous coating of vacuum grease to improve the thermal contact

between the thermometer and the shield.

Connected to the side of the He^ refrigerant chamber was

a He^ vapor pressure ceil which was also made of copper and was

hard-soldered to the chamber. The vapor pressure sensing tube,

which led from the cell, and the gas filling tube which led from

the calorimeter, were both vacuum-jacketed. Part of the way up

the vacuum jackets from the outer can cap, the tubes passed through


separate copper thermal shunts which were included to provide

a path of escape for the heat coming down the tubes and their

vacuum jackets into the He^ bath which sat in the inner Dewar.

The brass outer can and the copper shield can were mounted

to their respective caps using indium . *0 *-rings, according to the usual practice in this laboratory* The cryostat was mounted

inside two glass Dewars, the inner one containing liquid He^ and

the outer one containing liquid nitrogen.

2.3 - The Calorimeter.

The calorimeter (Fig. 2 ,3) was made almost entirely of copper. About 25 thin copper posts were located inside the body

of the calorimeter and hard soldered to the top and bottom to

assist in the rapid attainment of thermal equilibrium during heat

capacity determinations. A manganin heater (resistance about

500 ohms) was wound non-inductively around the side and was therm ally bonded to it by the usual method of using vacuum grease and

adhesive. The four heater leads were connected in pairs (current

and potential) to each end. of the heater wire. The soldered

electrical connections, which were separately coated with nail

varnish to provide electrical insulation, were anchored to the

side of the calorimeter by another layer of nail varnish which

was coated on a thin strip of cigarette paper. The germanium

resistance thermometer was screwed into a threaded well in the


Gas fHIing tube

Heater leods

Paper insulation

me®

Heat switch clamping plate

Heat switch plate support arm

Copper posts

Copper wall

Mangonin heater

Thermometerwell Copper posts

Suspension lugs— 8 o ff -

Germaniumthermometer

FIG. 2.3 THE CALORIMETERReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

base of the calorimeter, the threads of which were coated with

vacuum grease just before the thermometer was screwed into place.

The four thermometer leads were treated in the same way as those

of the heater. This procedure of thermally anchoring the lead

connections to the calorimeter helped prevent thermo-electric

emffs from entering the circuits to spoil the electrical measure

ments. In addition, because the thermometer leads from the ger

manium element were rather thick stranded copper (32 gauge), the total amount of the calorimeter system which was considered taking

part in the heat capacity determinations was more closely defined

when the complete length of copper leads was isothermal with the

calorimeter.

The heat switch plate was a piece of copper which was gold-

plated (see Sec. 2.4)• The switch plate was soldered with a small

quantity of Wood*s metal to the copper heat switch plate supporti ■»arm, which was itself hard—soldered to the calorimeter top. The

switch plate soldering was done with the calorimeter suspension

in tight adjustment and with the switch plate held rigidly between

the jaws. In this way, there resulted a minimum degree of vib

rations when the heat switch was opened, for the plate was then

centered perfectly inside the jaws. This method proved to be

much more convenient than trying to adjust the calorimeter

position with the switch plate already soldered to its arm.


Wood’s metal was used for the solder because of its low melting

point. It was feared that the use of a tin-lead solder might

perhaps have caused some of the other soldered joints on the cal

orimeter to break.

2".4 ~ The Heat Switch.

In a calorimeter cryostat, there must be some mechanism

which provides for the cooling of the calorimeter in a reasonable

time from room temperature, and then isolates it for the specific

heat measurements. Also, a direct thermal link between the

calorimeter and the vapor pressure cell must be established for

the purpose of the thermometer calibration. Such a mechanism

is generally called a heat switch and there are two main basic

designs in common use which satisfy these needs. One is by the

use of exchange gas (usually helium), and the other is by using

a mechanical switch which makes and breaks thermal contact between

the calorimeter and its cooling agent. The main disadvantage of

the former method is connected with the sorption effects of helium

gas, and because the undesirable features drastically increase in

importance below 1°K, it was decided to incorporate a mechanical

switch even though it too had an undesirable feature which was

important below 1° as well as posing major design difficulties.The switch is shown in Fig. 2.4 and the actual operation of

it in an experiment is described in Sec. 3*1* The construction of

the mechanism down to the jaws closely follows the design of


Interspace pumping hole

Stainless steel bellows

Heat switch clamping plate attached to calorim eter

Interspace pumping tube itch roo guideand heat swii

Outer can cap

Rotation of rod produces vertical bellows motion

Shield cap

Jow mechanism support tube

»

Heat switch jaws

Copper braid

Braid ends soldered to He3 vapor pressure cell

FIG 2.4 THE HEAT SWITCHReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

29

(3)FagGrstroem to whose thesis one is referred for a description,

of the entire switch down to this point. However, it was felt

that the method used by Fagerstroem of pressing the calorimeter

down onto the shield floor and raising it for isolation would

have introduced too much heat to the calorimeter by both friction

and vibration effects. The heating effects of friction result

when the heat switch is being opened, that is, when the touching

surfaces are made to part. The heating effects of vibration are

induced after the switch has been opened, where the calorimeter

is set into vibrational motion in its suspension. Therefore, it

was decided to keep the calorimeter firmly fixed in its suspension

and let the mechanism work movable jaws. In this way, it was hoped

to minimize the heating effects of friction which become important

below 1°K. The effort proved later to be successful ; whereas for

Fagerstroem*s switch the frictional heat imputs were of the order

1000 ergs, the switch used in this work gave heat imputs of the order 10 ergs. Also, by using a more rigid suspension system

than that of Fagerstroem, the temperature drifts due to the ext

ernal vibrations were reduced, but still remained the major factor

which limited the precision of the heat capacity measurements

below 1°K.The jaws work on the »lazy tongs * principle which has been

used before in this a p p l i c a t i o n ^2). They were opened and


30

closed by rotating a switch knob at the cryostat head. The touching

surfaces of the jaws and clamping plate were gold-plated. In( 2 )earlier designs of the switchv indium was used in place of

gold for it seemed to offer the advantage of providing a large

area of contact because of its malleability. However, it has been

recently pointed out^22 that to obtain full advantage of this it

is necessary to use a thick indium layer, in which case the poor

thermal conductivity of the superconducting indium would defeat

t le purpose of increasing the switch conductance. Also, it has

been shown(24) that the conductance of a switch depends mainly on the pressure across the jaws and this is independent of the

materials used in its construction. The reason for gold-plating

the switch was to prevent the copper from being oxidized which

would have caused the switch performance to deteriorate because

of the steady accumulation of the poorly conducting oxide over a

period of time.

Each of the switch jaws was thermally anchored separately

to the vapor pressure cell by lengths of copper braid which were

hard-soldered at each end. In this way, no use was made of the

rather poor conductance of the switch mechanism to effect thermal

equilibrium between the calorimeter and the shield. This was of

special importance when calibrating the thermometer because of the

necessity of keeping the conductance of the thermal link between


the calorimeter thermometer* and the vapor pressure cell as large as possible.

2.5 ~ The Vacuum Systems.

The space surrounding the calorimeter, that is, the shield

system, could be evacuated before pre-cooling the cryostat through

a pumping tube which also served as the vacuum Jacket for the He^

vapor pressure sensing tube. The shield system needed to be

pumped only to a rough vacuum before a run, because at low temp

eratures the residual air inside the system was frozen out. How

ever, the. system had., to be leak-tight to a very high degree

particularly to the interspace (the space between the shield and

the outer can) because if any exchange gas had come into contact

with the inside of the shield system at low temperatures, then

the specific heat results would have been in serious error because

of the sorption effects df the helium gas.

The interspace could also be evacuated but here the vacuum

requirements were much more strict. The need to evacuate the

interspace of helium exchange gas at 4»2°K in a reasonable time

determined the choice of a high speed pumping system. To provide

adequate evacuation, the pumping line from the interspace (which

also served as the guide for the heat switch rod) was a 3/8,r thin-

wall monel tube which was connected to a 1” streamline copper tube

through a coupling at the cryostat head. From there, the ln line


went to a valve with a large (about 1«) flow-through diameter.The other side of the valve was connected to a 2^”-throat oil

diffusion pvunp (Edwards Type 203) by a short length of 2" stream

line pipe. The total pumping length from the diffusion pump to

the cryostat head was about four feet, while the length of 3/8” tube inside the cryostat was about two feet. However, with a

considerable part of the latter tube being at a low temperature

throughout a run, the pumping speed of the system was not impaired

very much by its relative narrowness.

A cold cathode vacuum gauge was mounted in the pumping line

ahead of the valve. The gauge was used to monitor the exchange

gas pressure in the interspace. Provision was made for admitting

the exchange gas from the laboratory helium return line through

a by-pass line which was fitted with a needle valve.

2,6 — The Helium Three Systems.

1. The vapor pressure He3 system. The vapor pressure

system is rather simple . The vapor pressure cell was packed

with copper lathe turnings which helped improve the thermal

equilibrium inside it (see Sec. 4»4)« It is estimated that about

one half of the volume inside the cell was available for occup

ation by the He^ liquid, that is, about % ml. The quantity of

He^ inside the system was such that at the lowest temperature,

where the amount of liquid inside the cell was greatest, about


\ ml of liquid was formed. The pressure sensing tube leading

from the cell was l/l6” thin-wall stainless steel and it passed through the thermal shunt on its way up through the cryostat head

and on to the supply tank and manometer, from which was obtainede

the vapor pressure reading.

The manometer limbs were totally enclosed in a wooden case

which served to avoid sudden changes of temperature of the mercury

in the limbs* There was a perspex window in the front and a

fluorescent lamp in the back of the case. Mounted immediately

behind the limbs was a strip of fly-screening (with a mesh of

about 7 lines per cm) which helped to delineate more clearly the mercury meniscus when it was viewed through the cathetometer

telescope. A ground glass screen was placed in front of the

lamp to provide an even illumination of the meniscus over the whole

pressure range. To obtain a reliable reference pressure in one

of the limbs, a mercury diffusion pump with a cold trap was

connected to it. The reference pressure was then taken as zero

and the vapor pressure was given by the difference in height of

the two mercury levels, which were measured with a cathetometer

manufactured by Dumoulin-Froment, Paris.

In order to obtain accurate pressure readings, the density,

and hence the temperature of the mercury in the limbs had to be

known. For this purpose, a mercury—in—glass thermometer was


placed in a test-tube of mercury which was situated inside the

manometer case* The test-tube was made from the same piece of

glass from which the manometer limbs were taken* The limbs were

made from one length of 18 mm pyrex which w;as specially selected from the stock because it had the most perfect bore of all the

pieces. Precautions to find a ,good tube were necessary for two

reasons. First, the pressure readings can be spoiled by optical

refraction effects which occur in tubes which do not have uniform

wall thickness* Second, if the inner diameter is not.uniform

along the tube length, or if the tube is not perfectly round

everywhere inside, then the meniscus depression due to surface

tension may not be the same in either limb. Again, this may

result in errors in the measured pressure since the meniscus

depression was assumed to be the same in both limbs* To test

the bore roundness, several readings of the inner diameter at

each end were taken with a travelling microscope, and to test

the uniformity of the bore along the length of the tube, the

inner diameter readings from each end were compared.

The mercury used in the manometer was obtained from a

commercial source. When received, every bottle shotted a surface

scum despite the label *triple distilledT. To obtain a sample

which was suitable for manometer use, a quantity of the stock

was triple distilled again twice more, and kept under its own


35

vapor until the manometer was ready for filling. To maintain

this high level of cleanliness, the manometer limbs were

carefully cleaned by the usual technique using chromic acid, methyl hydrate, and distilled water.

2. The refrigerant He3 system. The refrigerant He3

system is a little more complex since the liquid He3 inside the chamber was pumped upon during a run. The 1” pumping line was

made from thin—wall monel and led up from the outer can to the

cryostat head where it was coupled by a short length of 2” streamline pipe to an oil diffusion pump (Edwards Type 203). The diff

usion pump was backed by a rotary roughing pump which was sub

merged in an oil bath in order to prevent leakage of the precious

He3 gas through the seals in the drive wheel shaft of the pump.The exhaust of the rotary pump went to aluminum storage tanks

fitted with valves which were closed to conserve the gas when the

system was being leak-tested. The rotary pump exhaust was fitted

with a dial pressure gauge which was used to give- an indication

of the amount of liquid He3 remaining in the refrigerant chamber during a run^ as well as to indicate leaks in the system between

runs.

The pumping tube was fitted with two radiation traps, one

at each of the can caps. The traps prevented room temperature

radiation from reaching the shield. If this precaution were not


taken, then a considerable heating of the low temperature

section might have occurred*

An attempt was made to isolate the cryostat from the

effects of the vibrations of the rotary pump. For this purpose,

the pump was mounted on its own stand separate from the apparatus

frame and some Sylphon bellows were placed in the intake and

exhaust lines. With these precautions the vibrations reaching

the cryostat head were found (by touch of the hand) to be quite

small. However, they were important enough to produce a notice

able heating of the calorimeter system during a run.

2.7 “ The Gas Handling System.

The sample gas handling system is almost identical to that

described by Fagerstroem in his Ph.D. Thesis from this laboratory

The central part of the system, namely, the calibrated volume

gas reservoirj was also part of this apparatus and will not be

described. The only addition was a dial pressure gauge which

indicated the vapor pressure of the condensed sample in the

calorimeter when the supply of gas from the calibrated tank was

shut off. It is noted here that the sample gas was cooled and

condensed in the calorimeter by providing a supply of liquid

nitrogen inside the inner I3ewar (a discussion of the experimental procedure is made in Sec. 3*1)* The gas filling tube leading


37

down from the cryostat head was made up from two tubes of

different diameters (A and B, Fig. 2,1), Initially, tube A

was a length of 0.040" 0D, 0,004" wall and tube B was a length

of 0.022" OD, 0,003" wall, both copper-nickel capillary. When

it was found that these tubes were too narrow to allow the sample

gas to collect as condensate in the calorimeter in reasonable

times, they were changed in favor of wider tubes. The new tubes

are: A - 1/8" 0D, 0,005" wall stainless steelj B - 0,040" OD,

0.004" wall copper-nickel. In addition, the new tube B (total

length, three feet) was coiled in order to increase the path

length for the heat being conducted down it to the calorimeter^

and was heated by a manganin heater (about 10 ohms) which was wound around the tube along its entire length. The heater was

used to accelerate the collection of condensed gas in the calor*?

imeter as well as to reduce the amount of condensate left in the

tube just before cooling the apparatus in preparation for an

experiment. Of course, with these wider tubes the heat leak due

to conduction down them was increased, This is always a consider

ation which must be weighed against the advantages when designing

calorimetric apparatus.

The thermal shunt through which the gas filling tube passed,

and the filling tube vacuum jacket above this point, were provided

with a manganin heater which was used to avoid the premature


condensation of the gas in the upper regions of the filling

tube. Such an occurrence would have plugged up the tube and

prevented any further filling of the calorimeter. Furthermore,

a quantity of condensed gas inside the filling tube would have

resulted in an over-estimate of the quantity actually inside the

calorimeter. While an accumulation of liquid here was not too

serious, it was found in the experiments on NO (which has itsto

normal freezing point above the liquid nitrogen boiling point),

that an accumulation of solid anywhere in the tube had to be

avoided because of the difficulty of unblocking the tube contain

ing solid. The temperature of the shunt was monitored with a

copper-eonstantan thermocouple.

2,8 — The Electrical Systems.

1. The germanium thermometer resistance measuring apparatus.

The resistance of the germanium thermometer was measured with an

isolating potential comparator, the design of which was first

developed by Dauphinee. The comparator used in this work closely

follows his design and to his original paper one is referred for

a full analysis of the p r i n c i p l e ^ . The outline of the basic

principle is as follows, reference being made to Fig. 2,5«The mechanical chopper C first connects the condenser A to

emf 1, where it is charged to the voltage The chopper contacts


C mechanical chopper A condenser G galvanometer

- (a) BASIC COMPAR ATOR -PRINCIPLE

FIG 2 .5 THE ISOLATING COMPARATOR CIRCUIT


45 volt *4r dry cell

Ammeter

w£z5

M |

C |

c2M, , M p , M s , double-^pole, double-throw mechanical

choppers driven at 3 5 c p s .

C|* C2» high quality polystyrene condensers.W standard resistance box S thermometer

(b) BASIC MEASURING CIRCUIT

FIG. 2;5 THE ISOLATING COMPARATORCIRCUIT


open and then close to the other side connecting the condenser

(with the same polarity) to emf 2 through the galvanometer G.

For e^ equal to e2, the total emf round the lopp is zero and

no current flows. If, however, e^ and e2 are not equal, thecondenser charges and discharges through the galvanometer since

the voltage across the condenser alternates between ej and e2.

This process repeats many times a second, since the chopper is

driven at a frequency of 35 cps, and if the galvanometer has asufficiently long time constant, there will appear on it a steady

deflection, indicating an out-of-balance condition. The emf*s

are never directly connected to each other} they are truly isolated *

and a considerable potential difference (V) may exist between

them without difficulty. At balance, there is no current flowing

in the potential leads and their resistance does not affect the

balance condition.

In the actual instrument used here, the voltages to be

compared were those across the thermometer and a reference stand

ard decade resistance box. The chopper chops the current

through the decade box and thermometer into a square wave AC

current. The choppers, and are connected to W and S as

shownj the condensers Cj and C2 bridging their contacts. Choppers

and M2 are driven synchronously with and are phased in such

a way that when the current through W and S is flowing in one


direction* condenser is connected to resistor W and condenser

C2 to S. When the current is reversed, the connections are

reversed also.

If we consider the action of the chopper M. (forgetting

for a moment the condenser C^), while is connected to W, it

charges to the voltage V .. On the next half of the cycle, the

chopper transfers to S, the current meanwhile being reversed.

When connected to S, C-j_ charges or discharges through the galvan

ometer G until it has the voltage -Vg. As this process is repeated

35 times per second, the difference between Vw and -Vs results in a pulsing current i^ through the galvanometer. Similarly, the

operation of M 2 results in a corresponding current i2 if Vw isnot equal to If we assume that for the moment there aresno thermal emf*s or other accidental voltages present in Vg or Vw ,

then any difference AVi = V - V giving rise to current i-, isW 6accompanied by a corresponding difference A V 2 = “*(VW - Vg) which in turn gives rise to the current i2 «

The pulsing currents flow in opposite directions and since

they have a phase difference of 180° in the chopper cycle, they add up to an alternating current i^ + 1 2 -*-n "the potential leads,

which is not blocked by the condenser The desired balance,

where A v ± = A v £ = 0, and hence = ±2 = 0, is indicated by a

null reading on the galvanometer. If now we allow thermal emfTs


in Vw or Vg, the balance condition is not affected; such emf *s

being essentially DC send direct current through C^, which can

only persist until is charged.

The instrument is now seen to combine the advantages of

AC and DC operation, for thermal emf*s are automatically balanced-

out, and on the other hand, it uses a DC galvanometer as null

indicator and the lead resistance does not affect the balance, both

features being characteristic of true potentiometric (DC) methods.

Also, because the voltages are compared under equilibrium cond

itions there is no reactive balancing and so only one reading

(from the resistance box) is required. For these reasons, it

was decided to use this instrument in place of the potentiometer

which has been used almost exclusively in this laboratory for the

purpose of accurate resistance thermometry. In view of its con

siderable advantages, it is surprising that the use of the instru

ment is not reported more often in the literature of low temperature

calorimetry. As included in the advantages mentioned above, the

fact that only one reading was necessary for each determination of

resistance led to a considerable reduction of work during a run,

end as such, represented one of the main factors which influenced

the choice of it for the thermometer measuring system.

The resistance decade box used for the standard reference in the comparator was a General Radio Type 1432-x with six decade


ranges from 0.1.Q- to 10K.fi-. To allow for the sensitivity ( IdR \V RdT )

of the germanium thermometer, it was necessary to read resistance

steps smaller than 0.1 ohm. For this reason, as well as to

provide a permanent continuous record of the thermometer resist

ance at all times during a run, the out of balance voltage across

the galvanometer was fed to an amplifier, the output of which

was taken to a Leeds and Northrup chart recorder. A rough balance

was first made by adjusting the dials on the decade box, and a

small correction to it was obtained from the trace on the chart

recorder in the usual way.

There was AC pick-up (mostly 60 Hz) in the circuit and this

was shown by a wavy trace on the chart. It was found necessary to

shield the current and potential leads in their separate pairs in

order to reduce the pick-up to an acceptable level. The lead pairs

were twisted together and taken through two l/l6” thin-wall tuhes leading from just inside the cryostat head down to the terminal

strip mounted just above the outer can. There was no attempt to

shield the leads from the terminal strip down to the thermometer.

However, the leads from the cryostat head (which passed through

9~pin kovar seals there) to the terminal strip outside the cryostat

were shielded. The leads f ro m this terminal strip to the comparator

were shielded cable. The shields w ere all inter-connected with a

copper w ir e which then passed to g ro u n d . In this w ay, by g ro u n d in g


the shields at only one common point, the possibility of ground

loops was prevented.

The lower limit of detection by the instrument was deter

mined by the noise ievel as displayed on the chart trace (most of

it actually originated in the comparator amplifier). Under normal

operating conditions, it was found, using a measuring current of

1.5 micro-amps, that at4«2°K, where the thermometer resistance was of the order 100 ohms, the noise level was 15 nanovolts. At

0.7°K, where the thermometer resistance was of the order 1000 ohms, the noise level was found to be 150 nanovolts. These levels

correspond to resistance fluctuations of the order 0.01# which correspond in turn to fluctuations in temperature, limiting the

precision of the estimation of the temperature drifts (see Sec. 3*3)*

2. The heater current supply. The specimen heater current

supply is shown in Fig. 2.6. Essentially, it consisted of a source

of emf (a 6-volt car battery) which fed a DC current through the voltage dividers V. By means of the switch S-j_, the current range

was selected, the continuous current adjustment within each range

being made by the potentiometers P. The switch S2 allowed current to flow through either the specimen heater or a dummy heater which

was mounted outside the cryostat. When the specimen had been

heated for a specific heat determination, the current was switched through the dummy heater. This procedure helped stablize the


o o o o

LsJ o


FIG

.2.6

HEAT

ER

CURR

ENT

SUPP

LY

current for the following specimen heating cycle. The switch S~w

allowed the electric clock C (Standard Electric Time Co., Type S10)

to be turned on and off synchronously with the heater current. The

potential difference across the heater was measured with a Tinsley

microvolt potentiometer. The heater resistance was obtained by

comparing the potential difference across the standard 500 ohm Kelvin wire-xyound resistor ¥ with that across the heater, the two

resistors being connected in series. The standard resistor was

stabilized by immersing it in an oil bath.

The leads from the specimen heater to the- terminal strip

above the outer can were lead—plated manganin. For the section

of the leads inside the shield, each lead was made up from about

three feet of wire and was coiled in the usual way. This kind of

wire was used because it had small thermal conductance and also

because the lead coating prevented any joule heating in the heater

current leads which might have spoiled the specific heat measure

ments. The thermometer leads were also made of the same kind of

wire and were coiled in the same way.

3. Other circuits. The shield carbon thermometer current

and potential leads were of un-plated manganin and led up from the

thermometer through the pusite seals in the outer can cap and on

to the terminal strip where they joined other manganin wires lead

ing outside the cryostat to the external circuit. The thermometer


circuit was very straight-forward, for it consisted essentially

of an emf source (a 6-volt car battery) connected to the series combination of the thermometer, a standard resistor^ and a voltage

divider (potentiometer). The resistance of the thermometer was

obtained by noting the potential differences across the standard

resistor and the thermometer.

The gas filling tube shunt heater was powered with AC current

from the mains, the voltage adjustment being made with a Variac.

Because the leads to the heater carried large currents (about 1

amp), they were made of copper. The lower tube heater was connected

to two thicker manganin wires which led up through the pusite seals

in both can caps to the terminal strip above the outer can. The

length of these leads was kept as small as possible to prevent

them from burning out. The leads from the terminal strip were of

copper and led out of the cryostat to a 6-volt car battery fund rheostat.

The filling tube thermocouple leads were taken up through

the cryostat to an ice bath and from there to the potentiometer,

where the thermocouple emf was read.The maximum error in the quantity of electrical heat Q which

was added to the calorimeter system for a heat capacity determin

ation (see Sec. 3.3 and Sec. 5.1) Is estimated to be +0.5%•


46

CHAPTER 3

PERFORMANCE OF THE APPARATUS

3.1 - Preparation of -the Samples.

The system used for obtaining condensed gas samples in jthe calorimeter* ready for measurements at low temperature was

essentially the same as that used by Fagerstroemvoy for his

measurements on condensed gases in and above the liquid He4

temperature range, with the exception that there was, in the

present apparatus, no low temperature filling tube valve which

had to be warmed while condensing the gas* Distillation effects,

which were reduced in Fagerstroem*s apparatus by positioning this

valve as close to the calorimeter as possible, was not a consider

ation in the present investigation since the vapor pressures of all

the solid gases used were entirely negligible in the experimental

range of temperatures. This enabled an important reduction in the

complexity of the gas handling system to be made. Because the

sample preparation procedure follows closely that of Fagerstroem,

only an outline of it will be described.


47

I After* a thorough leak-test, the entire gas handling system

I was evacuated, using a rotary pump, for periodSwhich varied from

run to run from a few days to 24 hours. Then the entire system

was twice flushed with small amounts (a few cm Hg pressure) of

the gas and then re—evacuated for at least one day* The pumps1Ff were then shut off and the system was filled with sample gas

through rubber hose leading from the steel cylinder containing the

gas supply. The gas pressure and the temperature of the reservoir

water bath were read after they were allowed to stabilize. A

mercury-in-glass thermometer was used to measure the water temper-

ture and a Wallace and Tieman mercury manometer (with vernier

divisions of 0.1 mm Hg) indicated the gas pressure. To within the

limits of experimental accuracy, the perfect gas law was sufficient

to give the molar quantity of gas inside the system from a knowledge

of the gas temperature, the volume of the system, and the gas

pressure. The gas filling pressures were always adjusted to be

as close to the maximum reading on the manometer (800 mm Hg) as

possible, since the greater the filling, pressure, the greater the

amount of condensed sample that could be collected in the calori

meter.At this point, with the heatsvitch closed, some liquid

nitrogen was placed in both the outer and inner B^wars, While the

amount in the outer Bev/ar was not important, the level in the inner


IJcwsr wss kept below the level of the outer cun cap bscEuss it

wes found, that the rate of condensation could be better controlled

in this way., rather than having the level above the cap closer

to the filling tube thermal shunt. While the apparatus was

cooling, the filling tube heaters were switched on, the air in the

interspace was replaced by a few microns pressure of helium exchange

gas, and the shield system was evacuated and then sealed by closing

a valve. After a few hours (typically, three), the calorimeter

was at the temperature of the normal boiling point of liquid nitro

gen with a snail quantity of condensate in it at the vapor pressure

corresponding to that temperature.

In all cases except that of NO, the calorimeter temperature

was further reduced in order to collect more condensate in the

calorimeter. The reduction of temperature was accomplished by

pumping on the liquid nitrogen bath in the inner Dewar. In the

case of NO, where its normal boiling point (121°K) and normal

melting point (110°K) lie considerably above the liquid nitrogen

boiling point (77°K), it was not necessary to pump on the nitrogen

bath since the vapor pressure of NO at the nitrogen boiling point

was sufficiently low to provide, ideally, a sample large enough

for good measurements. Unfortunately, because of the high melting

point of NO, it was found impossible to collect a very large

sample since the filling tube became blocked easily by the solid,


presumably in the regions of the tube which lay between successive

heater windings on Tube B. At one point, liquid air was tried in

place of nitrogen because . f its slightly higher boiling point,

but this did not result in an easier condensation. In practice,

the filling of the calorimeter NO was controlled in the easiest

way by simply adjusting the inner Dewar nitrogen level so as to

just touch the bottom of the outer can and simultaneously adjusting

the tube heater currents until the tube became unblocked, being

careful not to overheat parts of the tube which could cause solder

joints to break. The most vulnerable joint was that connecting

the two filling tubes A and B; actually this joint was overheated

during one run so as to melt the solder and cause a leak. The

unblocking of the tube could be seen quite easily in most cases

by the change on the manometer from a steady pressure to a slow

decrease. Sometimes, the tube constrictions were not completely

removed by heating the tube and the decrease in pressure took place

very slowly. The condensation of NO was the worst in this respect

and turned out to be the most difficult and tedious of all the

samples condensed, mainly because of the need to keep all parts of

the filling tube and calorimeter at temperatures in tha liquid NO

range, which only extends over about 12 degrees. The difficulties

were so formidable that it took about 50 hours of almost continual adjustment of the filling tube heater currents and nitrogen levels


50

to collect 0.2 moles (representing about 6 ml of liquid).The condensation of the other gases was much easier, with

the collection of 0.5 moles taking about 12 hours. The filling

tube heater currents and nitrogen levels could be set once and

for all, thereby avoiding the need to supervise the condensation

continually. When enough condensate had been collected and with

the valve in the gas filling line closed, the pressure of the gas

remaining in the glass reservoir and the temperature of its water

bath were read. The perfect gas law was again used to evaluate

the molar quantity of gas inside the reservoir and a subtraction

of the amounts before and after the condensation gave the amount of

condensate inside the calorimeter, assuming that there was no

appreciable quantity remaining in the filling tube, an assumption

which is discussed in Sec. 6.2 in the light of its effect on the error in the measured heat capacities.

With a sufficient quantity inside the calorimeter and the

gas supply to it shut off, the nitrogen bath in the inner Dewar

was exchanged for liquid helium. Either the nitrogen was boiled

off under reduced pressure (in the case of NO), or the Dewars

were taken down and the liquid simply poured out. It was essent

ial that this operation was done as rapidly as possible in order that the liquid helium might be transferred quickly to the inner Dowar so as to prevent the calorimeter contents from evaporating.

During the period that the nitrogen was being exchanged by the


51

helium bciths, the pressure on the dial gauge connected to the

calorimeter system was carefully watched in an attempt to detect

a rapid pressure increase which would have indicated evaporation

of the calorimeter contents.

During the first stages of the liquid helium transfer, the

tube heaters were left on in order to avoid distillation of the

condensate from the calorimeter into the regions of the filling

tube which were cooling faster than the calorimeter. When liquid

helium was starting to collect in the inner Dewar, the tube heaters

were switched off and the pressure on the dial gauge was observed

to fall very rapidly * The time taken for the transfer varied

somewhat depending on the sample heat capacity, but on the averageo„about four hours were needed to cool the calorimeter down to 4*2 K.

The transfer techniques used were the standard ones used in this

laboratory and they will not be described.

The sample was then in readiness, with the heat switch closed,

to be cooled to lower temperatures in preparation for the specific

heat measurements and this procedure is described in the following

section.

3-2 — Preparations for the Specific Heat Measurements.

With the solid gas inside the calorimeter at the temperature

of 4■2°K, it was then cooled in two stages to temperatures below 1 in the following way. First, the He^ bath temperature was


reduced by pumping on it with the laboratory Kinney pump 'to a

temperature of about 1*2 , then the He^ stage was brought in to provide further cooling to below 1° by pumping on the liquid He3 in the refrigerant chamber. A description of this procedure is

described in this section.

With the heat switch still closed, the exchange gas was

removed from the interspace using the diffusion pump. Because of

sorption effects, the removal of He gas becomes more and more

difficult as the temperature is lowered and so the interspace

evacuation was done at 4.2°, even though its removal resulted in a reduction in the calorimeter cooling rate during the subsequent

cooling stage. This choice was made because it was considered

more important to avoid the presence of helium gas when trying to

cool the calorimeter below 1° with the He3 stage. For the initial

runs, a mass spectrometer leak detector, which collected the

diffusion pump exhaust, monitored the rate of removal of the inter

space gas. It was found that only after a period of about two

hours did the leak detector indicate a sufficiently low helium

level, even though the cold cathode gauge indicated a pressure of

less than 10~^ torr only a few moments after the diffusion pump started to work. For the later runs, the leak detector was not

used and a period of four hours was employed for the interspace

evacuation before proceeding with any further cooling.


After this time, with the interspace pumps still working,

the he^ gas was allowed to enter the refrigerant chamber from

the storage tanks and then the temperature of the liquid He^

bath was lowered. The temperatures of the shield and calorimeter

were observed to fall very slowly, the heat from them being taken

to the He^ bath mainly by conduction through the He^ gas in the

pumping tube. When the He^ became cold enough, it started to

condense on the parts of the pumping tube which were in directX Acontact with the liquid He . As the liquid He*3 formed, it trickled

down to the warm refrigerant chamber where it was immediately

evaporated, thereby cooling the shield and calorimeter by taking

up its latent heat of evaporation. Eventually, after a period of

about two hours, all parts of the cryostat were at 1 .2° with sufficient liquid He in the chamber to provide further cooling

below 1°. It was estimated that the chamber contained, at most,

only about 1 ml liquid, but this was sufficent to cool all the samples without needing to re—cycle the gas, a technique that has

been developed for use i t» very long experiments with large heat

capacity samples.At this point, the heat switch was opened and then re—closed

with a lighter pressure on the jaws, because it was found that if the switch was opened only at the lowest temperature, prior to

taking measurements, then a very large amount of heat would be


developed, presumably because the strains, which were induced in

the cryostat by cooling from room temperature (where the switch

was originally closed), had altered the proper alignment of the

heat switch components. Xn addition, the heat switch was closed

rather tightly before cooling in order to accelerate the calorim

eter cooling rate, and undoubtedly this would have led to a large

frictional heating effect on opening the switch.

While the switch was open, it was discovered that the

calorimeter temperature was rising quite rapidly and continued to

rise for a considerable time. At first, this temperature drift

was something of a puzzle, since the drift caused by vibrational

heating^which was always present and which had been investigated

beforehand, was much too_ small to account for it. The first

observation of it appeared with CO and was initially thought to

be connected with a phase transition which was suspected to be

displayed by this solid gas. However, when the effect was found

with the other solid gases (N£> 02, NO), another explanation was

sought. It now seems most likely that the effect was caused by

the existence of solid gas crystals which were not in good thermal

contact with any part of the calorimeter and hence did not cool as

rapidly as the others. In support of this viewpoint, it is known

that many molecular solids, including those investigated, possess

one or more crystal structure modifications between which there may


be large changes in the density. If the temperature is lowered

quickly through such a phase change, then considerable strains

are set up throughout the solid and the result is the formationt

of a fsnow* of small crystallites. This explains why it is yery

difficult, experimentally, to obtain large single crystals in a

low temperature phase of these materials. It is then reasonable

to suppose there is poor thermal contact between adjacent

crystallites; therefore, the cooling of a snow would appear to

be very sluggish and thus the observed calorimeter warming effect

is thought to be the result of the system attaining thermal equil

ibrium. . Now this lack of thermal equilibrium did not necessarily

exist when making the specific heat measurements because all

regions were at low temperature. On the contrary, because the

specific heat becomes very small at low temperatures, any local

heating^which tends to establish unwelcome temperature gradients}

will be rapidly dissipated through the snow. Confirmation of

this was found from two sources; from the reaction of the calor

imeter thermometer to changes in the input heating rates, and

from the behavior of the measured specific heat to changes in the

deliberate heating rate. No effects were observed which could be

ascribed to a non—equilibrium condition of the calorimeter system

Generally, the heat switch was kept closed for about an hour in

order to let the calorimeter system attain thermal equilibrium


with the He^ bath at 1.2°K.

Then, with the interspace evacuated and liquid He^ inside the refrigerant chamber, the He^ roughing pump was switched on

and the valve connecting the chamber to the pump was opened slowly

so as to prevent a surge of gas in the pumping system. After a

few moments, the valve was fully open and the calorimeter temper

ature was observed to fall steadily. After a period of about 20 minutes the calorimeter was at the temperature of about 0.7° and because further cooling below this temperature was much slower, an

initial series of specific heat measurements was made, leaving

the temperature range below 0,7° to be covered in a subsequent series.

The heat switch was then very slowly opened by rotating the

switch rod knob at the cryostat head. Great pains were taken to

avoid sudden switch movements, since these were found to produce

large amounts of heat in the calorimeter system. With care, the

heat generated could be reduced to as little as 5 ergs in some cases, although a heating of 20 ergs was more normal. This amount

was estimated from the observed resistance change before and after

opening the switch, the thermometer, sensitivity (to obtain the

corresponding temperature rise), and the heat capacity as measured

in the experiment. In practice however, the temperature drift

after opening the switch, which was caused by vibrational heating,


. was so large as "to overwhelm the temperature rise mentioned above.

Typically, this vibrational heating rate was of the order 100 ergs per minute at the lowest temperature. Therefore, no attempt was

made to adjust the shield temperature during the course of the

run as is required by the methods of adiabatic calorimetry. Thus,

as the calorimeter temperature was rising during the run, the

heat leak the calorimeter, which was conducted along the

electrical leads and suspension threads from the shield, increased

in magnitude as the temperature gradient along the thermal path

increased.

Eventually, the warming trend was observed to change to

one of cooling, where the state of zero temperature drift corres

ponded to the condition of balance between the heat generated by

the vibrations and the heat taken away by conduction along the

wires. The calorimeter temperature at which this condition existed

was found, as expected, to depend mainly on the magnitude of the

calorimeter system heat capacity, since the vibrational heating

rate was least for the massive samples which were rigidly supported

and greatest for the light samples for which the amplitude of

vibration was quite considerable. The temperature at which there

was heat leak balance varied from about for the light samples

(for example, the empty calorimeter), to about 3- for the heavy

samples.


5 8

Then, with the heat switch open, the heat capacity measure-

ments were ready to be made and the procedure is described in the following section.

3.3 - Procedure of the Measurements.

With the heat switch open, the out-of-balance signal from

the thermometer comparator was displayed on the chart recorder

and the temperature drift of the calorimeter was observed for a

period of about two minutes. The comparator resistance box setting

was changed in the meantime in order to determine the chart"

sensitivity (the measure of the change in thermometer resistance

per unit division of length across the chart paper). The amplif

ication of the output signal from the comparator was adjusted such

that the recorder pen traversed the full width of the chart in

little over two minutes, provided that the noise level of the

trace was sufficiently low. With the comparator adjusted and

with the potentiometer (for the heater voltage measurements) balanced

against the standard cell, the heater current and timer were

switched on together. During the heating period, which was usually

30 seconds long, the voltage across the heater was noted, being

careful to make sure the voltage remained steady during the entire

period. If this were not the case, then that specific heat deter

mination was discarded.


59

At the end. of the heating period, the thermometer resistance

was again charted. on the recorder for about two minutes and

another sensitivity check was made. The resistance box settings

were written on the chart alongside the corresponding trace and

the temperature increment for the heating cycle was obtained from

the chart in the manner described in Sec. 5.1* This procedure

was repeated to obtain each specific heat pointy generally up to

a temperature of about 1.3°. In all, the first series of measure

ments took about an hour to complete. The heater voltage was

increased periodically to obtain progressively larger temperature

increments in a larger heat capacity sample.

Although the He^ roughing pump had been working all the time,

there was sufficient liquid He remaining to re-cool the cryostat O oto below 1 * The heat switch was then closed, and the He'3 diffusion

pump switched on. After about an hour, with the calorimeter at a

temperature near 0,5°>the heat switch was re—opened and a new

series of measurements was begun. Because the operation of the

He roughing pump contributed to the vibrational heating of thei

calorimeter,, the He^ pumps were switched off for this series.

During the course of the measurements the shield temperature

gradually increased because of the heat leaks reaching the shield by conduction along the tubes from the outer can. Thus, as the

experiment progressed, the heat leak to the calorimeter by conduc-


tion from the shield decreased until the shield was steady at the

-temperature of the He^ bath. This, coupled with the reduction

in the vibrational heat leak, had the beneficial effect of reduc

ing the calorimeter temperature drift during this series.

On the average, about 70 specific heat determinations were

made over the temperature range from 0,6° up to about 4°*

Host of the points were taken in the region below 1.5° since the

accuracy of the measurements was not as good in this region as it owas above 1,5 > especially for the pure solid gas samples which

had small specific heats.


61

CHAPTER 4

THERMOMETER CALIBRATION AND TEMPERATURE SCALES

4.1 - Introduction.

The calorimeter thermometer was calibrated against then 3 4He0 vapor, pressure scale. The advantages of He over He^ in

vapor pressure thermometry, have been substantiated in recent

years because it has been found that the film reflux effects

in He-II are very difficult to take into account. Another

consideration which favors the choice of He^ over He^ is the

fact that the He^ vapor pressure is very much greater than that

of He^ at the same temperature. This is especially important

below 2°K since the film flow corrections for He-II may represent

a considerable (uncertain) fraction of the total pressure. In

addition, the greater He^ vapor pressure may be read on the mano

meter with greater precision than that of He^,

4.2 - Considerations of the Choice of Calorimeter Thermometer.

Tlie sample thermometer was chosen very carefully, since


upon it depended very much the reliability of the specific heat

measurements* Some of the features of a thermometer which were

considered necessary are; fast response to changes in calorimeter

temperature, small mass, good sensitivity^ and good reproducibility

between temperature cyclings from room temperature to low temper

atures. For these reasons, resistance thermometers have been

used almost exclusively and while carbon radio resistors have

enjoyed almost a monopoly in low temperature calorimetry in the

past, the use of commercial germanium resistance thermometers is

growing rapidly, especially for the He^ temperature range.

The main advantage of germanium over carbon thermometers is

that their resistance-temperature characteristics appear to be

much more reproducible between temperature cyclings, although not

every thermometer from all manufactuers is satisfactory in this

respect It, seems that the better reproducibility of germanium

is connected with the fact that these thermometers use one single

crystal as the temperature sensing element, while the carbon radio

resistors are made from many small grains. The temperature

cyclings probably cause the grains to change their detailed state

of contact with each other through thermal expansion effects, thus

altering the bulk resistance of the resistor. Therefore, with

carbon thermometers, new calibrations are required every time they

are cooled down to low temperatures, while for reproducible


63

germanium thermometers, the calibration need only be done once and for all.

The main disadvantage of germanium thermometers is that

there is no simple interpolation formula which represents

satisfactorily the thermometer resistance as a function of!

temperature. Such is not the case for carbon thermometers however, I

for the well-known semi-empirical formula of Clement and Quinnell^2?) !3

works well for most thermometers. ]

After weighing these considerations, it was decided to usei\a commercially made germanium resistance thermometer. Actually, ]

several thermometers from different manufacturers were tried before

making the final choice, but few of them were suitable for calor-

imetry in the He*5 temperature range, for their resistance rose too

rapidly as the temperature was lowered. The need to keep the joule

heating of the thermometer as low as possible, consistent with the

sensitivity requirements of the resistance measuring apparatus,

places an upper limit on the maximum acceptable resistance. Also,

since the calibration of resistance thermometers depends to some

extent on the magnitude of the measuring current, it is advisable

to use a thermometer which does not have a very large change of

resistance over the range of temperatures used in the experiments,

In. this way, it becomes necessary to use only one measuring current

ln the entire work, both for the calibration and the specific heat


measurements. This avoids the complication of having to split

the calibration up into several temperature ranges, one for each measuring current.

The final choice was a Cryocal. Inc. germanium resistor

(serial #117), with a 4»2°K resistance of 130 ohms and a 1.0°K

resistance of 620 ohms, A resistance measuring current of 1.5

microamps was used throughout the work, thus satisfying the

requirements of keeping the joule heating low (a maximum of 5.10~9

watts), as well as being sufficient current to produce an accept

able sensitivity in the measuring apparatus throughout the whole

temperature range * When the thermometer was ordered from the

manufacturer, the copper case containing the germanium crystal

unit was specified not to contain helium exchange gas, which is

usually used. It was feared that the gas might have given a

spurious contribution to the measured specific heat because of

sorption effects.

4*3 - Thermometer Calibration Procedure.

With the heat switch tightly closed, the cryostat was cooled

m preparation for the calibration in exactly the same way as it was

for the specific heat measurements. Indeed, some calibration points

were taken at the completion of the measurements in some of the runs.

The experimental procedure for performing the calibration below

1.2°K (the lowest He^ bath temperature) was different from that used


65

above this temperature because of experimental difficulties. The

calibration procedures in the two regions will now be described

separately.

1. Calibration procedure below 1.2°K. With the He^ bath

at the temperature of 1*2° throughout and the interspace evacuated

of its helium exchange gas, the He** stage was put into operation.

It was found that the shield temperature, indicated by the shield

carbon thermometer, could be very easily controlled by making a

rough adjustment first of the He^ pumping valve and then balancing

the heat removed, by virtue of the evaporation of the He3, by the

addition of joule heating from the shield heater. In this way, a

very fast temperature response was obtained to changes in the heater

current which was adjusted to produce a steady shield temperature.

It was observed that the calorimeter temperature came to

equilibrium only a few moments after the shield temperature had

done so. However, no thermometer resistance reading was taken until

five or ten minutes had elapsed after the resistance had become

stable, in order to make .sure that the initial steady indication

was not spurious. When the resistance had become steady, the heights

of the two mercury columns of the He^ vapor pressure manometer were

read and the mercury temperature was noted. At the end of the

five or ten minute equilibration time, the vapor pressure was again

measured, making sure that the calorimeter theivnometer resistance


had not changed noticeably during this period. For the purpose

of the calibration, the arithmetic mean of the two pressure read

ings was used. The thermometer resistance was displayed on the

chart recorder in the usual way, while a measuring current of

1.5 micro amps was used throughout.

Then the pumping valve and the shield heater current were

adjusted to produce a slightly higher shield temperature and

another calibration point was taken. This procedure was repeated

to obtain several calibration points up to 1.2°. After this

temperature range had been covered the calibration above 1.2°K was

performed.

2. Calibration procedure above 1.2°. For this temperature

range, it was necessary to allow for the possibility of having

cold spots in the vapor pressure sensing tube. A cold spot is

simply a region of the tube which lies at a lower temperature than

that of the vapor pressure cell. They must be avoided at all costs

since the He^ rising from the liquid in the cell may re—condense

on the sides of the tube at a lower temperature than that of the

cell. Thus the pressure read on the manometer may then be too

low. Two alternative calibration procedures were considered to

avoid this effect.

F i r s t , i t was s u g g e s te d t h a t th e same te c h n iq u e as t h a t used

below 1 .2 ° m ig h t be u se d h e r e , w i t h th e He^ b a th m a in ta in e d s te a d y


67

at, say, 4.2°K. However, it was found, in attempting to control

the shield temperature (which was relatively easy below 1°), that

it was impossible to maintain the calorimeter temperature steady

for a sufficiently long period. Apparently this was a result of

the increasing time constant for temperature fluctuations in the

shield, as well as the need to adjust the He^ pumping valve in the

almost-closed position where the pumping speed was very sensitive

to fine adjustments.

The second alternative was to not use the He^ stage at all

and to let exchange gas into the interspace. Thus the shield

would be maintained isothermal with the He^ bath whose vapor

pressure could then be quite easily maintained steady for each

calibration point using the standard laboratory pumping line

equipment. This alternative was finally - adopted despite the

possible difficulties arising from the hydrostatic head pressure( 28)effect . This effect, which results in an increasing temper

ature with depth in a liquid, could possibly cause the formation

of cold spots in the vapor pressure sensing tube, since the He4

bath surface temperature may be lower than that of the He^ cell.

To avoid this possibility, the calibration points were taken only

when the liquid He^ bath surface had fallen to about the same

level as that of the outer can cap. In this way, the vapor pressure

sensing tube thermal shunt was not immersed in the bath liquid


during the calibration and it was expected that the heat' conducted

down the vapor pressure tube vacuum jacket from the cryostat head

would serve to raise the shunt temperature above that of the bath.

If this were so, then cold spots were unlikely to form anywhere

in the pressure sensing tube.

As a further precaution, the bath heater was used continuously

during the measurements in an attempt to stir the liquid and hence

reduce the residual temperature gradients in the bath. About 60

milliwatts was dissipated in the heater which was situated in the

bottom of the inner dewar. This was sufficient power to cause the

formation of tiny gas -bubbles which were seen to rise and pass

around the outer can. After making these adjustments to the helium

bath level and bath heater, the calibration was undertaken. The

manner in which the measurements were made was identical with that

used in the region below 1.2°.

Because the bath level was so low, it was feared that the

beat conducted down the various tubes from the cryostat head would

produce a considerable temperature rise in the shield. But with

beliu_m exchange gas in the interspace throughout the calibration

procedure, the calorimeter temperature seemed to agree with that of

the bath at all times. Evidently, the thermal coupling between the

shield and the bath through the interspace exchange gas was suffic

ient to prevent any undesirable heating of the shield from this

source.


4.4 - Discussion of Errors in the Temperature Scales.

1. Corrections to.the measured He* vapor pressure.

A. Thermomolecular pressure corrections. If two

regions of a closed gas system which are connected by a narrow tube

lie at different temperatures, then the condition for equilibrium

at high gas pressures is that the pressures in the two regions

shall be equal. However, as the pressures are lowered to a degree

where the mean free path of the gas molecules is of the same order

as that of the tube diameter, then the condition for equilibrium is

no longer that the two pressures shall be equal. Because a temper

ature gradient exists in the tube, there will be thermal transpir

ation effects which result in a mass flow from the cold to the warm

regions. This flow is balanced, in equilibrium, by an equal reverse

flow down the pressure gradient in the tube. Thus, at low pressures

and in equilibrium, there is a difference in pressure in the two

regions, the pressure in the cold region being lower than that in

the hot region. Although the pressure differences for perfect

gases may be calculated from a knowledge of the geometry of the gas

system, such is not the case for He^ (and He^) which behave in a

non-classical manner at the very low temperatures encountered in

the. present work.(29 QO )Experiments have shown 3 that the thermomolecular press—


ure ratio, which is defined as the ratio of the gas pressure in the

cold region (pc) to that in the warm region (pw ), may be expressed

as a function only of the product of the tube radius R and pw . In

the present application, pc is the vapor pressure just above the

surface of the liquid (the desired quantity), and pw is that meas

ured on the manometer. Unfortunately, the magnitude of the ratios

depends somewhat on the material of the tube; for example, on

whether glass^0) or stainless steel(31) is chosen, there being a

1 Q% difference (approximately) between them for the same R.pw value.

For the present work, the tables of Roberts and S y d o r i a k ^ 2 ) were

used (based on measurements using a vacuum—jacketed Inconel pressure

sensing tube), making linear interpolations between their tabulated

values of R.pw . For the size of tube used in this investigation,

the maximum correction to the cell temperature was 8 mdeg at the

lowest temperature, about 0.6°K,

B. Hydrostatic pressure head corrections. In a liquid,,

there tend to be set up temperature gradients due to the hydro

static head pressure effect. In the absence of convection, the

equilibrium boiling temperature increases with depth because the

hydrostatic pressure of the liquid must be added to that at the

liquid surface to obtain the equilibrium pressure. This correction

be easily calculated from the values of the liquid density.

However, in practice, convection currents are set up which tend


to reduce the temperature gradients and hence the magnitude of

the necessary temperature correction. Because it is difficult

to know the degree to which the gradients are altered by convec

tion, it is preferable to encourage the formation of convection

currents as much as possible to the point where corrections become unnecessary.

Because liquid He^ does not have a strong density-temperature dependence (compared with liquid He-I for example), there is

little natural stimulation for the formation of convection currents

in the presence of a temperature gradient. Thus, an attempt was

made in this work to reduce the temperature gradients in the liquid'iHe by allowing the liquid to come into contact with copper turn

ings inside the cell, as well as with the copper cell walls.

Although it has been found^33) -that using a copper cell alone the

hydrostatic head corrections may be reduced to a negligible amount

(less than 1 mdeg), it was decided to take the added precaution of

placing copper turnings inside the cell.

Table 4-1 shows the thermomolecular and hydrostatic head

temperature corrections which may be neccesary for the particular geometry used. The former corrections shown were actual]y employed

to compute the final corrected temperatures (see Appendix l), while

the latter ones were not used. The hydrostatic head corrections shown refer to the maximum possible (no convection) temperature


TABLE 4.1 CORRECTIONS TO THE MEASURED He3 VAPOR PRESSURES

TEMPERATURE CORRECTIONS (mdee)T°K P(He^) nun Hg HYDROSTATIC HEAD THERMOMOLECULAR

0.6 0.54 9 8

0.8 2.89 3 2

1.0 8,84 1.5

1.2 20.16 1 *■«

1.4 38.52 1

NOTE: The thermomolecular corrections must be subtracted

from -the cell temperature that is inferred from the

manometer readings.


difference between the surface of the liquid and the base of the

cell.

C. The effect of He^ impurity in the He3. The

effect of He 4 impurity is to overestimate the correct temperature

of the cell liquid from the measurements of the vapor pressure.

The necessary temperature correction, of course, depends on the

ratio of He4 to He^j for example, for a 0 ,0 5 % He4 content, a

correction of 0,3 mdeg at 3°K is necessary^34)4 temperature

is lowered, the correction falls to 0.1 mdeg at 1°K.

The gas used in this research was obtained from the Monsanto

Mound Laboratories and was their ’’Vapor Pressure Grade” for which

the supplier claimed the following analysis:

SupplierTs Analysis of the Vapor Pressure Grade He^ Gas

Greater than 99*98$ He^ in total helium

Greater than 99*9 % total helium

Less than 2,10~^% tritium

It is seen that the He 4 content is claimed to be less than 0,02$ in He^ giving a maximum temperature correction of 0.1 mdeg at 3°K.

This small correction was not applied to the present measurements. The 0.1% non-helium component undoubtedly gave a negligible contribution to the total vapor pressure and no temperature corrections


73

were necessary from this source.

Other corrections. Corrections to the measured pressures were necessary to reduce them to values in terms of

standard gravity (980.665 cm/sec ) and the density of mercury at 0°C, since the vapor pressure tables refer to these standard

conditions. The corrections for local gravity were not made because

they represented differences of only 0.002% to the pressures. The

mercury density corrections were made, however, because they repres

ented more significant pressure variations. The density values

were taken from the International Critical Tables and were plotted

on a graph as a function of temperature. For each thermometer

calibration point, the density at the measured mercury temperature

was read from the graph.

Calculations of the hydrostatic head effect in the gas column

in the pressure sensing tube were made and it was found that the

maximum temperature corrections were negligible,

E. Precision of the Calibration temperatures.

1 . He^ v a p o r p re s s u re m ea su re m en ts . A t lo w te m p e r

atures, where th e He^ v a p o r p re s s u re i s v e ry s m a l l , th e c a th e to m e te r

reading e r r o r re p re s e n ts th e g r e a te s t c o n t r ib u t io n t o th e u n c e r ta in t y

°f the te m p e ra tu re m easurem ents. A t w o rs t , a t 0.6°K, th e re a d in g

er ro r , w h ich i s e s t im a te d t o be + 0 .0 5 mm f o r th e com bined re a d in g s


on both limbs, represents +10 mdeg, and at 1.0°K, the reading

error represents +1.2 mdeg. The other major source of uncertainty

is the thermomolecular correction, but it is thoughtthat this

correction is correctly estimated to better than +1 mdeg. Above

1°K, the reading error produces small uncertainties in the temper

atures, since the slope (dP/dT.) of the vapor pressure curve

increases rapidly as the temperature is increased. Thus, the

cathetometer reading errors produce less than jKL mdeg uncertainty

in the temperatures. It is estimated that the temperatures above

1° are known to an overall accuracy of better than +1 mdeg. The

1962 He^ s c a l e d 34) was used to obtain temperatures from the meas-• • -

ured vapor pressure readings.

2. He^ vapor pressure measurements. The reading

error here is larger than that of the He° cathetometer and it is

estimated to be +0.1 mm. However, because the manometer was used

to read large vapor pressures, the resulting uncertainty of the

bath temperatures is thought to be quite small. At the lowest

temperature (2.6°K), the temperature uncertainty from this source

is estimated to be +0.5 mdeg, falling to ,+0.2 mdeg at 4*2°. No

other sources of error are thought to be important. The 1958 He^ s c a l e (37) was used to obtain temperatures from the measured vapor

pressure readings.


2. The reproducibility of the thermometer.

The germanium thermometer was relied upon to demonstrate

reproducible R,T characteristics from run to run. Some of the

evidence for the good reproducibility came from two sources:

1. During the course of every specific heat run,

the thermometer resistance at 4.2° (the temperature of the He^

bath under the pressure of the laboratory gas holder), and at

1.2° (the lowest temperature reached by pumping on the bath with

the laboratory Kinney pump) were noted. These temperatures were chosen for comparison because they could be obtained with very

little variation from run to run. In every run, the respective

resistances were found to agree with each other within the error

of estimating the bath temperature,

2, The entire set of calibration points was taken

in five different runs, the apparatus being warmed to room temper

ature between runs. The points fitted all the attempted calib

ration formulae (see Sec. 4.5) either with small random deviations

or with somewhat larger smooth systematic deviations. If the

thermometer were not reproducible, there would have been large

random deviations from the formulae.Of course, the critical test for the reproducibility of the

thermometer lies in its ability to give the specific heat of a standard substance (in this case, copper) in good agreement with

the accepted values, and to give identical results for the same


substance in . d i f f e r e n t r u n s . I n t h i s c o n n e c t io n , s p e c i f i c h e a t

measurements on c o p p e r w ere made (se e S ec, 5 * 3 ) , and tw o e x p e r im e n ts

were p e rfo rm e d on n i t r o g e n (see S ec. 6 . 2 ) ,

4,5 - The T he rm om ete r C a l ib r a t io n I n t e r p o l a t i o n F o rm u la e .

In the course of this work, approximately 2,000 temperature

values were needed for the specific heat data and it was decided

from the start to make use of the computer facilities in the

University for as much of the computation as possible. In order

to minimize the labour in computing temperatures from values of

the thermometer resistance, an analytic formula f(R,T) was sought

from which the data could be reduced with the aid of the computer,

instead of the much more tedious method of plotting the R, T points

on a large graph and reducing the data by hand. Essentially, the

method employed was to fit the thermometer calibration points,

by a least squares procedure, to various forms of. f(R, T) until a

satisfactory fit was found. Since the calibration points were

relatively widely separated, the interpolation formula (which must

give accurate R,T values in between the calibration points) had to

he chosen with care. The two criteria which were used for choosing

a satisfactory formula are;

1 . The d e v ia t io n s o f th e c a l i b r a t i o n p o in t s fro m th e v a lu e s

c a lc u la te d fro m th e fo rm u la had t o be s m a l l ,


77

2. The specific heat of a 1standard* substance had to beaccurately reproduced.

-forUnfortunately,/the purpose of a rigorous choice of a formula,

the first condition is difficult to apply (unless the deviations

are not random), while difficulties are encountered with the

second when choosing a standard substance* Possibly the easiest

metal to obtain in a high state of purity (the condition which

most often affects the specific heat) is copper, but when examin

ing the literature of the specific heat of this metal, considerable( 3 )disagreement is found even for quite recent work. For this

reason, it was proposed at the I964 Calorimetry Conference that a batch of uniformly high purity copper should be prepared and

samples from it distributed to various laboratories in order to

determine if a consistent specific heat value could be ascribed

to all the samples and if so, to publish this value as the

accepted copper specific heat. Such a batch of copper samples

was distributed^ unfortunately, when the sample received by this

laboratory was analysed it was found to contain important quantities1Iof Mn and Fe, both of which have disastrous effects on the specific

heat of copper, especially at low temperature. Actually, in prel

iminary experiments to test the cryostat, this sample and a diff

erent calorimeter from that used for the solid gas experiments did

show a low temperature specific heat characteristic of a sample


containing ferromagnetic impurities, although the beryllium-copper

alloy, which contains iron and which was used in the calorimeter

construction, may have been responsible. Because very precise

measurements were needed, it was decided not to use this copper

sample or the calorimeter, but to use instead a very high purity

copper sample which was very kindly furnished by Dr. D. L. Martin

of the National Research Council, Ottawa. This sample was attached

to the solid gas calorimeter for the specific heat measurements,

which are described in Sec. 5-3*

Several different forms of f(R,T) were tried* Originally,

the polynomials,

were tried, with n taking values, in turn, from 2 to 6.. None of

them gave a satisfactory fit over the entire temperature range,

although the fit at high temperatures (above 2°JC) was fairly good for them all. Therefore, it was decided to search for another

Because the Clement-Quiunell formula applicable to carbon

resistors is based on the low temperature variation of resistance

of semi-conductors, l/T InR, it was felt that a formula which also

uses this as a basis should represent fairly well the R,T relation

for germanium. The basic Clement-Quinnell equation, with various

formula which would give a good fit at low temperatures also.


extra terms was then tried;

JL = bc + b^lnR + b2

Clement—Quinne11 __formula

This equation, with n taking values, in -turn, from 2 to 4 was

tried, but again, none of them gave satisfactory fits over the

entire temperature range. The fit at low temperatures was good

for them all, but the high termperature fit was unsatisfactory.

Instead of searching for a polynomial containing more terms

(the approach which several workers have resorted to^^^), it was

decided to split the temperature range into two parts using^ in

each sub-range, a different calibration formula for the purpose

of computing the specific heats. For the low temperature region

(below 1.3°), the modified Clement-Quinnell formula was used,

while for the high temperature region (above 1.3° ) 9 a polynomial in lnR was used:

t-"n^bidnR)-11=3-

extra terms

i - a + a lnR + a_(lnR)2 + a.(lnR)3 + a. (lnR)4 — calibration- A T 3 4for T > 1.3°K

1 = b + b lnR + b + b . + b , + b ,T 1 2 3 „ —5. . c a l i b r a t i o n - B

lnR (lnR)2 (lnR)J (lnR)4for T < 1.3°K


A least squares analysis was performed separately on each

of the two formulae. For Calibration-A* input data from 1.27°

to 4.2° was employed (total of 22 points)* while for Calibration-B input data from the entire temperature range* 0.6° to 4.2° was used (total of 29 points). The deviation curve for the combined

calibration is shown in Fig. 4*1* The deviations were evaluated

in the computer program by obtaining from the least squares analysis

the best values of the constants. Then the calibration resistance

values were inserted back into the formula to obtain Tca^c. Tmeas

is the measured calibration temperature corresponding to that re

sistance value used to evaluate ^ca^c» Table 4*2 shows the constant

obtained from the least squares analyses* where eight figure accuracy

was used.oReferring to Fig. 4*1* "the deviations below 2 are seen to

be quite random* therefore* the specific heat errors from this

source should be random also. However* the deviations above 2

have a strong systematic trend which would likely result i.n system

atic errors in the specific heat in this region. If necessary* the

deviation curve may be used to adjust the specific heat values

computed from the calibration formula as follows.

The deviation between 3*2° and 3*6°>for example, changes by

about 4 mdeg* corresponding to a rate of change of 10 mdeg per degree 1/3- The slope of the deviation c u rv e is negative* hence theor


73CD■o-5oQ .CoCDQ .

■oCD

C/) (f)

Oo■ O

3.CD

CD■o-5oQ .Cao■oo

CDQ .

■oCD

C/)C/)

£FHCD•O'

0a

b ?

1oCDE

H

zo!§>L?JO

FROM 1962 He3 VAPOR PRESSURE SCALE meas FROM 1958 H VAPOR PRESSURE SCALEmeas

20

4 RANGE IN WHICH CAUB. (B) WAS USED TO COMPUTE SPECIFIC HEATS

• CALIBRATION (B) ▲ CALIBRATION (A)

RANGE IN WHIGH CALIB.(A) WAS USED TO COMPUTE SPECIFIC HEATS

(see text)

FIG 4.1 CHARACTERISTICS OF THE THERMOMETER CALIBRATION FORMULAE

TABLE 4.2 THE CONSTANTS APPEARING IN THE THERMOMETER CALIBRATIONFORMULAE

CALIBRATION ~ 6(low temperatures)

bQ = 0.268391 X 103

bx = - 0.839558 x 101

b2 = - 0.318890 x 104

b3 = O.I83888 x 105

b4 = ~ 0.523626 x 105

b5 = 0.593660 x 105

input d at a f r o m the entire t e m p e r a t u r e r a n g e

CALIBRATION ~ A (HIGH TEMPERATURES)

aQ = 0.406713 x 101

■= - 0.159592 x 101

a2 = 0.548157 x 10-1

a = 0.364853 x lO"13

a . = - 0.279514 x 10~24

INPUT DATA FROM 1.27° UP


81

specific heat values evaluated, from the calibration equation should be 1% too small in this region*- Thus, when the deviation

curve has a negative slope, the specific heat values are too

small, while for a positive slope, they are too large* Comparing

Figures 4-1 5*2,it is seen that the trend of the specific heat

is in rough agreement with this prediction, although the magnitude of the deviations is somewhat different from that expected on this

basis alone.When different calibration equations are used to compute

specific heats in contiguous temperature regions, the question arises as to how well the two sets of specific heat values (computed from each of the two calibrations) compare in the region where the calibrations meet.- Great care must be taken to. ensure

that there are no sharp discontinuities at the point where one calibration takes over from the other, in this case, at 1*3°»Fig. 4*2 shows the total heat capacity of the copper sample and

calorimeter computed from the two calibrations in the region around

!• 3°. The data are plotted as C/T against in order that the

differences may be seen more clearly than from a plot of C against T.

It is seen that the differences change sign when passing

through the temperature region around 1.3^* this region, the

differences represent only a few tenths percent (or less), which


Reproduced

with perm

ission of the

copyright ow

ner. Further

reproduction prohibited

without

permission.

3.0

2.

C/y (mJ/deg)

2.6

24

© CALIBRATION (A )-H IG H TEMPERATURE & CALIBRATION (B)-LO W TEMPERATURE

•mm m ■1 %1

-! i

THE LINE REPRESENTS THE EQUATION

—■ M8 + c*T2) . + tg + « T 2)_T Color 0 Copperwith 8 , oc values from the least squares analyses (see text).

T*J.3*K

1.0 2.0 T 2 (°K) 3.0

FIG.4.2 HEAT CAPACITY OF THE COPPER SAMPLE WITH CALORIMETER EVALUATED FROM EACH CALIBRATION FORMULA

is g e n e ra lly le s s th a n th e s c a t te r o f I n d iv id u a l p o in ts fro m th

le a s t squares f i t t o a s t r a ig h t l i n e . Thus, th e p ro ce d u re o f

d iv id in g th e e n t i r e te m p e ra tu re ra n g e in t o tw o d i f f e r e n t c a l ib -

ra t io n re g io n s i s fo u n d t o be s a t is f a c t o r y .


83

CHAPTER 5

THE DATA HANDLING-— THE CALORIMETER AND COPPER MEASUREMENTS

5,1 - R e d u c tio n o f t h e Raw D a ta .

I n o r d e r t o be i n a p o s i t i o n t o com pute th e s p e c i f i c h e a t

o f th e sam p les ( i n c lu d in g th e c a lo r im e t e r ) , i t was n e c e s s a ry t o

c a lc u la te , f ro m th e ra w d a ta , th e e l e c t r i c a l e n e rg y added t o th e

system and th e r e s u l t i n g te m p e ra tu re r i s e f o r e a ch h e a t in g c y c le .

The e l e c t r i c a l e n e rg y was o b ta in e d fro m th e m e a su re m e n ts o f th e

hea te r v o l ta g e , t h e h e a te r r e s is ta n c e , R ^ , and th e h e a t in g

p e r io d , t , and t h e te m p e ra tu re r i s e was o b ta in e d fro m th e th e rm o

meter r e s is ta n c e r e a d in g s . The r e d u c t io n o f th e d a ta a lo n g th e s e

lin e s i s now d e s c r ib e d .

S in ce th e te m p e ra tu re o f th e c a lo r im e t e r s ys te m was v e r y

ra re ly c o n s ta n t i n t im e , i t was n e c e s s a ry t o a l lo w f o r th e d r i f t

(dR /d t) o f th e th e rm o m e te r r e s is ta n c e t r a c e a c ro s s th e c h a r t r e

corder p a p e r . I n o r d e r t o o b ta in th e t r u e te m p e ra tu re r i s e f o r

the pu rpose o f c o m p u tin g th e h e a t c a p a c i t y , b o th th e r e s is ta n c e

d r i f t s b e fo re and a f t e r th e h e a t in g p e r io d w ere e x t r a p o la te d i n t o


84

th is re g io n . Because th e d r i f t s were a lw ays found to be co n s ta n t

in tim e between th e h e a t in g p e r io d s , s t r a ig h t l in e s were drawn

through th e d r i f t s and extended to th e m id -p o in t o f th e h e a tin g

pe riod . From th e re s is ta n c e v a lu e s fou nd a t t h i s p o in t (Rj-,

denoting th e b e fo re -h e a t in g , and Ra th e a f te r - h e a t in g v a lu e s ) , th e

corresponding te m p e ra tu re s , T^ and Ta were computed from th e

thermometer c a l ib r a t io n fo rm u la e and th e n th e d i f fe r e n c e , Ta-T^

gave the t r u e te m p e ra tu re r is e f o r t h a t p a r t ic u la r h e a t in g c y c le .

For t h is and a l l subsequen t c a lc u la t io n s , th e com puter was

employed in th e fo l lo w in g way.

From th e f i v e measurements f o r each h e a t in g c y c le , Rb, RQ,

Vh, Rj^, and t , th e h e a t c a p a c ity was com puted. Thus, a l l i n th e

same computer p ro g ra m , th e te m p e ra tu re r is e Ta-T j) , and th e mean

tem perature, Tm = (Ta + T ^ ) /2 were c a lc u la te d from R^ and R^, th e

e le c t r ic a l energy Q = . t /R ^ was computed, and th e h e a t c a p a c ity

c = f i / ( Ta~Tb^ o b ta in e d . T h is p ro ce ss was re p e a te d f o r each one

of the s e t o f h e a t in g c y c le s , th u s g iv in g a s e t o f h e a t c a p a c ity

po in ts e x te n d in g o v e r th e e x p e r im e n ta l te m p e ra tu re ra n g e . For

the purpose o f th e c a lc u la t io n s in v o lv in g th e te m p e ra tu re v a r ia t io n

of the hea t c a p a c ity , th e mean te m p e ra tu re T^ was used to in d ic a te

the tem pera tu re c o rre s p o n d in g to th e h e a t c a p a c ity p o in t found f o r

the h e a tin g c y c le . F o r th e s m a ll te m p e ra tu re in c re m e n ts used in

th is rese a rch , t h i s ass ignm en t was s u f f i c ie n t l y a c c u ra te and no


8 5

c o rre c tio n s w ere n e c e s s a ry t o a l lo w f o r th e n o n - l in e a r v a r ia t io n

of the h e a t c a p a c ity w i th te m p e ra tu re .

5.2 - The C a lo r im e te r H e a t C a p a c ity .

In o rd e r t o d e te rm in e th e s p e c i f ic h e a t o f th e copper and

s o lid gas sam p les , i t was n e c e s s a ry t o s u b t r a c t th e empty c a lo r«

im eter h e a t c a p a c ity fro m th e measured t o t a l . T h e re fo re , th e

empty c a lo r im e te r h e a t c a p a c ity was measured in a se p a ra te ru n .

The p rocedure and th e r e s u l t s a re d e s c r ib e d b e lo w .

A f te r e v a c u a t in g th e c a lo r im e te r system f o r ab ou t a day,

the c ry o s ta t was c o o le d i n th e manner d e s c r ib e d i n Sec, 3 .2 , and

the measurements made i n th e manner d e s c r ib e d in Sec. 3 .3 .

P a r t ic u la r a t t e n t io n was p a id t o th e m easurem ents be low 1 .3 ° K

because th e c a lo r im e te r h e a t c a p a c ity was v e ry s m a ll i n t h i s

reg ion . C o n s id e ra b le s c a t t e r i n th e m easurem ents was expected

because o f u n c e r t a in t ie s i n th e stray h e a t le a k s t o th e c a lo r im e te r ,

which co u ld re p re s e n t a c o n s id e ra b le f r a c t io n o f th e t o t a l h e a t in g

ra te d u r in g th e h e a t in g p e r io d s . F o r t h i s re a s o n , ab ou t 40 p o in ts

were taken be low 1 .3 ° , w h i le o n ly 20 p o in ts were ta k e n above t h i s

tem pera tu re .

The r e s u l t s a re shown in g r a p h ic a l fo rm in F ig . 5 -1 and th e

table o f r e s u l t s i s g iv e n i n A p p e nd ix 2 -A . I n F ig . 5.1> th e low

temperature r e s u l t s a re shown as a p lo t o f C /T against arid th e


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

0.8

0.7

•LOW TEMPERATURE- DATA (BELOW) C3

©o DEVIATION*

Gmeo5-teT+~T3)X 100%

fv-®-'meos

©o

T°K», FROM LEAST SQUARES

FIT TO THE ENTIRE SET OF DATA

THE HIGH TEMPERATURE RESULTS

C/T (m«J/deg2)

„L.

5%~ r

00

0THE LINE REPRESENTS THE LEAST SQUARES FIT TO C /T = (0 .6 6 2 + 0 .0 875 T 2 ) m J/deg2

( see text)

THE LOW TEMPERATURE RESULTS

0 1.0 j2 (°K)2 2.0FIG. 5.1 THE EMPTY CALORIMETER HEAT CAPACITY

high temperature results are shown in the form of a deviation plot.

The deviation is defined as the difference between the individual

specific heat values, C(Tm ), and the expression C = ^ T + ocj3

evaluated at each of the temperatures Tm belonging to the heating

cycles. The parameters and oc were obtained from a least squares

fit of the data to the above expression, the analysis being per

formed on the computer. It is seen that there is generally more

scatter in the low temperature points than in those at higher

temperatures for the reason mentioned above. It is noted that at

the lowest temperatures, the stray heat leak represented about 40$ of the total heating rate during the heating periods. However, as

the temperature increased during the course of the run, both the

magnitude of the stray heat leakis and its percentage contribution

to the total heating rate diminished together. Thus, the errors

in the estimation of the stray heating rate became less significant

with respect to the precision of the heat capacity measurements

and this is shown by the fact that there is less scatter in the

points as the temperature is increased. At high temperatures, the

deviations appear to be systematic rather than random and this is

thought to be a result of the dominant effect of the systematic

errors in the thermometer calibration formulae.

The heat capacity of the calorimeter was assumed to have a

temperature dependence of the form:


where the term ft T represents the heat capacity of the electrons3and the term oc T . that of the phonons. It was assumed that there

were no other contributions over the experimental range of temper

atures. To test the validity of using Eq. 5.1 to represent the

specific heat of the calorimeter, two separate least squares

analyses were performed, one oh the data from the entire temper-

ature range, 0.6° to 2.6°, and the other on the high temperature data, 1.3° to 2.6°. The results of the two analyses are shown in

Table 5 -1.

It is seen that the two linear terms agree very well and

although there is not such good agreement between the two cubic

terms (a disagreement of 1/0 , the agreement is thought to be

satisfactory in view of the 3% standard error in each of the two values.

Although the magnitude of the linear term may be accounted

for by the electronic specific heat corresponding to the amount

of copper (61 gm) contained in the calorimeter, such is not the

case for the cubic term, where the magnitude is about twice that

expected from the copper. Evidently, some other component of the

calorimeter makes a significant contribution to the heat capacity

and this contribution has a T3 temperature dependence. The


TABLE 5.1 RESULTS OF THE LEAST SQUARES ANALYSES ON THE EMPTYCALORIMETER DATA

■-

RESULTS OF THE TWO

ANALYSES

0 .6° - 2.6° 1.2° - 2.6'

g (mj/deg2) 0*662 0.662

OC (mj/deg^) 0.0875 0.0884

Standard error in ^ (%) + 1.0 + 1.3

Standard error' in OC (/£) + 3.1 + 2.6

RMS Deviation of the points from Eq. _5.1 (mj/deg)

0.032 0.017


possibility" of trapped gas inside the calorimeter being responsible (from incomplete evacuation^ for example), can be ruled out at

once* because about § litre STP gas would be required to account for the magnitude of the extra contribution. The materials used

in the calorimeter construction which could possibly account for

it are:

1. The vacuum grease (Apiezon T) used to anchor the

heater wire thermally to the side of the calorimeter, as well as

that used around the germanium resistance thermometer case, or

2, The Teflon covering of the thermometer leads

extending from the thermometer case to the soldered lead connect

ions which were anchored to the side of the calorimeter.

The specific heat of Apiezon T grease has been measured r e c e n t l y ( 3 8 ) ancj it has been found that below 4*2°K, the tempera

ture dependence of the specific heat has the following form:

C = 2.034T + 5.065T3 + 0.3547T5 - 0.0101T7 x 106cal/deg gm

Besides th e s p e c i f i c h e a t p o s s e s s in g s ig n i f i c a n t te rm s in powers

o f T o th e r th a n c u b ic , a b o u t 1 .5 gm w ou ld be re q u ir e d t o a cco u n t

fo r the m ag n itu de o f th e e x t r a c a lo r im e te r s p e c i f ic h e a t , w h ic h ,


89

in view of the rather low density of the grease, represents a

much larger volume than the estimated quantity which was used.

The specific heats of some polymers (Teflon, Kel-F nylon,

and some polyethylene samples) have been measured between 1° and 4 50 (39) and the results show that in all cases the specific heat

Ois proportional to TJ, as is required to explain the present

calorimeter results. The result for Teflon is as follows:

C = 0.045 T^ mj/deg gm

Taking the density as 2.2 gm/cm^, then about 0.5 cm^ would

he required to account for the magnitude of the extra calorimeter\

heat capacity which is just about the estimated quantity which

covers the thermometer leads* Thus, the anomalously high coeffic

ient of T^ in the calorimeter heat capacity may be explained by the

contribution of the Teflon.

Unfortunately, the calorimeter measurements do not extend up

to the temperatures obtained with the copper and solid gas samples,

that is, about 4°K. For the purpose of subtracting the calorim

eter heat capacity from the measurements on the copper and solid gas

samples, Eq. 5«1 was used to represent the calorimeter heat capacity

an the entire range of temperatures up to 4°K. However, in view

°f the fact that the measured calorimeter heat capacity below 2.6 may he accounted for by the contributions of the copper and Teflon

wsfpiReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

alone (which together contain only terms in T and T3), then the

method of extrapolating Eq. 5.1 out of the range of the calorimeter

measurements up to 4°K is thought to be satisfactory, for other

contributions are not expected to be important in the region of

extrapolation.

To subtract the calorimeter heat capacity from the measure

ments on the copper and solid gas samples, Eq* 5*1 was evaluated

at each of the temperatures Tm belonging to the corresponding heat

ing cycles of the sample measurements and then this calculated

value was subtracted from the total heat capacity found for the

set of heating cycles. The calculations were performed on the

computer in the same program as that used to evaluate the total

heat capacity.

5.3 - The Copper Measurements.

The specific heat of copper was measured for three main

reasons. They are:

1. To test the general operation of the cryostat. Among

the information required here was the heat switch performance

(sample cooling times, frictional heating on opening the switch),3he stage performance (minimum accessible temperature), operation

of the electrical circuits (no short circuits), and degree of

vibrational heating of the calorimeter system.


91

2. To test the degree of thermal isolation of the'calorimeter. During the measurements, the calorimeter was not perfectly isol

ated from its surroundings, there being the electrical leads and

the gas filling tube which provided a direct thermal link between

them. Therefore, some parts of the leads and tube undoubtedly

took part in the heating process to some extent, and only a direct

experimental test could termine whether this uncertainty contrib

uted an objectionable error to the measurements. Because the

conductance of the thermal link varied with temperature, it was

decided to reduce the conductance of the thermal link as much as

possible by choosing low conductivity materials for the leads

(manganin) and the filling tube (copper-nickel). Even so, it was

felt desirable to test the degree of isolation of the calorimeter

by making measurements on copper.

3. To test the thermometry. This has been described in Sec. 4.2.

If the copper measurements were found to be satisfactory,

then this would lend weight to the validity of the solid gas

measurements, since any unforseen systematic errors of the cryostat

arising from the sources mentioned above which could spoil the solid

gas measurements, would also spoil the copper measurements.

For the measurements, a blind hole was drilled in one end

of the cylindrical copper sample to allow for the germanium


thermometer and a small quantity of vacuum grease was smeared onto

the base of the calorimeter. The copper sample was then slung

underneath the calorimeter using fine copper wire as a suspension

cradle, with the sample in firm contact with the calorimeter base.

The excess grease which was squeezed out from the mating surfaces

was removed.

The results are shown in graphical form in Fig. 5.2 and in

tabular form in Appendix 2—B. Again, as for the display of the

empty calorimeter results, in Fig, 5«2 the results below 1.26° are

given as a plot of C/T against T^ and the results above 1.26° are

shown in the form of a deviation plot. The empty calorimeter heat

capacity was subtracted in the manner described in Sec, 5*2 and the

resulting heat capacity was then normalized to one mole of copper.

The data from the entire temperature range was fitted by a least

squares analysis to an expression of the form:

where the term linear in temperature represents the contribution of the electrons and the cubic term represents the contribution

of the phonons. It was assumed that there were no other contrib'

jj-T + 1944 j/mole deg Eq. 5.2

utions over the experimental range of temperatures. The results of the least squares analysis are shown in Table 5*2. The values

°f V" and Og are 0.6662 mj/mole deg^ and 336 respectively,


ZJCD-o—ioQ.coCDCl

-oCD3</>C/5o'3ooo■oV<cq'

3CDcp.

CD-o—5oQ.Cao3■ooCDQ.

"OCDC/5C/5

.+2xPl - l

F 0

0 - iQ -2

■LOW TEMPERATURE* DATA (BELOW) Crooner ~ (* T+ «»T )

DEVIATION = c p p p e r----------------------X 100%'copper

3 , T(°K) 4

l i) HIGH TEMPERATURE RESULTS

ff»«*0BTAINED FROM LEAST SQUARES FIT TO THE HIGH TEMPERATURE RESULTS ONLY

G/T(m J/mole deg2)

5 %..fUi) LOW TEMPERATURE RESULTS

THE LINE REPRESENTS THE EQUATION C / T = « + « T 2WITH *» «F R 0 M LEAST SQUARES FIT TO THE HIGH TEMPERATURE RESULTS

1 0.5

FIG 5.21 I

1.0 1.5 2.0THE MOLAR SPECIFIC HEAT OF COPPER T

in very poor agreement with the commonly accepted values; the

difference between them being 4% and 3% respectively. However,

it is possible that the low temperature results are in error

because of experimental difficulties which were encountered when

preparations were being made for the copper run. Since the

preparations were rather hurried, it was not possible under the

circumstances to re-arrange the calorimeter suspension threads

so as to prevent them coming into intimate contact with the side

of the copper sample. Consequently, there was set up a thermal

path of significant conductance between the copper sample and the

shield and as a result, there was stronger than normal thermal

coupling between them. Under these conditions, it is expected

that the heat leaks to the calorimeter would be more than normally

dependent on fluctuations in the shield temperature. Thus, the

shield temperature fluctuations could contribute a greater than

normal uncertainty to the temperature rise in a heating cycle.

This effect would be most important at low temperatures where

the heat capacity of the calorimeter system is very small.

The results above 1.26° were fitted by a least squares

analysis to an expression of the form of Eq. 5»2* The results of

the analysis are shown in Table 5.2. The values of and eg are

•7039 mj/mole deg2 and 346.1° respectively, and are in excellent agreement with the results of Martin^26). Prom the lower graph in


TABLE 5.2 THE SPECIFIC HEAT OF COPPER

( i ) RESULTS FROM THE ENTIRE TEMPERATURE RANGE

THIS WORK M A R T I N OSBORNE ET A L ^4 1 ^

^ (mj/mole deg) 0.6662 0.6961 0.6943

Oj) (°K) 336 345.6 344.5

Standard error in ^ (%) ± 1.5 + 0.78Standard error in O^(^) ± 3.5 + 1.0RMS Deviation of the points 0.052from Eq. 5.2 (mj/mole deg)

(ii) RESULTS ABOVE 1.26°K

^ (mj/mole deg) 0.7039

(°K) 346.1

Standard error in ^ (%) +0.43

Standard error in 0° (%) + 1.0RMS Deviation of -the points 0.008

from Eq. 5.2. (mj/mole deg)


Fig. 5*2 can be seen the disagreement, below 1°, between the high

and the low temperature data. Table 5.3 gives the impurity analysisand weight of the copper sample.

It is possible that Eq. $.2 is insufficient to represent

the copper specific heat and it may be necessary to invoke the

existence of an extra term in the lattice contribution of the formBt5(40). If so, then the constant B would have to be negative

to account for the trend of the measurements at low temperatures.

According to Martin^^^, a negative B may be required to bring

his 0j) value (and also that of this work) into agreement with the ©1Op derived from the elastic constants, However, a compilation

of recent data from different workers by Osborne et al^4-*-) has

resulted in the establishment of the Copper Reference Equation,

which is a representation of the specific heat of copper from all

the reliable sources so far available:

The Copper Reference Equationul

The best values of the constants A. are as follows:

A-l = 6.9434 x 10“1 A2 = 4-7548 x 10“2 A3 = 1.6314 x 10“6

A 4 = 9.4786 x io~8 A 5 =-1.3639 X 10“10A^ = 5.3898 X 1 0 ~ 1 4

(For Cp in inj/mole deg)


TABLE 5 . 3 THE IMPURITIES IN THE COPPER SAMPLE - THE SAMPLEWEIGHT

SEMI-QUANTITATIVE

QUANTITATIVE SPECTROGRAPHIC SPECTROGRAPHIC

ANALYSIS BY CARRIER METHOD ANALYSIS

PPM (by weight) ppM (by weight)

IMPURITY TOP BOTTOM TOP BOTTOM

... Fe 0.14 0.13

Mn not detected not detected

Mg 0.3-0.03 0.3-0.03

Si 0.1-1.0 0.1-1.0

Ag 0.3-0.03 0.1-0.01

WEIGHT OF SAMPLE 145.0 gm

ATOMIC WEIGHT OF COPPER 63.57 gm


It is seen that A3 (that is, our B) is positive, contrary to the present requirements. Also, the magnitude of the term

XTA T is so small below 4 K, "that it could not possibly account

for the disagreement between the low and the high temperature

results of this work.


CHAPTER 6

THE. OXYGEN AND NITROGEN RESULTS

6.1 - The Purpose of the Measurements *.

The specific heats of oxygen, and nitrogen were measured for the following reasons:

1. Oxygen. Oxygen was measured partly because the results

of Fagerstroem^} from this laboratory were in poor agreement with

the earlier results of Kostriukova and Strelkov 42)^ but mainly

because there was some interest in the behaviour of the specific

heat at temperatures below those which had been obtained previously.

When this research was started, there was some doubt as to the

magnetic behaviour of solid oxygen and there was speculation that

there might be magnetic dipole ordering below 1°K which would show UP as a specific heat anomaly. At that time, it was not definitely

established whether or not the low temperature transition OC~ p ,

which Fagerstroem had investigated, was connected with the trans

ition to the antiferromagnetic state, although most evidence


s u p p o r t e d this view (for discussion, and references, see Jamieson

M.A. Thesis^1"^). However, the matter was settled during the course

of this research when the neutron diffraction work of Collins^43)

showed that oxygen in the OC “phase is anti ferromagnetic and while

there is some degree of short-range order in the higher temperature

p -phase, there is no long-range anti ferromagnetic alignment of

the dipoles. Thus, the cC-p transition was found to coincide with

the onset of magnetic ordering and no specific heat anomaly from

this source could be expected other than that at the oc- p transition

temperature, 23.8°K. As expected, no anomalous behavior was found

in the entire temperature range of this work.

2. Nitrogen. Nitrogen was measured simply because specific

heat measurements have not been reported before for this solid gas

in the low temperature range obtainable with the present apparatus.

The only other reported data is that of Giauque and Clayton

down to 15°K and of K« Clusius et al^'*^ down to 10°K. Because

the characteristic temperature of molecular solids is a rapidly

varying function of temperature when the temperature is increased

out of the range where the Debye approximation is valid, that is,

where the characteristic temperature takes the constant value 0 , then the estimation of 0° from measurements at high temperatures (10° or 15°K) is liable to be in serious error. Since the low


temperature limit & is often used in theoretical discussions^

it was thought useful to obtain values of the characteristic

temperature to as low a temperature as possible. With these low

temperature results, a much more reliable estimate of G° may then

be made. In addition, accurate specific heat values of pure

nitrogen were required in order to estimate the contribution of

the oxygen impurity to the dilute oxygen-nitrogen mixture results

(see Chapter 8).

6.2 - Presentation and Discussion of the Results.

The results are shown in graphical form in Figures 6.1 and6.2 and in tabular form in Appendices 2—C and 2—D for oxygen and nitrogen respectively. The results shown in the Figures are plots

of the characteristic temperature which was obtained from the

measurements as follows. The heat capacity of the calorimeter

was subtracted from that of the complete system in the manner

described in Sec. 5.2 and the resulting heat capacity of the solid

gas was normalized to one mole, where the mole was considered to

contain 2N atoms, N representing Avogadrofs Number. The molar A Aspecific heat was then equated to the expression 1944 (T/o)^

J/mole deg in accordance with the Debye theory• The specific heat

values and then the 9 values upwards from 2.4°K were corrected for tlie systematic deviations in the thermometer calibration formula-A

as described in Sec. 4.3.


Reproduced

with perm

ission of the

copyright ow

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without

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LEGEND: • PRESENT RESEARCH* FAGERSTROEM (1965)

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6:

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FIG. 6.1

2.0 3.0T(°K)

THE CHARACTERISTIC TEMPERATURE ®(T) OF OXYGEN

4.(

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o o E E <j- in o> m sj- too o

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Implicit in this procedure is the assumption that the

diatomic molecules do not rotate and where only the molecule as a whole takes part in the lattice vibrations, there being no other

contribution to the specific heat. This assumption may not be

correct, but in the absence of any detailed knowledge of molecular

rotation in solids, especially for the diatomic molecular solids,

it is considered preferable to consider only the motion of the

centre of gravity of the molecules as taking part in the lattice

vibrations. It is pointed out that in the solid state and as

the temperature is lowered, free molecular rotation is suppressed

and gives way to libration (torsional oscillations about equilib

rium positions), but in the temperature range covered in this

research, both the librational and inter-molecular vibrational

frequencies were thought to be too high for these modes to

contribute significantly to the specific heat. This expectation

was borne out by the results, since the characteristic temper

atures for both oxygen and nitrogen appeared to be fairly constant

over the experimental, temperature range»

The calculation of the amount of condensed gas inside the

caloi’imeter from the calibrated gas reservoir pressures and temp

eratures may have given an overestimate if some of the condensate

had been situated in the filling tube where it may not have taken

part in the heating process from which was determined the specific


100

heat. Since there was no way of directly knowing the amount

contained in the tube, the resulting errors in the solid gas

specific heat are difficult to estimate. However, an upper limit

may be placed on this error since the maximum error would be

obtained when the whole of the filling tube B (the lower tube)

contained condensed gas, making the reasonable assumption that

the upper tube A did not contain an appreciable amount. With this

assumption, the maximum overestimate of the amount of solid gas

inside the calorimeter is about 1%, However, because the tubes

were heated during the condensation process, it \As= most unlikely

that an error of this magnitude could occur. Thus, the resulting

errors in the characteristic temperature values from this source

are thought to be very small, less than 0 .2$, for both the oxygen and nitrogen runs.

1. Oxygen. Reference to Fig. 6.1 shows that at low temper

atures there is considerable scatter to the measurements. At

higher temperatures (above 1.8°K) the scatter is very small, being

less than 0.5$. The low temperature scatter is a result of having

to subtract two quantities of similar magnitude — the calorimeter

and total specific heats — where the errors in the two values

combine to produce a considerable uncertainty in the solid gas

specific heat. As the temperature is increased, the solid gas

specific heat quickly outgrowsthat of the calorimeter and the


resulting errors in the solid gas specific heat rapidly diminish*

The results of Fagerstroem are also shown in the Figure

and a comparison between the two sets of results shows that the

agreement is satisfactory above 2 K. Below 2 K a comparison may

not be profitably made in view of the rather large scatter in both

sets of results. The estimates of the low temperature limit of

the characteristic temperature from the data above about 2°K are

as follows:

KostriukovaThis Work Fagerstroem^ 3 ) and Strelkov(4-2)

©°(°K) 104.5 + 1.0 104 ± 2.0 100

Comparing the results* it is thought that the Russian value is in error.

2. Nitrogen. Reference to Fig, 6.2 shows that the low

temperature points have considerable scatter* but those above 1 .8° are seen to have a slight systematic trend even after correcting

for the deviations in the thermometer calibration formula. Note that the scale in this Figure is very much larger than that used

to display the oxygen results and the deviations show up more clearly. The Figure shows results from two separate runs using

different amounts of solid nitrogen in each run.


The low temperature results of the two runs seem to have

opposite trends. Thus, most of the points of Series~I turn upwards

while most of those of Series~II turn downwards. The reason for

this divergent behavior is not at all clear. Fig, 6.3 shows the total heat capacity (including the calorimeter) from both

runs at low temperatures, plotted as C/T against T2. Because the

divergent, trend is seen to show up in the total heat capacity also,

then it cannot be the result of faulty subtraction of the calorim

eter heat capacity or of errors in the estimation of the quantity

of solid nitrogen in the calorimeter, where these two errors are

though to be the main sources of uncertainty in the solid gas

specific heat. In addition, it is unlikely that the behavior is

a result of poor thermometer calibration, since this would affect

the results from both runs equally.


© SERIES I “ 0 .4 9 4 moles a SERIES I I “ 0. 6 5 5 moles

THE LINES REPRESENT THE BEST STRAIGHT LINES THROUGH THE HIGH TEMPERATURE POINTS WHERE THE, LINES ARE FORCED TO PASS THROUGH THE POINT ( arc0|» 0 )

A A

FIG 6.3 TOTAL HEAT CAPACITY C/Tvs T2 FOR THE TWO NITROGEN RUNS

1 0 3

CHAPTER 7

THE CARBON MONOXIDE AND N ITRIC OXIDE RESULTS

7,1 - P re s e n ta t io n and D is c u s s io n o f th e Carbon M onoxide R e s u lts .

The CO results are shown in graphical form in Pig. 7»1

and the table of values is given in Appendix 2~E. The results

shown in the Figure are plots of the characteristic temperature

which was computed from the specific heat data using the same

method as that used for the N£ and O2 results (see Sec. 6.2). Corrections for the thermometer calibration deviations were

applied to the specific heat data above 2.4 K. in the manner des<~ cribed in Sec. 4.3. The maximum error in the specific heat res

ulting from an overestimate of the quantity of solid gas inside

the calorimeter (see Sec. 6.2) is estimated to be, at most, 0,5%>

The r e s u l t s o f G i l l and Morrison(9 ) a re a ls o shown i n F ig .

7.1, where t h e i r ta b u la te d s p e c i f i c h e a t v a lu e s were reduced to

values o f th e c h a r a c t e r i s t i c te m p e ra tu re u s in g th e same assu m ptio ns

as those used i n t h i s w o rk . The maximum d is a g re e m e n t, w h ich o c cu rs

at about 3.5°K, i s e q u x v a le n t t o a 5% d i f f e r e n c e in th e s p e c i f ic


® Present research$ ° I FGFNfV, uctttwu. A GILL and MORRISON (1966)

1 0 5 •” •. .

(T) • ’ •

100

95

* A

Characteristic Temperature ©,given by Cps I944.( ,g ’ J/mole deg

1_____________L_2 3 4

T(°K)FIG 7.1 THE CHARACTERISTIC TEMPERATURE © (T)OF CARBON MONOXIDE

1 0 4

heat, which is equal to the sum of the maximum errors in* both sets of results.

It is seen from Fig. 7.1 that no anomalous specific heat behavior is apparent down to the lowest temperature, 0.7°K. This

result is discussed in the light of the application of the Third

Law to CO in Chapter 9 *

7.2 - Presentation and Discussion of the Nitric Oxide Results

The NO results are shown in graphical form in Fig. 7.2

and the table of values is given in the Appendix 2-F. Again,

the Figure shows the characteristic temperature plotted as a

function of temperature, with corrections for the thermometer

calibration deviations having been made. The maximum error in

the specific heat resulting from an over-estimate of the quantity

of solid gas inside the calorimeter is rather larger here than

for the other specimens. Because of experimental difficulties,

only a small amount of condensed NO could be collected. Hence,

a given amount remaining in the filling tube represents a much

larger fraction of the total quantity of condensate in the NO run.

It is estimated that the maximum error in the specific heat from

this source is about 1,5%<Reference to Fig, 7*2 shows that, as for the other solid gas

samples, there is considerable scatter in the data below 1.8 K

tut above this temperature the characteristic temperature is seen


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FIG. 7

.2 TH

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(T)

OF NI

TRIC

OX

IDE

to r is e w ith in c r e a s in g te m p e ra tu re and then rem ains s teady up to

4°K. The amount o f th e r is e i s e q u iv a le n t to a re d u c t io n o f about

10# in th e s p e c i f ic h e a t . T h is i s v e ry unusual b e h a v io r , s in ce

the c h a r a c te r is t ic te m p e ra tu re f a l l s from i t s lo w te m p e ra tu re l i m i t

fo r most m o le c u la r s o l id s . There i s th e p o s s ib i l i t y t h a t th e re i s

a low te m p e ra tu re c o n t r ib u t io n t o th e s p e c if ic h e a t o th e r th a n

tha t from th e l a t t i c e v ib r a t io n s w h ich , because i t was n o t taken

in to account when com p u ting th e c h a r a c te r is t ic te m p e ra tu re , would

produce th e obse rved b e h a v io r . I n any case, even i f t h i s were so,

then th e e n tro p y a s s o c ia te d w i th th e anomaly w ould be in s u f f i c ie n t

to account f o r th e r e s id u a l e n tro p y .


CHAPTER 8

MEASUREMENTS ON THE. IMPURE SAMPLES

8,1 - Account of the Chronological Sequence of the Experiments.The specific heat measurements on the first CO sample

received from the manufacturer revealed an anomaly, not accounted

for by the Debye theory, whose maximum was of the same order of

magnitude as the expected lattice contribution (see Fig. 8.2,

points labelled CO + 0.14$ ^ first, the anomaly was thought

to be connected with a change of phase to a third low temperature

crystal modification. The broadness of the anomaly suggested that

there might be a sluggish transition to the low temperature phase

similar, for example, to a martensite transition. If so, a time-

effect study of the anomaly might reveal its nature. Such a study

was undertaken and the chronological order of the experiments is as follows:

1. In an attempt to suppress the transition, the sample was cooled quickly from the freczing point to 2.5°h. and measurements


were made up to 3.2 . If the attempt were successful, the measure

ments should have reproduced the data of Gill and Morrison^9^,

which were taken upwards from 2.5°. Then the sample was quickly

cooled to 1° and further measurements were made up to 3.2°.2. The sample was held at 4.2° for about 40 hours and the

above sequence of measurements was repeated.

3. The sample was held for about 40 hours at a high temper

ature (about 25°K), which was well above the region of the anomaly, in an attempt to anneal the sample. The sample was cooled again

and the same sequence of measurements was followed, with the excep

tion that for one series the deliberate heating rate was decreased

by a factor ten below that normally used. For some types of

transitions, the specific heat may depend on the magnitude of the

heating rate and it was hoped to observe such a dependence.

The results of these experiments are shown in Fig. S.l .

where the total measured heat capacity of the sample and calor

imeter is plotted in the form C/T against T2. The results show

that the specific heat is entirely reproducible and that no time—

effects are apparent.Soon after these experiments, the gas was sent for analysis

which revealed that it was contaminated with large quantities of

C02 and air (sec Table 8.1). At first, it was thought that the


• •o»

111

© CO

CM

o>■

• o*S

•a CM

•• •

CMlO

""Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

FIG. 8

.1 TH

E TIM

E EF

FECT

ST

UDY

AND

THE

EFFE

CT

OF AD

DING

Cog

TO

THE

IMPU

RE

Co SA

MPL

E

TABLE 8.1 IMPURITY ANALYSIS OF THE IMPURE CARBON MONOXIDE SAMPLE

IMPURITY % (by Volume)

C02 0.108

N2 0.39

°2 0.137

DEW POINT -30°F


CC>2 might; be responsible for the anomaly because the characteristic rotational temperature o^^the CO2 molecule was consistent with the temperature of the maximum, and in addition, the amount of CO,,

present was consistent with the entropy under the anomaly. Therefore,

the obvious experiment was to add some more to the sample. If

the above explanation were correct, then the magnitude of the

anomaly would be increased in proportion to the extra amount of C02.

About 0.57% C02 was added to the sample already containing 0.108/S C02 and the results are shown in Fig. 8.1, again as a plot of the total measured heat capacity C/T against T . The results show that

no discernable effect may be ascribed to the added C02#To continue the investigation, it was decided to study the

effect of adding more oxygen to the sample. The procedure was to

evacuate the entire gas handling system for about a day and then

admit a small amount of oxygen from the cylinder to the reservoir

where the amount was measured by noting the reservoir pressure

reading. The host gas was then admitted to the reservoir to bring

the total pressure up to about 800 mm Hg. The mole fraction of

the oxygen was equated to the ratio of the two pressure readings

before and after admitting the host gas. It is estimated that the

mole fraction was obtained with an uncertainty of no less than +5%.

An amount of 0.22% 02 was added to the CO sample already containing 0'-l37/£ 02 and the results are shown in Fig. 8.2 (points labelled


C0+ 0.47:5 02). The anomalous specific heat, Canom, was’obtained

from the measurements in the following way. First, the calorimeter

heat capacity was., subtracted from the total and then the result

was normalized to one mole of CO. Then, .the molar specific heat

of pure CO was subtracted and the result was normalized to one

mole of 02 to obtain C^nom. All calculations were performed in

one computer program where the specific heat of pure CO was rep

resented by its characteristic temperature which was taken as a

constant of value 103° over the entire temperature range, 0.6° to 3.5°K.

The final experiment to study the effect of adding oxygen

was to add about 0.55% P2 to a pure nitrogen host. This experiment

was undertaken in order to try to find a difference in the size or

shape of the anomaly which could be ascribed to a difference in

the surroundings of the oxygen molecule. The anomalous specific

heat for this sample is shown in Fig. ,8.2, and the table of results

is given in Appendix 2-H. For the purpose of subtracting the

specific heat of pure N2, the characteristic temperature was token

as a constant of value 83.5° over the entire temperature range,°.6° to 4°K.

The temperature of the maximum of the anomaly is estimated

to be, Tmax = (1.95 ± 0,05)°K for both the N2 -I* 0.55% 0% samplethe CO + 0.14% 02 sample, and TjnaX - (1.90 + 0.05)°K for the

I- 0.47% 0 sample.


with perm

ission of the

copyright ow

ner. Further


without

permission.

■o-5oQ.

LEGEND: ’ Ne+0.55%0** CO+O.14% Oe• ca+O47%02

■SCHOTTKY ANOMALY FROM TWO-LEVEL, Q|/Qo"2»

6-5.14* SCHEME

o*a>T3 SCHOTTKY ANOMALY FROM FREE MOLECULE’ SCHEME • y

SCHOTTKY ANOMALY FROM-TWO LEVEL NON-DEGENERATE SCHEME

0 3

FIG 8.2 THE ANOMALOUS SPECIFIC HEAT C on^lT ) OF THE SAMPLES CONTAINING OXYGEN IMPURITY

8.2 - Discussion of the Results.

The results show quite clearly that the anomalous heat

c a p a c i t y of the original CO sample is due to the presence of

oxygsn and that the magnitude of the anomaly is linearly propor

tional to the amount of oxygen present. The last result is of

great significance because it rules out the possibility of an

interpretation based upon exchange effects between the oxygen

molecules. This matter is discussed in more detail in Sec. 9.2.

For the CO sample containing 0.47% 0^, the anomalous specific

heat is seen to fall faster on the high temperature side than the

other two results and eventually goes negative (the negative

results are not shown in Fig. 8.2). For this sample, Canom is

everywhere less than that of the other two samples. Also, the

temperature of the maximum lies below that of the other two. This

behavior suggests that the lattice heat capacity (that of pure CO)

subtracted from the total was over-estimated for this sample. To

explain the observed difference between the two CO—O2 results, it is necessary to introduce a difference of about 2% in the character

istic temperatures. If this interpretation is correct, then it

becomes necessary to explain why an impurity of only 0.5% of 0 in

CO modifies the lattice vibrational frequency spectrum to correspond

to a change in the characteristic temperature of this magnitude,

furthermore, in the case of N2 it appears that the same proportion


of 02 (0.5%) does not modify the frequency spectrum to the samedegree. From a study of the thermodynamic properties of CO +

(o)2,6/s Gill and Morrisonw 7 concluded that the specific heat behavior at low temperatures of this mixture was no different

from that of pure CO. If so, then it is difficult to see that

the above interpretation of the present negative results is just

ified, since the perturbing effect of small additions of 0 and

N to a host lattice should be about the same because of the 2similarity in their molecular weights.


CHAPTER 9

FURTHER DISCUSSION OF THE RESULTS

9.1 - The Residual Entropy of CO and NO,The results of this research show that both CO and NO

display no anomalous specific heat behavior down to about 0.6°Kj

therefore, the residual entropies of these substances remain.

However, the case of CO needs re-examination because of the incorrect value of the characteristic temperature used by Clayton

and Giauque(l0) to estimate the calorimetric entropy beneath the

range of their measurements, that is, below 12°K. The value used

by Clayton and Giauque was 80°, whereas the average value m the

region below 12°K found by Gill and Morrison^9 is about 100

Therefore, Clayton and GiauqueTs estimate of the entropy mregion of extrapolation is in error by a factor of about two. The

j • ~ ,-*TH-r>r>nv found by the two groups calculations of the calorimetric en py

is given belo\\r.


113

CO

12° - O K

12 — B.Pt. (includingall latent heats)

Entropy of real gas at 1 atm, B.Pt.

ENTROPY cal/mole deg

Clayton and^10^Gill a n d ^ Giauque Morrison

0.46 0.23(extrapolated) (measured)

3 6 . 5 5

37.01

3 6 . 5 5

3 6 . 7 8

Spectroscopic Entropy of real gas at 1 atm, B.Pt. 3 7 . 8

Residual Entropy = 1.0 + 0.2

NO

(12)Residual Entropy obtained by Johnston and Giauque = 0.75cal/mole deg


As pointed out above, if the usual explanation of the

residual entropy* which is based on the idea of Tfrozen-int

disorder of the molecules in the crystal* is correct* then it

is necessary to say that molecular rotation in the solid is

impossible. Infrared studies^^^* spin-lattice studies (4 7)

dielectric studies(48 )# antj thermodynamic studies (49) show

that* to some extent* molecular rotation is possible in many solids

even at temperatures well below the freezing point* although the

rotation may not be as free as it is in the gas. There may be

energy barriers which oppose rotation in the solid* but some

degree of molecular re-orientation is very often possible. In

an extreme case* the molecule may be regarded as performing

torsional oscillations about one of several possible equilibrium

orientations where the molecule may tunnel through the potential

barriers to switch from one orientation to another. Unfortunately*

there seems to have been no detailed study made of this problem

in the cases of CO and NO* although from our limited knowledge we

have no reason to suppose that they show any essential differences

from those solids for which molecular r e —orientation has been(50 )definitely established* for* example* CH^ .

The CO molecule has a small electric dipole moment (0.11 debye)

and it is possible that dipole ordering will occur at low temper

atures such as to remove the residual entropy. On a Curxe-Wexss


scheme, the transition, temperature is given by:

T ^ Nyx2 3k

where N = number of molecules/cm^

= dipole moment

k = Boltzmann*s constant.

Evaluating this expression, it is found that dipole ordering

may occur at around 0.6°K. (I am grateful to Dr, J. A. Morrison

for suggesting this idea.) From a theoretical, study of the

various interactions in some molecular c r y s t a l s t h e dipole

orientation energy of CO molecules at 0°K was calculated to be

-3.4 cal/mole,* Equating this energy to kT, one finds for the

temperature at which dipole orientation becomes important a value of about 2°K. Thus, it is possible that dipole ordering occurs

below the present range of measurements where a transition to a

third.low temperature crystal modification takes place analogous,

for example, to the transition in oxygen^4J'. Some evidence

for a third low temperature phase of CO has been provided by an

X-ray study of the CO-Ar phase diagram^'1" , although it appears

that there have been no subsequent investigations which could

improve on this limited amount of information. It is interesting

jReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

116

that nitrous oxide ^ 0, which has a linear structure NNO, also has a weak dipole moment (0.17 debye) and may therefore be discussed along similar lines.

Nitric oxide may present us with a different situation,

however. Prior to this work, there had been no specific heatomeasurements on the solid below 14 K; therefore, before any

further investigations are made, it is suggested that the temper

ature range between 4° and 14°K be examined. It is possible that

quadrupole interactions between the dimers may produce an

ordering effect similar to the oc— j-S transition in both CO and

N2 ~* t although the present uncertainty regarding the structure

of the N2O2 dimer^4) may preclude any useful analysis along these lines.

Unfortunately, from thermodynamic studies alone it is only

rarely possible to determine the mechanism responsible for any

observed anomalies. In the cases under discussion here, we would

like to have at hand extensive infrared absorption, neutron

scattering, dielectric relaxation, spin-lattice relaxation, and

perhaps further thermodynamic data - such as studies of the effects

of pressure - to very low temperatures (below 1°K) in order to be

in a better position to make a decision regarding the origin of

the re si-dual entropy.


117

9*2 — Tlie Effect of Oxygen as an Impurity.

The results from the three dilute oxygen mixtures described

in Chapter 8 show the effect of small additions of oxygen to twohosts, CO and N • To understand the results, a satisfactory theory2must explain the following observations:

1. The temperature of the maximum of the anomaly is about

2°K and -is the same value for all of the mixtures.

2. The magnitude of the anomaly is linearly proportional

to the amount of oxygen present.

3. The size and shape of the anomaly does not differ accord

ing to whether the host is CO or N2 •4* Additions of other gases such as and N2 (see

Table 8.1) to CO is without measurable effect on the specific heat.

5. No irreversible or other non-equilibrium effects are

apparent for one of the mixtures (see Sec. 8.1 ).

From observations 1 and 2 we may at once rule out an inter

pretation based upon exchange effects between the oxygen molecule

spins. This is so for two reasons: First, if exchange forces

were operating, we would expect the magnitude of the anomaly to

depend on a higher* than linear power of the density of the oxygen

impurity because of the co-operative nature of exchange forces.

Second, the temperature of the maximum would depend on the


118

concentration of oxygen; decreasing the concentration would shift

the maximum to lower temperatures. For example, in the case of (53 )solid air (20^ 0^ in N^) the antiferromagnetic transition is

found to be displaced to about 3°K from the OC- transition of

23-8°K in pure oxygen.

Some independent evidence from which we may conclude that

exchange effects are unimportant is that provided by a study of

the magnetic susceptibility of oxygen molecules enclosed in a

clathrate compound from room temperature down to 0.25° K ^ 4> 55) <The experiments showed that exchange effects were negligible in

the entire temperature range, for the observed variations from the

magnetic behavior expected from isolated freely-rotating molecules

could be successfully explained in terms of a hindered rotator

model, where the rotation of the molecules was perturbed at low

temperatures by the internal crystal field of the clathrate.

Whereas the average separation of the oxygen molecules in the clath-orate materials was 8 A, the average separation in the mixtures used

in the present study was never less than 4-0 X . Clearly, on this

basis it is improbable that exchange effects are important in the

present work.

It is now thought that the results may be interpreted in terms

of a model in which the oxygen molecule has a set of low-lying

molecular energy levels (see Sec. 1.2). Immediately, there is no


119

difficulty choosing a suitable set of levels, for the3 -vimolecule has an unusual electronic ground state, , which is

split into three closely-spaced levels by the interaction of the

spin with the rotation of the molecule (see Fig. l.l). Clearly,

we can now explain observation 4 because the molecules C02, H20,3 -vand do not possess a ground state and hence cannot give

rise to a low temperature anomaly in the specific heat.

An attempt must now be made to fit a Schottky-type specific

heat anomaly corresponding to a particular set of energy levels

to the observed anomaly. When an agreement has been found between

the observed anomaly and a Schottky anomaly, we may conclude that

the energy level scheme which resulted in the agreement is that

actually in operation in the molecule. It must be remembered that

from the form of the Schottky specific heat we may obtain the

energy level spacings and degeneracies, but only as a ratio of the

degeneracy of the ground state. The first step was to evaluate

the Schottky anomaly corresponding to the ground state triplet of

the free oxygen molecule (see Fig. l.l). The result is shown in

Fig. 8,2. It is seen that the agreement with the observed anomaly

is very poor; clearly, the free molecule energy 3.evel scheme has

been modified by the surroundings of the host molecules in the solid

state. .

As a basis for obtaining an agreement with a particular energy


level scheme, it was decided to use only those schemes for which

the corresponding anomaly had the same temperature of the maximum

as that of the observed anomaly, that is, the energy level spacings

were adjusted so as to obtain this agreement. The first proposed

scheme was a simple two-level, non-degenerate system. The energy

spacing was put at 4«66°.in order to obtain agreement with the

maximum of the observed anomaly. The result of the calculations

shown in Fig. 8.2 is in very poor agreement with the observed

anomaly. Thus, there is no two-level, non-degenerate scheme which

could fit the observed results.

The final attempt at a fit was to evaluate the Schottky anom

aly for a two-level system where the degeneracy ratio, upper to

lower, was set at 2:1. The spacing .of the levels was put at

5 * 14° order to obtain agreement with the maximum of the observed

anomaly. Figure 8.2 shows that the agreement with the experimental

results is excellent in view of the rather large uncertainty (about

5%) in the estimation of the mole fractions of oxygen in the samples.

The above energy level scheme (shown in Fig. 9*1) which is operating

in the molecule in these experiments differs considerably from the

free molecule scheme. Moreover, the observed scheme does not

differ according to whether the host molecule is either CO or N2 (observational and 3).

The fact that excellent agreement is found with a Schottky

/


FIG 9.1

g =5.14

9 |/9 0 *2

9q

THE EXPERIMENTALLY-DETERMINED ENERGY LEVEL SCHEME OF THE OXYGEN MOLECULE IN ITS GROUND ROTATIONAL STATE


121

expression may appear remarkable at first sight. However, it

seems that we must accept the fact that the 0^ molecules are

rotating at these low temperatures, although not necessarily as

freely as in the gas. It is thought that the rotational ground

state degeneracy is still lifted by the interaction of.the spin

with the molecular rotation, but because the rotation is most

likely hindered, the magnitude of the interaction is expected to

be different.from that in the freely-rotating molecule. This

accounts for the difference in the energy level schemes between

that observed in these experiments and that of the free molecule.

Curiously, we are brought up against the very same problem

as that encountered when trying to interpret the residual entropy

of CO and NO; that is, the question of molecular rotation at low

temperatures. As discussed in Sec. 9.1, we would like to have

some more direct evidence relating to this question in the case

of the dilute oxygen mixtures, such as infrared absorption and

spin-lattice relaxation data. Perhaps the most profitable line

of- enquiry would be through a spin-lattice relaxation study of

dilute 0^0-^-enriched oxygen mixtures.


ACKNOWLEDGEMENTS

I wish, "to thank Dr, G. M. Graham for the supervision ofi*

this research and for his helpful suggestions concerning the

interpretation of the dilute oxygen mixture results.

My deepest gratitude is due Dr_ J. A, Morrison at whose

suggestion this research program was initiated. His kind

consideration throughout our correspondence was a constant

stimulus to me*

I wish to thank Dr. D. L. Martin who very kindly provided

the copper sample and its impurity analysis and various cryogenic

materials which were used in the apparatus construction.

Many thanks are due Dr. F. D, Manchester who very generously

loaned to me his isolating potential comparator and furnished the

vapor pressure He° gas from his own short supply.

Finally, I wish to acknowledge the generous financial support

provided by the National Research Council by whom I was awarded a

Studentship in the years 1965-1967*


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I. W.Nernst, Kgl. Ges, d. Wiss. Gott., 1 (1906)■ 2. F.E.Sirnon, Z.Phys. JJL, 806 (1927)3. C.-H. Fagerstroem, Ph.D. Thesis, University of Toronto (1965)4. W.F.Giauque and R.Overstreet, J.Am.Chem.Soc. jy, 1731 (1932)

5. A.Frank and K.Clusius, Z.Phys. Chem. B36. 291 (1937)6. J.H.Colwell, E.K.Gill, and J.A.Morrison, J.Chem.Phys. 39.

635 (1963)7. R.W.Hill and B.W.Ricketson, Phil.Mag. yfj, 277 (1954)

8. J.Wilks, The Third Law of Thermodynamics. Oxford UniversityPress, (1961)

9. E.K.Gill and J .A .Morrison, J.Chem.Phys. 1585 (1966)

10. J.O.Clayton and W.F.Giauque, J.Am.Chem.Soc. jy, -2610 (1932)

II. C.S.Barrett and L.Meyer, J.Chem.Phys.' 43. 3502 (1965)12. H.L.Johnston and W.F.Giauque, J.Am.Chem.Soc. juL, 3194 (1929)

13. E.S.R.Gopal, Specific Heats at Low Temperatures. Plenum Press,New York (1966)

14. W.J.Dulmage, E,A.Meyers, and W.N.Lipscomb, Acta Cryst. .6,760 (1953)

15. H .C .Jamieson, M.A.Thesis, University of Toronto (1965)16. G.Herzberg, Molecular Spectra and Molecular Structure.

Van Nostrand, 2nd. Ed. (3.965)

17. H.A.Kramers, Z.Phys. jy, 422 (1929)18. R.Sch.lapp, Phys.Rev. j£l, 342 (1937)


19.20.

21.22.

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24.

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M.Mizushima and R.M.Hill, Phys.Rev. 5L3j 745 (1954)H.M.Rosenberg, Row Temperature Solid State Physics.

Oxford University Press (1963)R.G.Scurlock and E.M.Wray, J.Sci.Inst. J4J,. 421 (1965)

R.W.Hill and G.R.Pickett, Proc.Low Temp.Calorimetry Conf.,Helsinki (1966)

F.D.Manchester, Can.J.Phys. 2lL> 989 (1959)R.Berman, J.Appl, Phys. 318 (1956)

T.M.Dauphinee, Can.J.Phys. jy., 577 (1953)

D.L,Martin, Phys.Rev. 141. 576 (1966)J.R.Clement and E.H.Quinnell, Rev.Sci.Inst. 213 (1952)

F.E.Hoare and J.E.Zimmerman, Rev.Sci.Inst. I84 (1959)S.Weber et al, Leiden Comm. 246A. (1936)S.Weber and G.Schmidt, Leiden Comm. 246C (1936)

G.T.McConville, R.A.Watkins, and W.L.Taylor, Proc.Low Temp.Calorimetry Conf., Helsinki (1966)

T.R.Roberts and S.G.Sydoriak, Phys.Rev. 102. 3O4 (1956)S.G.Sydoriak and T.R.Roberts, Phys.Rev. 106, 175 (1957)

S.G.Sydoriak and R.H.Sherman, J.Res.Nat.Bur.Stand. 68A,547 (1964)

N .E .Phillips, Phys.Rev. 134. 385 (1964)

J.T.Schriempf, Cryogenics .6, 362 (1 9 6 6)H.Van Dijk et al, J.Res.Nat.Bur.Stand. 64A, 1 (i9 6 0)E.F.Westrum et al, Cryogenics _2, 43 (1967)

W,Reese and J.E.Tucker, J .Chcm.Phys. _43? 105 (1965)


40. T.K.H,Barron and J.A.Morrison, Can,J.Phys. 799 (1957)41. D#W.Osborne, H.E.Flotow, and F.Schrein.er, Rev.Sci.Inst. 58.

159 (1967)42. M.O.Kostriukova and P.G.Strelkov, Dokl.Akad.Nauk.SSR, 90,

525 (1953)43. ,M.F.Collins, Proc.Roy„Soc. 8<), 415 (1966)

44. W.F.Giauque and J.O.Clayton, J .Am.Chem.Soc. Jj£j 4 875 (1933)45. K.Clusius et al, Z.Naturforsch. 14a. 793 (1959)

46. W.C.Price and G.R.Wilkinson, J.Phys.Chem.Solids ,18, 74 (1961)

47* E.R.Andrew, J.Phys.Chem.Solids ,18, 9 (1961)48. C.P.Smyth, J.Phy s. Chem. Solids lj$, 40 (I96I)49. L.A.K.Staveley, J.Phy s. Chem. Solids lj3, 46 (I96I)50. G.A.de Wit, Thesis, University of British Columbia (1966)51. L.Jansen,A.Michels, and J.M.Lupton, Physica j20, 1235 (1954)

52. M.W.Melhuish and R.L.Scott, J.Phys.Chem. 68, 2301 (1964)53* W.H.Lien and N .E .Phillips, J .Chem.Phys. 3^ , 1073 (1961)

54* A.H.Cooke et al, Proc.Roy,Soc. 225. 112 (1954)

55* H.Meyer, M.C.M.0*Brien, and J.H.Van Vleck, Proc.Roy.Soc.2Al, 414 (1957)


APPENDIX 1

THERMOMETER CALIBRATION DATA

He^ vaporpressure: i mm (meas)

h mm corr for P He Po/Pw pc (corr) mm

T°K R ohms

0.71J

0.71 0.927 0.658 0.618 1436.42.20 2.19 O.984 2.15 0.757 972.06.29 6.26 0.997 6 .24 O.93O 693.6

11.54 11.48 1 11.48 1.058 573.514.73 14 • 66 1.116 534.016.48 I6.40 1.145 516.2518.93 18.84 1.181 495.5020.78 20.68 1.207 480.4021.75 21.65 1.22.0 472.8025.55 25.42 1.267 449.0028.20 28.06 1.297 437.4528.22 28.08 1.297 437.0831.87 31.72 1.335 421.3032.90 32.74 1.346 417.3139.42 39.23 I.406 395.6344.98 44.76 1.453 379.7551.43 51.18 1.503 365.2566.08 65.76 1.603 339.4365.90 65.58 I.601 339.2578.10 77.72 1.673 322.85

(Continued....)


86.1195.19

85.69

94.73

1.716 313.261.763 304.68

NOTES:

u 4He vapor pressure:h mm (meas) T°K R ohms

67.5 2.431 217.9795.45 2.610 204.60

149.4 2.872 187.05

231.1 3.166 170.86

317.1 3.406 159.57

426.7 3.655 149.38

536.2 3.865 141.79

642.2 4.041 135.96

771.1 4.231 130.15

1. The therinomolecular pressure ratios, pc/pw were

obtained from the tables of Roberts and Sydoriak2. The tempei’atures vrere obtained from the 1962 He^

vapor pressure scale and the 1958 He^ vaporpressure scale(37).

(32).


APPENDIX 2

TABLES OF SPECIFIC HEAT RESULTS

APPENDIX 2—A THE EMPTY CALORIMETER

1, Range in which Calibration-B was used to compute specific heats

T°K AT (mdeff) C (mj/dee) T°K AT (mdes) C (mj/dee:)

0.6535 23.5 0.497 0.6904 23.7 0.5120.7282 20.7 0.564 0.7699 22 .6 0.538

078132 32.1 0 583 0.8653 29.7 0.615

O .8887 29.2 0.644 0.9615 46.0 O.7420.9445 26.5 0.710 1.0502 44.3 0.770

0.9903 25.0 0.752 0.6889 42.4 0.440

1.0435 44.7 0.763 0.7944 56.0 0.595

1.1023 39.3 0.867 0.9219 47.7 0.7001.1764 40.1 0.850 I.OO48 43.7 0.7621.2717 32.7 1.04 1.0825 41.9 0.796

1.2609 32.7 1.01 1.1575 39.4 O.8460.6485 26.4 O.404 1.2199 34.6 0.965

0.7994 27.5 0.609 1.2890 56.3 1.06

0.8655 26.4 0.637 0.8476 31.3 0.596

0.9368 26.7 0.630 0.9228 28.7 0.6640.9869 24.I 0.698 1.0002 44.0 0.7881.0313 23.1 0.732 1.0745 41.9 0. 8311.0851 42.0 0.787 1.1465 38.4 0.906

(Conti nuecl. . . . )


1.1382 39.8 0.830 1.2100 36.1 0.9641.1881 34.6 0.955 1.2669 36.4 0.956

Range in which Calibration-A was used to compute specific heats

. 1 ►3 O A T (mdeja?) C (mj/deg) T°K AT (rndeg) C (mj/deg)

1.2742 32.5 1.05 2.0692 83.4 2.16

1.2634 32.5 1.02 2.1330 79.6 2.261.3330 55.8 1.08 2.2109 117 2.461.4060 52.1 1.16 2.3031 111 2.61

1.4672 49.5 1.22 2.3884 103 2.82

1.5320 45.8 1.32 2.4811 131 2.98

1.9310 73.4 1.37 2.5755 121 3.23

1.6735 66.7 1.51 1.2913 56.0 1.07

.1.7387 63.5 1.59 1.3621 51.2 1.171.7960 59.6 1.69 1.4239 48.2 1.24

1.9523 58.5 1.72 1.4777 47.6 1.26

1.9205 95.7 1.88 1.2694 36.2 0.962

1.9989 88.1 2.04

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission

APPENDIX 2-B. THE COPPER RESULTS

1. Range in which Calibra-fcion-B was used to compute specific heats

T°K A T fmdee) C (mJ/des) T°K A T (mdee) C (ml/dee)

0.707 11.0 1.32 0.623 9.71 1.600.730 12.6 1.75 0.647 9.61 1.620.753 12.8 1.71 0.667 12.5 1.25

0.775 12.3 1.78 0.692 18.3 1.530.801 20.3 1.88 0.718 17.0 1.640.832 19.6 1.96 0.746 17.5 1.590.858 18.2 2.10 0.775 25.8 1.76

0.889 33.9 2.07 0.811 23.8 1.920.928 31.6

* J-

2.22 O.840 22.4- 2.03

O.964 31.0 2.26 O.867 23.0 1.98

0.997 31.2 2.25 0.900 36.4 2.121.026 29.1 2.41 0.939 35.0 2.201.061 49.1 2.60 0.976 31.8 2.43

1.103 45.3 2.81 1.007 30.5 2.53

1.142 44.6 2.86 I.050 49.9 2.581.179 42.8 2.98 1.105 46.I 2.79

1.228 75.0 3.19 1.153 44.4 2.891.290 70.6 3.38 1.198 41.1 3.13

0.567 . 53.4 1.09 1.257 70.4 3-24

0.604 7.26 1.38


Range in. which Calibration-A was used to compute specific heatsT°IC A T (mdeg) C (mj/deg) T°K AT (mdeg) C (mj/<

1.326 65.3 3.50 2.439 163 8.471.386 62.3 3.66 2.581 148 9.30

1.441 58.4 3.91 2.736 189 10.41.518 106 4.15 2.905 172 11.51.608 97.7 4.51 3.058 157 12.61.692 90.7 4.86 3.229 211 14.01.767 85.7 5.15 3.412 187 15.7I.876 156 5.61 3.579 171 17.1

2.015 140 6.22 3.777 2 55 19.22.137 128 6.82 3.998 224 21.92.281 182 7.58

NOTES: 1, The Table shows the total heat capacity of the

copper sample and calorimeter.

2. Number of moles of copper = 2.315


APPENDIX 2~C. THE OXYGEN RESULTS

1. Ranee in w h ich C a l ib r a t io n - B was used to compute s p e c i f ic h e a ts

MOLAR MOLAR MOLARSPECIFIC HEAT SPECIFIC HEAT SPECIFIC HEAT

T°K (m j/m o le dee) T°K (m j/m o le dee) T°K (mj/mole deg)

o. 848 0.859 1.116 1.886 1.094 1.8720.919 1.040 1.178 2.331 1.173 2.496

0.985 1.150 0.917 1.003 1.233 2.963

1.052 1.509 1.046 1.395

2 . Ranee in w h ich C a l ib r a t io n - A was used to compute 's p e c i f ic h e a ts

1.289 3.510 1.812 10.29 2.630 31.08

1.343 3.857 2.013 13.94 2.752 35.58

1.397 4.565 2.089 15.69 2.861 39.51

1.451 5.176 2.156 17.19 2.958 43.55

1.514 5.749 2.218 18.57 3.O64 48.631.569 6. 384 2.285 20.34 3.179 54.37

1.628 7.169 2.360 22.34 1.285 3.0081.690 7.707 2.439 24.84 1.336 3.738

1.744 8.888 2.524 27.35 1 .390 4.567

(Continued....)


1.447 4.731 1.827 10.50 2.447 25.021.490 5.277 1.888 11,45 2.553 28.471.532 5.866 1.954 12.81 2.648 31.64

1.571 6.246 2.013 13.92 2.758 35.29

1.614 7.029 2.088 15.71 2,883 40.841.662 7.808 2.174 17.69 2.996 45.60

1.715 8.845 2.261 19.71

1.774 9.563 2.349 22.25

NOTES: 1. Number of moles of oxygen = 0.555

2. The gas was obtained from Matheson Ltd,

and was their Extra Dry Grade.


APPENDIX 2-D . THE NITROGEN RESULTS - SERIES-1

1. Range in which Calibration*-!} was used to compute specific heats

oT K

MOLAR SPECIFIC HEAT (mj/mole deg) T°K

MOLAR SPECIFIC HEAT (mj/mole deg)

MOLAR SPECIFIC HEAT

T°K (mj/mole deg)

0.943 2.260 0.811 1.545 1.123 4.371

1.032 3.560 O.89I 2,147 1.168 5.295

1.116 4.338 0.949 2.613 1.214 5.857

1.175 4.9H 1.017 3.296 1.258 6.5661.236 6.310 1.078 3.998

2. Range in which 12alibration-A was used to compute specific heats

1.302 7.454 2.415 47.14 1.664 15.531.369 8.191 2.552 55.08 1.736 17.50

1.450 10.35 2.681 64.09 1.800 19.50

1.536 12.20 2.855 77.44 1.897 23.15

1.605 13.95 3.055 94.34 2,020 27.70

1.667 15.39 3.225 110.8 2.124 32.141.722 16.91 3.372 126.1 2.215 36.27

1.794 19.39 1.306 71.60 2.322 42 .20

(Continued....)


1.879 22.26.1.954 25.10

2.064 29.662.192 35.50

2.294 40.69

1.360 8.471

1.419 9.676

1.481 11.09

1.543 12.34

1.599 13.79

2.442 48.5-8

2.585 . 57.69

2.745 68.762.882 79.37

3.045 93.81

1 . Range i n w h ich C a lib ra t io n - 'B was used S E R IE S -II t o compute s p e c i f ic h e a ts_____________

MOLAR MOMR MOLARSPECIFIC HEAT SPECIFIC HEAT SPECIFIC HEAT

T°K (m J/m ole deg) T°K (m j/m o le deg) T°K (m j/m o le -deg)

0.713 1.437 I.I48 5.253 0.923 2.5910.758 1.517 1.199 5.998 0.956 3.0990.812 1. 880 0.598 .8343 0.991 3.267

0.851 2.269 0.643 1.027 1.028 3.815

0.886 2.392 0,681 1.101 1.063 4.199

0.920 2.614 0.724 1.200 1.102 4.590

0.954 2. 938 0.755 I.638 1.139 5.095

0.990 3.491 0.788 1.788 1.171 5.437

1.023 3.586 0.824 2.011 1.220 6.2121.063 4.165 0.859 2.144

1.102 4.572 0.893 2.412

(Continued...)


2. Range in which Calibration.~A was used to compute specific beats

MOLAR SPECIFIC HEAT

T°K (mj/mole deg)

1.247 6.6821.303 7.583

1.368 8.792

1.423 9.8381.270 7.070

1.323 8.010I.38I 9.065

1.434 10.05

1.489 11.25

1.546 12.62

MOLAR SPECIFIC HEAT

T°K (mj/mole

1.597 13.72

1.660 15.37

1.732 17.47

1.795 19.30

1.880 22.27

1.983 25.98

2.070 29.682.412 46.55

2.509 52.54

2.649 61.79

MOLAR SPECIFIC HEAT

T°K (mj/mole deg)

2.823 74.08

2.969 86.033.096 97.773.261 113.23.454 135.43.618 158.1

3.804 I83.24.004 216.3

4.175 251.8

NOTES: 1. Number of moles in Series-I = 0.494

Number of moles in Series-II = 0.536

2. The gas was obtained from Matheson Ltd.

and was their Extra Dry Grade.


APPENDIX 2-E. THE CARBON MONOXIDE RESULTS.

1, Range in which Calibration-B was used to compute specific heats

T°K

MOLAR SPECIFIC HEAT (mj/mole dee) T°K

MOLAR SPECIFIC HEAT fmJ/mole deg) 1-3 0 *

i


0.725 0.7H 1.094 2.301 0.923 1.560

0.774 0.666 1.138 2.417 0.959 1.301

0.818 I.032 1.191 2.897 0.991 1.7510.8 58 0.962 1.245 3.357 1.033 1.809

O.904 1.326 O.683 Q.574 1.073 2.125

O.964 1.516 0.779 0.701 1.111 2.081

1.001 1.709 0.824 O.836 1.152 2.853

1.045 1.880 0.886 1.238 1.238 3.499

2. Range in which Calibration--A was used to compute specific heats

1.296 3.705 1.620 7.763 2.686 35.24

1.350 4.257 1.683 8.456 2.799 39.29

1.410 4.882 1.751 9.666 2. 896 43.79

1.465 5.461 1.824 10.92 3.025 50.01

(Continued....)


T°K

MOLAR SPECIFIC HEAT (mj/mole deg) T°K

MOLAR SPECIFIC HEAT (mj/mole deg) 0 T K


1.515 6.451 1.917 12.75 3.182 58.481.555 6.645 2.027 14.88 3.326 66.731.282 3.799 2.121 17.09 3.461 74.991.325 - 4.070 2.228 19.91 3.633 88.261.377 4.796 2.344 23.20 3.831 105.41.438 5.369 2.445 26.14 4.042 124.81.553 6.797 2.559 30.18 4.263 152.9

NOTES: 1. . Number of moles = 0.536

2. The gas was obtained from Matheson Ltd,

and was their C.P.Grade.


APPENDIX 2-F. THE NITRIC OXIDE RESULTS

1. Range in which Calibration-B was used to coraoute SDecific heats

MOLAR SPECIFIC HEAT

T°K (mj/mole deg) T°K

MOLAR SPECIFIC HEAT (mj/mole deg) 0 T K


0.768 '' 0.343 1.266 1.860 1.125 1.2960.840 0.688 0.719 0.544 1.181 1.741O.968 0.712 0.778 0.344 1.242 1.7201.160 1.226 1.012 0.667

1.211 2.091 1.079 1.106 -

2. Range in which (Salibration-A was used to compute specific heats

1.319 2.197 1.412 2.994 2.551 17.45

1.373 3.016 1.491 3.325 2 .664 19.48

1.431 3.092 1.557 4.118 2.798 22.45

1.482 3.555 1.636 4.778 2.953 26.05

1.537 3.823 1.721 4-770 3.093 29.76

1.595 4.076 1.790 6.116 3.232 33.96

1.649 4*668 1.873 7.021 3.378 38.25

1.719 5.468 1.980 8.476 3.506 43.05

(Continued....)



T°K (mj/mole deg) T°K (mj/mole deg) T°K (mj/mole deg)1.802 6.351 2.070 9.566 3.666 48.691.876 7.240 2.186 11.25 3.847 57.44

1.304 2.213 2.322 13.24 4.OO3 65.121.353 2.884 2.436 15.08 4.143 71.65

NOTES: 1. Number of moles = 0.203

2. The gas was obtained from Matheson Ltd.

and was their regular grade.


APPENDIX 2-G THE C0-02 RESULTS. - SERIES-I

1. Range in which Calibration-B was used to compute specific heats


T°K (mj/mole dee) T°K (mJ/mole dee) T°K (mj/mole dee)

0.707 1.283 1.175 8.024 0,930 4.030

0.745 1.868 1.228 8,8 55 O.963 4.360

0.777 2.234 1.275 9.625 0.995 4.8910.827 2 .616 0.588 0.433 1.025 5.580

0.871 3.099 0.632 0.925 1.055 5.977

0.909 3.527 0.681 1.015 1.084 6.329

0.955 4.567 O.718 1.474 1.123 -7.2080.994 5.089 O.76O 1.834 1.169 7.863

1.037 5.560 0.796 2.140 1.212 8.651

I.O84 6.295 0.873 3.382 1.250 9.058

1.128 7.090 0.900 3.429

2. Ranee in which Calibration-A was used to compute specific heats

1.321 10.55 1.392 11.62 2.653 39.85

1.376 11.38 1.445 12.58 2.787 44.85(Continued.,.,)


T°K

MOLAR SPECIFIC HEAT (m j/m o le de ff) T°K

MOLAR SPECIFIC HEAT (m j/m o le des:) T°K

m o la :SPECIFIC(m j/m o le

1.438 12.49 1.504 13.44 2.930 50.23

1.497 13.35 1.560 14.46 3.055 55.911.585 14.97 1.612 15.24 3.190 62.881.697 16.87 1.689 16.68 3.333 69.481.796 18.52 1.788 18.31 3.508 81.591.886 20.22 1.876 20.05 3.707 95.00

1.967 21. 88 1.981 22.07 3.877 109.32.066 23.91 2.098 24.46 4.081 129.1

2.179 26.40 2.237 27.72 4.308 155.6

1.297 10.06 2.394 31.85

1.346 10.94 2.532 35.84

S E R IE S -Ii;

1 . Range in w h ich C a l ib r a t io n - B was used t o compute s p e c i f ic h e a ts

0.700 2.463 1.095 12.20 0. 828 5.398

0.735 3.367 1.130 13.55 0.856 5.809

0.773 3.724 1.170 14.53 0.895 7.0610.804 4.788 1.216 15.81 0.921 7.364

0.833 5.541 1.258 16.99 0.952 8.374 (Continued,..)


T°K

MOLAR SPECIFIC HEAT (m j/m o le dee) T°K

MOLAR SPECIFIC HEAT (m j/m o le dee) T°K

MOLAR SPECIFIC HEAT (m j/m o le dee)

0,863 5.853 0.603 1.163 0.990 9.471

0.899 6.848 0.639 1.845 1.024 10.260.929 7.803 0.680 2.207 1.065 11.50

0.960 8.354 0.711 3.044 1.098 12.32

0.993 9.282 0.736 3.369 1.139 13.70

1.023 10.19 0.764 3.796 1.187 15.17

1.057 11.22 0.802 4.507 1.240 16.47

2 . , Ranee i n w h ich C a l ib r a t io n - A w as 'use d to compute s p e c i f ic h e a ts

1.310 1.465 2.884 51.70 1.465 22.161.364 1.531 3.023 56.15 1.531 23.69

1.4U 1.624 3.176 62,08 1.624 25.44

1.482 1.730 3.340 69.33 1.730 27.55

1.565 1.829 3.487 76.17 1.829 29.14

1,642 1.949 3.653 85.15 1.949 31.43

1.734 2.084 3.862 99.20 2.084 33.66

I .844 2.209 4.073 116.7 2.209 35.72

1.298 2.370 4.257 133.1 2.370 38.74

1.347 2.560 2.560 43.00

(C o n tin u e d , . . . )


NOTES:

MOLAR MOLARSPECIFIC HEAT SPECIFIC HEAT

T°K (mj/mole deg) T°K (mj/mole deg)

1.403 20.92 2.730 47.17

1. Number of moles of CO in Series-I = 0.519

” »» 11 » » « Series-II = 0.513

2. Mole fraction of oxygen in Series-I = 0.14$« ' n « Series-II = 0.34^

3. The specific heat values shown in the Table

refer to one mole of CO.


APPENDIX 2—H. THE N2 - 02 RESULTS.

1. Range in which Calibration-B was used to compute specific heats

MOLAR SPECIFIC HEAT

T°K (raj/mole deg)

0.700 5.115

0.748

0.767 0.788 0.811

0.831 0.856 O.883 0.908

0.940

0.977 1.011 1.050

6.528

7.3978 , 2 6 6

9.506

10.3011.82 12.86

14.36

15.9317.72

19.42

21.77

MOLAR SPECIFIC HEAT

T°K (mj/mole deg)

1.093 24.03

1.133 26.131.174 28.52

1.217 30.64

1.257 32.600.571 I.8360.603 2.684

0.635 3.452

0.660 4.1110.709 5.481

0.734 6.7020.759 7.298

0.781 8.222

MOLAR SPECIFIC HEAT

T K (mj/mole cleg)

0.808 9.705

0.840 11.03

0.869 12,46

0.903

0.937

0.967

0.995

I.O641.099

1.1311.168

1.207

1.258

14.20 15.74

- 17.58

18.8022.73

24.17 24.0228.17

30.37

32.99

( C o n t i n u e d . . . . )


2. Range in which Calibration-^ was used to compute specific heats

MOLAR SPECIFIC HEAT

MOLAR SPECIFIC HEAT

MOLAR SPECIFIC HEAT

T°K (mj/mole deg) T°K (mj/mole deg) T°K (mj/mole deg)

1.321 36.18 1.861 58.48 3.014 118.71.376 38.73 1.937 61.46 a .166 131.31.441 41.67 2.036 65.11 3.334 147.91.515 44.98 2.156 69.76 3.514 169.3

1.583 47.94 2.268 74.41 3.720 195.6

I.648 50.95 2.438 82.65 3.943 230.5

1.709 52.81 2.656 94.51 4.133 267.11.780 55.43 2.846 106.8

NOTES: 1. Number of moles of N^ = 0.;

2. Mole fraction of oxygen = 0

566

.55%3. The specific heat values shown in the Table

refer to one mole of ^ •


SPECIFIC HEAT MEASUREMENTS OF SOME SOLID … · Chapter 9 FURTHER DISCUSSION OF THE RESULTS 9-1 —...

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