1 5.6 Isobaric thermal expansion and isothermal compression (Hiroshi Matsuoka)

5.6.5 The isothermal compressibility as a response function to a pressure change

The isothermal compressibility

!

"T is basically a measure for the response of a system’s

volume to an increase in the system’s pressure: it tells us how a material changes its volume as

its pressure is varied. It tells us simply how hard the material is:

Larger ! T : softer

Smaller ! T : harder

Typical orders of magnitude for

!

"T are:

Gases:

!

"T #10$5 Pa$1

Liquids:

!

"T #10$10 ~ 10$9 Pa$1

Solids:

!

"T #10$12 ~ 10$11 Pa$1

In engineering applications, we also use a quantity called “bulk modulus,” which is directly

related to ! T by B ! 1 " T .

Inter-molecular forces determine ! T

The values for ! T for solids are similar to those for liquids and the values for ! T for solids

and liquids are much smaller than those for gases. The values of ! T , like those for

!

" , are

basically determined by inter-atomic or inter-molecular forces on the microscopic level. Inside a

solid or a liquid, atoms are close together so that a small increase in the external pressure does

not allow the atoms to decrease their inter-atomic distances very much because of strong

repulsive forces that define the atomic or molecular size. Inside a gas, the atoms are far apart

and hardly exert forces on each other so that they can decrease its volume most readily in

response to an increase in the external pressure.

!

"T for low-density gases

The ideal gas law or v = RT P leads to

2

!

"T = #1v$v$P%

& '

(

) * T

=1P

(HW#5.6.8: show this)

At P ! 105 Pa , we then find ! T " 10#5 Pa#1 . This

!

"T also decreases as P increases, which

implies that it gets harder to compress a low-density gas at higher pressures. Note also that

!

"T

remains constant if the pressure is kept constant.

!

"T for solids and liquids

!

"T usually increases as T is increased except for water between 0°C and 40°C .

Substance ! T 300 K( ) Pa-1( ) Ethanol 7.8 !10"10

Water 5.0 !10"10

Diamond 1.9 !10"12

Copper 7.8 !10"12

NaCl 4.2 !10"11

How do we measure

!

"T ?

To measure

!

"T , we first measure adiabatic (i.e., with no heat transfer) compressibility

defined by

! S = "1v

#v#P$ % &

' ( )

No heat transfer.

To estimate

!

"T , we then use the following relation:

!

"T ="S + Tv#2

cP,

where cP is what we call molar heat capacity at constant P, which we will introduce later. As

!

cP > 0 always, we find

!

"T #"S .

3 How do we measure

!

"S for gases and liquids?

To measure

!

"S for a fluid (a gas or a liquid), we measure the speed of sound “w” in the fluid

and use the following relation:

!

"S =1

#massw2 ,

where !mass is the mass density of the fluid.

How do we measure

!

"S for crystalline solids?

To measure

!

"S for a crystalline solid, we measure the speeds of the both longitudinal (i.e.,

sound) waves and transverse lattice waves in the solid. From these speeds, we can estimate what

we call elastic constants, from which we can estimate

!

"S .

For most solids at room temperature or at 300 K, ! T ~ !S ~O 10"11 Pa-1( ) For a rough estimate of

!

"S for a solid, we can use the equation for

!

"S for a fluid:

!

"S =1

#massw2 .

For 1 mole of solid, we find

v ~O 1 cm3 mol( ) ~ O 10!6 m3 mol( )

M ~ O 10 g mol( ) ~ O 10!2 kg mol( )

!mass =Mv

~O 104 kg m3( )

w ~ O 103 !104 m s( ) so that

! S ~O 1"massw

2

#

$ % %

&

' ( ( ~O 10)12 )10)1 0 Pa-1( ) ~O 10)11 Pa-1( ) .

4 We also find

T ~O 102K( )

! ~O 10"5K"1( ) cP ~ R ~O 10 J mol !K( )( )

so that

! T " ! S = Tv# 2

cP~ Tv#

2

R~O 10"15 Pa"1( ) << ! S .

(HW#5.6.9: show this)

Therefore,

! T ~ !S ~O 10"11 Pa-1( ) .

In contrast, for low-density gases, we find

!

"T #"S = Tv$2

cP~ Tv

1 T( )2

R~ vRT

=1P

~ O 10#5 Pa#1( ) ~ "T

so that for low-density gases, the difference between ! T and ! S is significant.

5.6.6 Phenomenology of

!

"T for solids at

!

P =1 atm

Under the atmospheric pressure, the isothermal compressibility

!

"T of a solid becomes a

function of its temperature T only:

!

"T ="T T,P =1 atm( ). For solids,

!

"T T,P =1 atm( ) as a

function of T has the following two common features (see the figures on the next page):

•

!

"T T,P =1 atm( ) approaches a non-zero value when T is decreased toward absolute zero.

•

!

"T T,P =1 atm( ) increases as T is increased.

With further examination of data for

!

"T T,P =1 atm( ) of various solids, we can identify two

universality classes according to low-temperature behaviors of

!

"T T,P =1 atm( ).

5 Insulators as the universality class

Solid insulators such as sodium chloride (NaCl) share the same temperature dependence of

!

"T T,P =1 atm( ) at low temperatures (see the figure below). More specifically, at low

temperatures, we find

!

"T ="T T = 0 K,P =1 atm( ) 1+ # A T 4( ) at low T( ) ,

where a positive constant

!

" A varies from one insulator to another. This result is consistent with

the pressure dependence of the coefficient of thermal expansion

!

" as

!

"#T"T

$

% &

'

( ) P

= *"+"P$

% &

'

( ) T

= *""P

A P( )T 3{ },

- . /

0 1 T= *

dAdP$

% &

'

( ) T 3 .

As you can see in the figure below, because of the

!

T 4 term, the slope of

!

"T as a function of T is

zero at

!

T = 0 and

!

"T increases very slowly as T is increased near

!

T = 0.

3.5 10-11

4 10-11

4.5 10-11

0 50 100 150 200 250 300

NaCl

!T (P

a-1)

T (K)

“Simple” metals as the universality class

!

"T T,P =1 atm( ) for “simple” metals such as alkali metals (e.g., sodium, etc.) and noble

metals (e.g., copper) behaves, at low temperatures, as

!

"T ="T T = 0 K,P =1 atm( ) 1+ # A T 2 + # B T 4( ) at low T( ) ,

6 where the constants

!

" A and

!

" B vary from one metal to another, and the

!

T 2-term is due to free

electrons in simple metals. This result is consistent with the pressure dependence of the

coefficient of thermal expansion

!

" as

!

"#T"T

$

% &

'

( ) P

= *"+"P$

% &

'

( ) T

= *""P

A P( )T + B P( )T 3{ },

- . /

0 1 T= *

dAdP$

% &

'

( ) T *

dBdP$

% &

'

( ) T 3.

As you can see in the figure below, because of the

!

T 2 term, the slope of

!

"T as a function of T is

zero at

!

T = 0 and

!

"T increases relatively sharply compared with

!

"T for NaCl as T is increased

near

!

T = 0.

6 10-12

8 10-12

1 10-11

1.2 10-11

0 200 400 600 800 1000 1200 1400

Cu

!T (P

a-1)

T (K) The Gruneisen model as a minimal model for phonons

Using the Gruneisen model, which is an extension of the Debye model for the phonons in

solids, we can show the observed temperature dependence of the coefficient of thermal

expansion and the isothermal compressibility due to the phonons at low temperatures:

!

" # $12% 4

5&

' (

)

* + R,T T = 0 K, Patm( )v T = 0 K, Patm( )

T-

&

' ( )

* +

3

and

!

"T #"T T = 0 K, Patm( ) 1+3$ 4

5%

& '

(

) * + 3+ ,1( )R-

"T T = 0 K, Patm( )v T = 0 K, Patm( )

T-

%

& ' (

) *

4. / 0

1 2 3

,

7 where

!

" is the Debye temperature of a particular solid and is defined by

!

" #!wkB

6$ 2 NAvogadro

v T = 0 K, Patm( )

%

& '

(

) *

1/ 3

,

where w is the speed of sound, which also depends on

!

v T = 0 K, Patm( ) .

!

" is the Gruneisen

parameter for the particular solid defined by

!

" # $v T = 0 K, Patm( )

%d%

dv T = 0 K, Patm( )

&

' (

)

* + = $

d ln%d lnv T = 0 K, Patm( )

,

and is on the order of 1. Note that both

!

" and

!

"T are expressed in their scaling forms or as

functions of the reduced temperature

!

ˆ T " T # .

The free electron gas model as a minimal model for conduction electrons in simple metals

Using the free electron gas model for the conduction electrons in simple metals, we can also

show the observed temperature dependence of the coefficient of thermal expansion and the

isothermal compressibility due to the electrons at low temperatures:

!

" #$ 2

2TFTTF

%

& '

(

) *

and

!

"T #32$

% & '

( ) v T = 0 K, Patm( )

RTF1+

* 2

3TTF

$

% &

'

( )

2+ , -

. -

/ 0 -

1 - ,

where

!

TF is the Fermi temperature of a particular metal defined by

!

TF =!2

2mkB3" 2 NAvogadro

v T = 0 K, Patm( )

#

$ %

&

' (

2 3

.

Note that both

!

" and

!

"T are expressed in their scaling forms or as functions of the reduced

temperature

!

ˆ T " T TF .

8 SUMMARY FOR SEC.5.6.5 AND SEC.5.6.6

1. Typical orders of magnitude for

!

"T are: Gases:

!

"T #10$5 Pa$1 Liquids:

!

"T #10$10 ~ 10$9 Pa$1 Solids:

!

"T #10$12 ~ 10$11 Pa$1 2. For low-density gases:

!

"T = #1v$v$P%

& '

(

) * T

=1P

.

3. We measure the adiabatic compressibility defined by

!

"S # $1v%v%P&

' (

)

* +

No heat transfer

and calculate

!

"T by

!

"T ="S + Tv#2

cP.

4. For a fluid (a gas or a liquid), we can estimate

!

"S by measuring its speed of sound w and using

!

"S =1

#massw2 .

5. For a crystalline solid, we measure the speeds of the both longitudinal (i.e., sound) waves

and transverse lattice waves in the solid. From these speeds, we can estimate what we call elastic constants, from which we can estimate

!

"S . 6. For solid insulators such as NaCl,

!

"T ="T T = 0 K,P =1 atm( ) 1+ AT 4( ) at low temperatures. 7. For simple metals such as Cu,

!

"T ="T T = 0 K,P =1 atm( ) 1+ AT 2 + BT 4( ) at low temperatures.

Answers for the homework questions in Sec.5.6.5

HW#5.6.8

! T = "1v

#v#P$ % &

' ( ) T

= "1v

##P

RTP

$ % & '

( )

* + ,

- . / T

=RTvP2 =

PvvP2 =

1P

HW#5.6.9

! T " ! S = Tv#2

cP~ Tv#

2

R~O

102 K( ) 10"6 m3( ) 10"5 K"1( )2

10J K( )

$

% & &

'

( ) ) ~O 10"15 Pa"1( ) << ! S