CALORIC THEORY OF HEAT
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Transcript of CALORIC THEORY OF HEAT
CALORIC THEORY OF HEAT
Jiří J. Mareš & Jaroslav Šesták Institute of Physics ASCR, v. v. i.
Prague - 2007
Motivation
Paradoxes encountered by treatment of relativistic and/or quantum phenomena inconsistency of conceptual basis of classical thermodynamics.
Main flaw (?)
Principle of equivalence of energy and heat
An alternative approach which is free of such a
postulate = Caloric theory of heat.
An elementary exposition of this phenomenological theory is given.
Subject of the lecture
Two aspects of thermal phenomena are reflected by a couple of quantities (J. Black)
Intensive quantity (temperature, or T)
Extensive quantity (heat, )
Thermometry
Theory of heat engines
(= Sources of any theory of thermal physics)
Fixed thermometric points - baths
There exist by a definite way prepared bodies (“baths”), which, being in diathermic contact with another test body (= thermoscope), bring it into a reproducible state. These baths are called
fixed thermometric points.
The prescription for a fixed point bears the character of an “Inventarnummer” (= inventory entry, Mach)
Empirical properties of fixed points
Fixed points can be ordered
To every fixed point can be ever found a fixed point which is lower or higher
An interlying fixed point can be ever constructed
A body changing its thermal state from A to E has to pass through all interlying fixed points
Postulate of hotness manifold
There exists an ordered continuous manifold of a
property intrinsic to all bodies called hotness
(= Mach’s Wärmezustand = thermal state) manifold.
The hotness manifold is an open continuous set
without lower or upper bound, topologically equivalent
to a set of real numbers.
Important scholion
According to the aforementioned postulate, in nature
there is only hotness, i.e. ordered continuum of thermal
states of every body, and the
concept of temperature
exists only through our
arbitrary definitions and constructions!
Construction of an empirical temperature scale
The locus in X-Y plane of
a thermoscope which is in
thermal equilibrium with a
fixed-point-bath is called
isotherm.
Keeping Y = Y0, one-to-one mapping between variable X
and set of fixed thermometric points can be defined
Existence of continuous function = (X), called
empirical temperature scale ,
which reflects properties of hotness manifold and is
simultaneously accessible to (indirect !) measurement.
“Absolute” temperature scales
G. Amontons (1703), Existence of l’extrême froid
(“absolute zero temperature”, = Fiction !)
Definition: Assuming the existence of the greatest lower
bound of the values of , we can confine the range of
scales to 0. These temperature scales are then
called “absolute” temperature scales.
(Quite an arbitrary concept, cf. “proofs” of inaccessibility
of absolute zero temperature)
Theory of heat engines
Carnot’s principle (postulate) and its mathematical formulation (1824)
“ The motive power of heat is independent of the agents
set at work to realize it; its quantity is fixed solely by the
temperatures of the bodies between which, in the final
result, the transfer of the heat occurs.”
Mathematical formulation: (sign convention!)
L = F(1, 2), (1)
where variable means the quantity of heat regardless of
the method of its measurement, L is the motive power (i.e.
work done) and 1 and 2 are empirical temperatures of
heater and cooler respectively.
The unknown function F(1,2) should be determined by
experiment.
Carnot’s function
Assuming that 2 is fixed at an arbitrary value and 1 = ,
relation (1) may be rewritten in a differential form (not so biased
by additional assumptions as the integral form)
dL = F’( ) d,
(2)
where F’( ) is called Carnot’s function.
Since this function is the same for all substances, it depends
only on the empirical temperature scale used.
Kelvin’s proposition
Mutatis mutandis, Kelvin proposed (1848) to define an
“absolute” temperature scale just by choosing a proper
analytical form of F’( ).
There is, however, an infinite number of possibilities
how the form of F’( ) can be chosen.
Necessity of rational auxiliary criterion
A corner stone of classical
thermodynamics
Experiments of B. Count of Rumford (1789) and Joule’s
paddle-wheel experiment (1850) have reputedly proved
the
equivalence of energy and heat
(or of universal “mechanical equivalent of heat”,
J 0, J 4.185 J/cal) Clausius’s programme
“die Art der Bewegung, die wir Wärme nennen”
Dynamical (or kinetic) theory of heat
Actual significance of Joule’s experiment
In fact, postulating the principle of equivalence of work
and heat , Joule (and later others) determined at a single
temperature conversion factor between two energy
units, one used in mechanics [J],
the other in calorimetry [cal.].
J became an universal factor by circular reasoning!
Calibration of Carnot’ function for ideal gas
Isothermal expansion
V1 V2 of Boyle’s gas
pV = f( ) (3)
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L = f( ) F’( ) / f ’( ) (5)
This relation is independent of units or method of heat measurement and of empirical temperature scale .
It has universal validity because Carnot’s postulate (2) is
valid for any agent (working substance).
Using then ideal gas temperature scale for which
f( ) RT, the equation (5) can be rewritten as
L = T F’(T) (6)
Carnot’s function in dynamical theory of heat (thermodynamics)
The dynamic theory of heat “postulates”
the equivalence of work and heat ( = “heat”)
L = J (7)
(J is mechanical equivalent of heat, J 0)
F’(T) = J / T (8)
Consequences of “equivalence principle”
Degradation of generality of energy concept (exclusivity of heat energy, limited transformation into another form of energy)
Temperature and heat are not conjugate quantities i.e.
[ ] [T ] [Energy]
Appearance of entropy [J/K] – an integral quantity (uncertainty of integrating constant) without clear phenomenological meaning in thermodynamics
Carnot’s function in caloric theory
In caloric theory of heat ( = “caloric”) Carnot’s function is reduced to dimensionless constant = 1
(the simplest chose)
F’(T) = 1 (9)From (5)
L = T (10)
SI unit of heat-caloric is 1 “Carnot” 1 Cr
[Cr] = [J/K], (unit of entropy in thermodynamics)
Interpretation of caloric
Relation (9) fits well with general prescription for
energy in other branches of physics, viz.
[Energy] = [ ] [T ]
Amount of caloric “substance “
at “thermal potential”(= temperature) T
represents total thermal energy T.
Cyclic process and Reversibility
Permanently working engine Closed path in e.g.
X-T plane (bringing the system into an identical state) is
called cyclic process.
Definition:
If the caloric is conserved ( = const.) in a cyclic
process, the process is called reversible
integrability
Integration of Carnot’s equation for a reversible process
For reversible process = const.
L = F’(T)dT As F’(T) =1
L= (T1 – T2) (11)
The production of work from heat by a reversible process is not due to the consumption of caloric but rather to its transfer from higher to a lower temperature (water-mill analogy)
1
2
T
T
Dissipative processes and “wasted” motive power (Carnot’s conjecture)
The power “wasted” or lost due to the heat leakage = conduction and/or friction is also given by (11)
Lw = w (T1T2)
The only possible form in which it is re-established is the
thermal energy of caloric enhancement ’ which
appears at T2 eq.(12)
T2 (w +’) = w T2 + T2{w (T1 T2)/T2} = w T1
Irreversible process and related statements
Definition: A process in which enhancement of caloric takes place is called irreversible.
Corollary: By thermal conduction the energy flux remains constant (basis of calorimetry)
Theorem: ( “Second law”) Caloric cannot be annihilated in any real thermal process.
! cf. redundancy of the “First law”
Measurement of caloric
Caloric may be measured or dosed:
Indirectly, by determining corresponding thermal energy at given temperature (thermal energy = T )
“Directly”, utilizing the changes of latent caloric
(connection with fixed points)
Caloric syringe , Ice calorimeter
Caloric syringe
= A tube with a pistonand diathermic bottom,filled with ideal gas.
According to eqs. (3) and (4)to the volume change V1 V2 corresponds (per mol) dose of caloric
= R ln(V2 / V1)
Bunsen’s ice calorimeter
“Entropymeter”
(As the caloric is
exchanged at constant
temperature)
= V ( V1 V2)
V 1.35102 Cr/m3
12. Efficiency of reversible heat engine
Since the Carnot’s efficiency C is defined as the ratio L/ we immediately obtain from (11)
C = (T1T2) (13)
Replacing entering caloric by its thermal energy Kelvin’s dimensionless efficiency
K = {1(T2 / T1)} (14)
These formulae are important for theory of reversible processes but useless for real (irreversible) systems
Efficiency of the optimized heat engine
L = ( + d) (T T2)
T ( + d) = T1
C = T1{1(T2/T)}
If Lu and u are work and caloric per unite time Fourier relation for thermal conductor is
u T1 = (T1 T)
Lu = (T T2) (T1 T)/T
Optimum for output power dLu/ dT = 0 T = (T1T2)
C = {T1 (T1T2 )} (15)
K = {1 (T2/ T1)} (Curzon, Ahlborn)
Conclusions
It has been shown that the freedom in
construction of conceptual basis of thermal physics is
larger than it is usually meant.
This fact enables one to substitute the
Caloric theory of heat for the Thermodynamics.
As we hope, the paradoxes which are due to the
incorporation of postulate of equivalency of heat and
energy into classical thermodynamics will thus
disappear.
Thank you for your attention
Confinement to the two-parameter systems
The state of any body is determined at least bya pair of conjugate variables:
X, generalized displacement (extensive quantity, e.g. volume)
Y, generalized force (intensive quantity, e.g. pressure)
[Energy] = [X ] [Y ]
Diathermic contact
Correlation test of diathermic contact
The two, mechanically decoupled systems (X,Y) and (X’,Y’), are called to be in the
diathermic contact
just if the change of (X,Y) induces a change of (X’,Y’) and vice versa.
Non-diathermic = adiabatic (limiting case)
Zeroth Law of Thermometry
There exists a scalar quantity called temperature which
is a property of all bodies, such that temperature equality
is a necessary and sufficient condition for thermal
equilibrium.
Thermal equilibrium may be defined without explicit
reference to the temperature concept, viz
Thermal equilibrium
Any thermal state of a body in which conjugate
coordinates X and Y have definite values that remain
constant so long as the external conditions are
unchanged is called equilibrium state.
If two bodies having diathermic contact are both
in equilibrium state, they are in thermal equilibrium.
Maxwell’s formulation
Taking into account these definitions, the original
Maxwell’s formulation (1872) of the Zeroth law can be
proved as a corollary.
Bodies whose temperatures are equal to that
of the same body have themselves equal
temperatures.
Constitutive relations
Equation of state in V-T plane
= (V,T )
d = V (V,T ) dV + V (V,T )dT ()
Constitutive relations
V = (L/V)T / T Latent caloric (with respect to V)
V = (L/T)V / T Sensible caloric capacity (at constant V)
“wasted motive power” Lw
dLw = (L/V)T dV + (L/T)V dT
From eq. ()
dLw = T d
An example - relativistic transformation of temperature
Von Mosengeil’s theory (1907) (Einstein 1908)
Q = Q0(1 2), T = T0 (1 2),
invariance of Wien’s law, ( /T) = inv. Invariance of entropy S = S0 (Planck)
(”moving thermometer reads low”)
Ott’s theory (1963) (Einstein 1952)
Q = Q0/(1 2), T = T0 /(1 2),
(”moving thermometer reads high”)
Jaynes (1957) T = T0 (NO DEFINITE SOLUTION !)