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 !)