CALORIC THEORY OF HEAT

1. CALORIC THEORY OF HEAT Jiří J. Mareš & Jaroslav Šesták Institute of Physics ASCR, v. v. i. Prague - 2007

2. 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.

3. 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)

4. 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)

5. 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 

6. 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.

7. 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!

8. 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.

9. 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.

10. “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)

11. 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.” 

12. 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.

13. 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.

14. 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

15. 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

16. 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!

17. Calibration of Carnot’ function for ideal gas Isothermal expansion V1V2 of Boyle’s gas pV = f( ) (3) 

18. 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)

19. 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)

20. 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

21. 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)

22. 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.

23. 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

24. 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)

25. 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(T1T2) 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+’) = wT2 + T2{w(T1 T2)/T2} = w T1

26. 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”

27. 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

28. Caloric syringe = A tube with a piston and 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)

29. Bunsen’s ice calorimeter “Entropymeter” (As the caloric is exchanged at constant temperature)  = V ( V1  V2) V 1.35102 Cr/m3

30. 12. Efficiency of reversible heat engine Since the Carnot’s efficiency Cis defined as the ratio L/ we immediately obtain from (11) C = (T1T2) (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

31. Efficiency of the optimized heat engine L = ( + d) (T T2) T ( + d) =  T1  C = T1{1(T2/T)}

32. If Lu and u are work and caloric per unite time  Fourier relation for thermal conductor  is uT1 = (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)

33. 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.

34. Thank you for your attention

35. Confinement to the two-parameter systems The state of any body is determined at least by a pair of conjugate variables: X,  generalized displacement (extensive quantity, e.g. volume) Y,  generalized force (intensive quantity, e.g. pressure) [Energy] = [X ]  [Y ]

36. 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)

37. Zeroth Law of Thermometry There exists a scalar quantity called temperaturewhich 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

38. 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.

39. 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.

40. 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)

41. “wasted motive power” Lw dLw = (L/V)T dV +(L/T)V dT From eq. ()  dLw = T d

42. 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 !)