CALORIC THEORY OF HEAT

1
CALORIC THEORY OF HEAT

• Jirí J. Mare Jaroslav esták
• 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
• ( Machs 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 lextrême 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

• Carnots 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

• Carnots 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 Carnots function.
• Since this function is the same for all
  substances, it depends only on the empirical
  temperature scale ? used.

14
Kelvins 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
  Joules 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) ? Clausiuss
  programme ?
• die Art der Bewegung, die wir Wärme
  nennen
• ? Dynamical (or kinetic) theory of heat

16
Actual significance of Joules 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 Boyles 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 Carnots
  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
Carnots 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
Carnots function in caloric theory

• In caloric theory of heat (? caloric)
  Carnots 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 Carnots 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
(Carnots 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 ?) ?w T2 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
Bunsens 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 Carnots efficiency ?C is defined as
  the ratio
• L/? we immediately obtain from (11)
• ?C (T1?T2) (13)
• Replacing entering caloric by its thermal energy
  ? Kelvins 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 T11?(T2/T)

32

• 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)

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
  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

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
Maxwells formulation

• Taking into account these definitions, the
  original Maxwells 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 Mosengeils theory (1907) (Einstein 1908)
• Q Q0?(1?? 2), T T0 ?(1?? 2),
• invariance of Wiens law, (? /T) inv.
• Invariance of entropy S S0 (Planck)
• (?moving thermometer reads low)
• Otts theory (1963) (Einstein 1952)
• Q Q0/?(1?? 2), T T0 /?(1?? 2),
• (?moving thermometer reads high)
• Jaynes (1957) T T0 (NO DEFINITE SOLUTION
  !)