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Final Answers
© 2000-2014   Gérard P. Michon, Ph.D.

Classical & Relativistic
Thermodynamics

 Benjamin Count Rumford 
 1753-1814  Sadi Carnot 
 1796-1832
La chaleur est la force vive qui résulte des
mouvements insensibles des molécules d'un corps
.
"Mémoire sur la chaleur" (1780)   Lavoisier & Laplace
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Related articles:

Related Links  (Outside this Site)

Heat and Work  (historical viewpoint)  Physics Hypertextbook  by Glenn Elert.
History of Thermodynamics  (Wikipedia).
About Temperature  at  Project Skymath.
 
Heat and Thermodynamics  by  Pr.  Jeremy B. Tatum  (retired from Uvic).
Notes:   1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 18 | 16 | 17 | 18
 
Thermodynamic Asymmetry in Time  by  Craig Callender  (Pr. of Philosophy).
Joule Expansion, Joule-Thomson Expansion  by  W. Ron Salzman  (2004).
Jonathan Oppenheim  (Cambridge)   |   Black Hole Thermodynamics
Black Hole Information Loss  by  Warren G. Anderson  (1996)
 
On the Weight of Heat and Thermal Equilibrium in General Relativity  (1930)
Possibilities in relativistic thermodynamics for irreversible processes without exhaustion of free energy  by  Richard C. Tolman  (1932)
Relativistic thermodynamics with angular momentum and its application to
blackbody radiation
  by  Nakamura, T.K.  and  Okamoto, R.  (August 2004)
Statistical mechanics of generally covariant quantum theories
by  Merced Montesinos  and  Carlo Rovelli
On Special and General Relativistic Thermodynamics
by  Horst-Heino von Borzeszkowski  and  Thoralf Chrobok
Thermodynamics as a Finsler Space with Torsion  by  Robert M. Kiehn.
 
Relativistic Thermodynamics  at  IHP
(Institut Henri Poincaré, founded in 1928) :
Louis de Broglie (1892-1987; Nobel 1929),
Olivier Costa de Beauregard (1911-2007)
Abdelmalek Guessous, Jean Fronteau, etc.

DMOZ: Thermodynamics

Videos :   Ludwig Boltzmann (1844-1906).  Dangerous Knowledge: 4  |  5  |  6
Information Paradox (Hawking vs. Susskind & Maldacena):   1  |  2  |  3  |  4  |  5
Thermodynamic Temperature
How hot can it get?  by  Michael Stevens   (Vsauce).
Negative temperatures (1956)  by  Philip Moriarty  (filmed by  Brady Haran).

 
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Thermodynamics, Heat, Temperature


 Vibrating 
 Piston  Bouncing Particle
(2003-11-11)     The Concept of Temperature
How random energy is transferred...

Consider the situation pictured above:  A vibrating piston is retained by a spring at one end of a narrow cavity containing a single bouncing particle.  We'll assume that the particle and the piston only have horizontal motions.  Everything is frictionless and undamped, as would be the case at the molecular level...

If the center of mass of two bodies has a (vectorial) velocity  v,  an exchange of a momentum  dp  between them translates into the following transfer of energy  dE,  in an  elastic collision  (this expression remains valid relativistically).

dE   =   v . dp

We would like to characterize an ultimate "equilibrium", in which the average energy exchange is zero whenever the piston and the particle interact.

Let's call V the velocity of the piston, M its mass and x the elongation of the spring that holds it.  The force holding back the piston is -Mwx for some constant w, so that, in the absence of shocks with the particle, the motion is a sinusoidal function of the time t and the total energy E given below remains constant:

x   =   A sin (wt + f)             V   =   Aw cos (wt + f)
E   =   ½ M ( V 2 + w 2 x 2 )

When the particle, of mass m and velocity v, collides with the piston, we have:

V > v     and     X  +  v t   =   A sin (wt + f)

The first condition expresses the fact that the particle moves towards the piston just before the shock.  The second relation involves a random initial position X, uniformly distributed in the cavity.  Only the value of X modulo 2A  [i.e., the remainder when 2A is divided into X]  is relevant, and if we consider that the length of the cavity is many times greater than 2A, we may assume that X is uniformly distributed in some interval of length 2A.

 Come back later, we're
 still working on this one...

For a perfect gas, the thermodynamical temperature  T  can be defined by the following formula, which remains true in the relativistic case (provided that the averaging denoted by angle brackets is understood to be at  constant time  in the observer's coordinates).  This appears as equation 14-4 in the 1967 doctoral dissertation of Abdelmalek Guessous  (supervised by Louis de Broglie).

3 k T   =   < ( v - <v> ) . ( p - <p> ) >

In this, v and p are, respectively, the speed and momentum of each individual (monoatomic) molecule.

 Come back later, we're
 still working on this one...


(2003-11-09)     The First Law: Energy is Conserved

Arguably, thermodynamics became a science in 1850, when Rudolf Clausius (1822-1888) published a modern form of the  first law of thermodynamics:

In any process, energy can be changed from one form to another,
including heat and work, but it is never created or destroyed
.

 Benjamin Thompson,
Count Rumford (1753-1814)

This summarized the observations of several pioneers who helped put an end to the antiquated  caloric theory, which was once prevalent:

 Sir Humphry Davy 
 1778-1829
  • Benjamin Count Rumford (1753-1814):  Boring cannons, 1798.
  • Sir Humphry Davy (1778-1829):  1799.
  • Sadi Carnot (1796-1832):  Puissance motrice du feu, 1824.
  • James Prescott Joule (1818-1899):  Paddle-wheel experiment, 1840.  Hermann von Helmholtz 
1821-1894
  • Julius Robert Mayer (1814-1878):  Stirring paper pulp, 1842.
  • Hermann Helmholtz (1821-1894):  On "animal heat", 1847.


(2003-11-09)   The Second Law: Entropy Increases
 Sadi Carnot 
 1796-1832; X1812
Sadi Carnot, X1812

Heat travels only from hot to cold.  Carnot's principle.

In 1824, Sadi Carnot gave a fundamental limitation of steam engines by analyzing the ideal engine now named after him, which turns out to be the most efficient of all possible heat engines.  This result is probably best expressed with the fundamental thermodynamical concepts which were fully developped  after  Carnot's pioneering work, namely  internal energy  (U)  and  entropy  (S).

After the work of Carnot, the concept of entropy was pioneered by Rudolf Clausius (1822-1888),  William John Macquorn Rankine (1820-1872)  and  Lord Kelvin (1824-1907).

Entropy (S) is an  extensive  quantity because the entropy of the whole is the sum of the entropies of the parts.  However, entropy is  not  conserved as time goes on:  It increases in any transformation which is not reversible.

 Temperature vs. Entropy 
in a Carnot Cycle
 
Temperature as a function of Entropy

Carnot's Ideal Engine:  The Carnot Cycle.

  • Hot (A to B) slow isothermal expansion.  The hot gas performs work and receives a quantity of heat  TDS.
  • Cooling (B to C) adiabatic expansion.  The gas keeps working without any exchange of heat. 
  • Cold (C to D) slow isothermal compression.  The gas receives work and gives off (wasted) heat TDS.
  • Heating (D to A) adiabatic compression from outside work (flywheel) returns the gas to its initial state A.

As the internal energy (U) depends only on the state of the system, its total change is zero in  any  true cycle.  So, the total work done by the engine in Carnot's cycle is equal to the net quantity of heat it receives:  (T1-T0DS.

For any simple hydrostatic system, that quantity would be the area enclosed by the loop which describes the system's evolution in the above S-T diagram (which gives the system's supposedly uniform temperature as a function of its entropy).  This same mechanical work is also the area of the corresponding loop in the V-p diagram (pressure as a function of volume).  This latter viewpoint may look more practical, but it's far more obscure when it comes to discussing efficiency limits...

Efficiency of a Heat Engine

Carnot was primarily concerned with steam engines and the mechanical power which could be obtained from the fire heating up the hot reservoir (the cold reservoir being provided from the surroundings at "no cost", from a water stream or from atmospheric air).  He thus defined the efficiency of an engine as the ratio of the work done to the quantity of heat transferred  from the hot source.  For Carnot's ideal engine, the above shows that this ratio boils down to the following quantity, known as  Carnot's limit :  Sadi Carnot 
 1796-1832

-  T /  T1

The unavoidable "waste" is the ratio of the extreme temperatures involved.

Refrigeration Efficiency

The primary purpose of a refrigerator (or an air-conditioning unit) is to extract heat from the cold source (to make it cooler).  Its efficiency is thus usefully defined as the ratio of that heat to the mechanical power used to produce the transfer.  So defined, the efficiency of a Carnot engine driven (backwards) as a refrigerator is:

( T /  T0  -  1 ) -1

This is [much] more than 100%, except for extreme refrigeration, which would  divide by two or more  the ambient temperature above absolute zero.

The rated efficiency of commercial cooling units (the "coefficient of performance" COP) is somewhat lower, because it's defined in terms of the  electrical  power which drives the motor  (taking into account any wasted electrical energy).

Efficiency of a Heat Pump

A heat pump is driven like a refrigeration unit, but its useful output is the heat transferred to the hot side (to make it warmer).  A little heat comes from the electrical power not converted into mechanical work, the rest is "pumped" at an efficiency which always exceeds (by far) 100% of the mechanical work.  For a Carnot engine, this latter efficiency is:

( 1  -  T /  T1 ) -1


(2006-09-12)   What's worth knowing about a thermodynamical system
The two flavors of state variables: extensive and intensive.

Thermodynamics is based on the statement (or belief) that almost all details about large physical systems are irrelevant or impossible to describe.  There would be no point in tracking individual molecules in a bottle of gas, even if this was practical.  Only a small number of statistical features are relevant.

A substantial part of thermodynamics need not even be based on statistical physics.  Once the interesting quantities are identified, their mutual relations may not be obvious and they repay study...  This global approach to thermodynamics starts with the notion of  internal energy  (U)  and/or with other  thermodynamical potentials  (H, F, G ...)  measured in energy units.

As the name implies, the variation of the internal energy  (U)  of a system is an accurate account of all forms of energy it exchanges with the rest of the Universe.  In the simplest cases, the variation  (dU)  in internal energy boils down to the mechanical work  (dW)  done to the system and the quantity of heat  (dQ)  which it receives.  The first law of thermodynamics then reads:

dU   =   dQ + dW

U  is a function of the system's state variables, so its variation is a  differential form  of these  (as denoted by a straight "d")  whereas the same need not be true of  Q  and  W,  which may depend separately on other external conditions  (that's what the greek  d  is a reminder of).

A few exchanges of energy can be traced to obvious changes in quantities which are called  extensive  (loosely speaking, a physical quantity is called  extensive  when the measure of the whole is the sum of the measures of the parts).  One example of an extensive quantity is the volume (V) of a system.  A small change in an extensive quantity entails a proportional change in energy.  The coefficient of proportionality is the associated  intensive  quantity.  The intensive quantity associated to volume is  pressure...

dW   =   - pe dV  Internal vs. External Pressure

That relation comes from the fact that the mechanical  work done  by a force is the (scalar) product of that force by its displacement  (i.e., the infinitesimal motion of the point which yields to it).

In the illustrated case of a "system" consisting of the blue gas and the red piston,  we must (at the very least) consider the kinetic energy of the piston, whose speed will change because of the net force which results from any difference between the external pressure (pe) and the internal pressure (p).

However, in the very special case of extremely slow changes  (a  quasistatic  transformation)  the kinetic energy of the piston is utterly negligible and the internal pressure (p) remains nearly equal to the slowly evolving external pressure:

dU   =   - p dV

Now, we can't give a general expression for  dU  valid for more general transformations  unless  some new extensive variable is involved  (besides volume).  Our intial explanation involving the piston's momentum is certainly valid  (momentum is an extensive variable)  but it can't be the "final" one, since common experience shows that the piston will eventually stop.  The piston's energy and/or momentum must have been "dissipated" into something commonly called  heat.

Could  heat  itself be the extensive quantity involved in the energy balance of an infinitesimal energy balance for an irreversible transformation?  The answer is a resounding  no Benjamin Count Rumford 
 1753-1814  This misguided explanation would essentially be equivalent to considering heat as some kind of conserved fluid  (formerly dubbed "caloric").  The naive  caloric theory  was first shown to be untenable by Rumford in 1798.

The pioneering work of Carnot (1824) was only reconciled with the first law by Rudolf Clausius in 1854, as he recognized the importance of the ratio  dQ/T  of the quantity of heat  dQ  transferred at a certain temperature T.  In 1865, this ratio was equated to a change  dS  in the relevant fundamental extensive quantity for which Clausius himself coined the word  entropy.

Like volume (V) is associated with pressure (p), entropy (S) is associated with the intensive quantity called  thermodynamical temperature  (T)  or "temperature above absolute zero", which can be defined as the reciprocal of the  integrating factor  of heat...  This is a linear function of (modern) customary measurements of temperature.  The SI unit of thermodynamical temperature is the  kelvin  (not capitalized).  It's abbreviated K, and should not be used with the word "degree" or the "°" symbol  (unlike the related "degree Celsius" which refers to a scale originating at the  ice point,  at  0°C  or  273.15 K, instead of the  absolute zero  at  0 K  or  -273.15°C).

Nowadays, 0°C  is  exactly  equal to  273.15 K  and  approximately  equal to the ice point  (the temperature of melting ice under  1 atm  of pressure).  By definition of the Kelvin scale, the temperature of the triple point of water is exactly  273.16 K  or  0.01°C  [currently].

A system at temperature  T  that receives a quantity of heat  dQ  (from an external source at temperature  T)  undergoes a variation  dS  in its entropy:

dS   =   dQ / T

Conversely, the source "receives" a quantity of heat  - dQ  and its own entropy varies by   - dQ / Te.  The total change in entropy thus entailed is:

dQ   ( 1/T - 1/Te )

The  second law of thermodynamic  states that this is always a nonnegative quantity  (total entropy never decreases)...  The system can receive a positive quantity of heat  (dQ > 0)  only from a warmer source  (Te > T).

A statistical definition of entropy (Boltzmann's relation) was first given by Ludwig Boltzmann (1844-1906) in 1877, using the constant  k  now named after him.  In 1948, Boltzmann's definition of entropy was properly generalized by Claude Shannon (1916-2001) in the newer context of  information theory.

The general expression for infinitesimal transformations (reversible or not) in the case of an homogeneous gas  (i.e., an  hydrostatic  system)  is simply:

dU   =   T dS  -  p dV

In a  quasistatic  transformation  (p = pe)  the two components of  dU  can be identified with  heat transferred  and  work done  ( to  the system) :

dQ  =  T dS               dW  =  - p dV

The above applies to an  hydrostatic  system involving only two  extensive  quantities (entropy and volume) but it generalizes nicely according to the following pattern which gives the variation (dU) in internal energy as a sum of the variations of several extensive quantities, each weighted by its associated intensive quantity:

Extensive: Entropy (S) Volume Area Length etc.
dU   =   T dS -  p dV a dA +  f dL +  ...
Intensive: Temperature Pressure Surface
tension
Tensile
force
 

For a system whose state at equilibrium is described by N extensive variables (including entropy) the right-hand side of the above includes N terms (N=2 for an hydrostatic system).  This equation is the differential version of the relation which gives internal energy in term of N variables, including entropy.  That same relation may also be viewed as giving entropy in terms of N variables, including internal energy.  A state of equilibrium is characterized by a  maximal  value of entropy.

Relativistic Thermodynamics:  Total Hamiltonian Energy (E)

If the system is in (relativistic) motion at velocity  v  and momentum  p, it may also receive some mechanical work  v.dp  which translates into a change of its overall kinetic energy and, thus, of its total  Hamiltonian  energy  (E) :

dE   =   v.dp

Albert Einstein used that relation to establish the basic variance of  E, namely:

Hamiltonian Energy
E   =    E0
vinculum
space
vinculum
Ö  1 - v2/c2

On the other hand, the  internal energy  U  is best  defined  as follows.

U   =   E  -  v . p

This gives  U  and other  thermodynamical potentials  (H, F, G, etc.) the same variance as a quantity of heat, a temperature or a Lagrangian:

Internal Energy
Vinculum
    U     =     U0   Ö  1 - v2/c2    

For more details about other equations of relativistic thermodynamics and the historical controversies about them, see below.


(2005-06-14)     1948:  Claude Shannon's Statistical Entropy (S)
From here to certainty, entropy measures the lack of information.    ;-)

Consider an uncertain situation described by  W  distinct mutually exclusive elementary events whose probabilities add up to 1.  If the  n-th  such event has probability  pn , then  Shannon  defined the  statistical entropy  S  as:

 
S ( p1 , p2 , ... , pW )   =    
 
W
å
n = 1
 
  - k  pn  Log (pn )
 

In this, k is an arbitrary positive constant, which effectively defines the unit of entropy.  In physics, entropy is normally expressed in joules per kelvin (J/K), logarithms are  natural logarithms,  and k is  Boltzmann's constant.  Besides other energy-to-temperature ratios of units, entropy may also be expressed in  units of information, as discussed below.

Apparently, only two units of entropy have ever been given a specific name but neither became popular:  The  boltzmann  is the unit which makes  k  equal to unity in the above defining relation  (one  boltzmann  is about  1.38 10-23 J/K).  The  clausius  is a practical unit best defined as  1000 thermochemical calories per degree Celsius, namely 4184 J/K  (exactly).

In practice, the "clausius" has always been  much  less popular than either the  cal/K  (in the old days)  or the  J/K  (nowadays).

So defined, the statistical entropy  S  is nonnegative.  It's minimal  (S = 0)  when one of the elementary events is certain.  For a given  W,  the entropy  S  is maximal when every elementary event has the same probability, in which case   S =  k Log (W).

The relation   S =  k Log (W)   is known as  Boltzmann's Relation.  It's named after Ludwig Boltzmann (1844-1906) who introduced it in a context where W was the number of possible states within a small interval of energy of fixed width, near equilibrium (where all states are, indeed, nearly equiprobable).  As the width of such an interval is arbitrary, an arbitrary additive quantity was involved in Boltzmann's original definition  (before quantum theory removed that ambiguity).  Boltzmann's constant  k = R/N  is the ratio of the ideal gas constant (R) to Avogadro's number (N).

The above definition of entropy is due to Claude Elwood Shannon (1916-2001).  Up to a positive factor, it yields the only nonnegative continuous function which can be consistently computed by splitting events into two sets and equating  S  to the  sum  of three terms, namely the two-event entropy for the split and the two conditional entropies weighted by the respective probabilities of the splitting sets:

p   =    p1 + p2 + ... + pn
q= pn+1 + pn+2 + ... + pW    =    1- p
S ( p1 ... pW ) = S (p,q)  +  p S (p1 /p ... pn /p)  +  q S (pn+1 /q ... pW /q)


Units used in Computer Science and/or Information Theory :

In information theory, the unit of entropy and/or information is the bit, namely the information given by a single  binary digit  (0 or 1).  In the above, this means k = 1/Log(2).  [  k = 1 if logarithms are understood to be in base 2, but "lg(x)" is a better notation for the binary logarithm of x, following D.E. Knuth. ]  When it's necessary to avoid the confusion between a binary digit (bit) and the information it conveys, this unit of information is best called a  shannon  (symbol Sh).

The word "bit" itself was coined around 1950, at Bell Labs, by the Princeton mathematician John W. Tukey.  Werner Buchholz started calling 8 bits a "byte" in 1956.  Units obtained by multiplying either of these by a power of 1024 have since become very popular...

Other odd units of information are virtually unused.  For the record, this includes the  hartley  (symbol Hart) which is to a decimal digit what the  shannon  (Sh)  is to a binary digit;  the ratio of the former to the latter is:

Log(10)/Log(2)   =   lg(10)   =   3.32192809488736234787...

Therefore, 1 Hart is about 3.322 Sh.  This unit, based on decimal logarithms, was named after Ralph V.L. Hartley  (of  radio oscillator  fame)  who introduced information as a physical quantity in 1927-1928, more than 20 years before the development of  Information Theory  by Claude Shannon.

The "nat" or "nit" is another unused unit of information  (preferably called a  boltzmann  as a physical unit of entropy)  which is obtained by letting k = 1 in the above defining equation, while retaining the use of natural logarithms:  One  nat  is about 1.44 Sh = 1 boltzmann  (and a  nit  is about 1.44 bits).

If you must know, a bit (or, rather, a shannon) is about  9.57 10-24 J/K... A  picojoule per kelvin  (pJ/K) is about 12 gigabytes  (12.1646 GB).

The  hoaxy  brontobyte  would be about  1.7668 J/K.

Shannon-Hartley channel capacity theorem   |   Shannon-Nyquist sampling theorem


(2005-06-19)     The Third Law:  Nernst's Principle (1906)
On the inaccessible state where both entropy and temperature are zero.

The definition of statistical entropy in absolute terms makes it clear that S would be zero only in some perfectly well-defined pure quantum state.  Other physical definitions of entropy fail to be so specific and leave open the possibility that an arbitrary constant can be added to S.

The principle of Nernst (sometimes called the "third law" of thermodynamics) reconciles the two perspectives by stating that entropy must be zero at zero temperature.  Various forms of this law were stated by Walther Hermann Nernst (1864-1941; Nobel 1920) between 1906 and 1912. 

A consequence of this statement is the fact that nothing can be cooled down to the absolute zero of temperature (or else, there would be a prior cooling apparatus with negative temperature and/or entropy).  In the limited context of classical thermodynamics, the principle of Nernst thus justifies the very existence of the absolute zero, as a  lower limit  for thermodynamic temperatures.

Violations of Nernst's Principle :

From a quantum viewpoint, the principle of Nernst would be rigorously true only if the ground state of every system was nondegenerate  (i.e., if there was always only one quantum state of lowest energy).  Although this is not the case, there are normally very few quantum states of lowest energy, among many other states whose energy is almost as low.  Therefore, the statistical entropy at zero temperature is always extremely small, even when it's not strictly equal to zero.

In practice,  metastable  conditions present a much more annoying problem:  For example, although crystals have a lower entropy than glasses, some glasses transform extremely slowly into crystals and may appear absolutely stable...  In such a case, a substance may be observed to retain a significant positive entropy at a temperature very near the absolute zero of temperature.  This is a practical violation of the principle of Nernst, albeit not a theoretical one...


(2006-09-18)     Thermodynamic Potentials
Ad hoc substitutes for internal energy or entropy.

Thermodymic potentials  are functions of the state of the system obtained by subtracting from the  internal energy  (U)  some products of conjugate quantities  (pairs of intensive and extensive quantities, like -p and V).

They have interesting physical interpretations in common circumstances.  For example, under constant (atmospheric) pressure  enthalpy  (H=U+pV)  describes all energy exchanges except mechanical work.  That's why chemists focus on changes in enthalpy for chemical reactions,  in order to rule out whatever irrelevant mechanical work is exchanged with the atmosphere in a chemical explosion  (involving a substantial change in V).

As illustrated  belowfree enthalpy  (Gibbs' function, denoted G)  is a convenient way to deal with a  phase transition,  since such a transformation leaves  G  unchanged, because it takes place at constant temperature and constant pressure.  More generally, the difference in free enthalpy between two states of equilibrium is the least amount of useful energy (excluding both heat and pressure work) which the system must exchange with the outside to go from one state to the other.

 Battery Depletion One non-hydrostatic example is a battery of electromotive force  e  (i.e.,  e  is the voltage at the electrodes when no current flows)  and internal resistance R as it delivers a charge q in a time t.  The longer the time t, the closer we are to the quasistatic conditions which make the transfer of energy approach the lower limit imposed by the change in G, according to the following inequality:

DU   =   (VA - VB ) i t   =   (Ri - e) q   >   -e q   =   DG

Thermodynamic Potentials for an Hydrostatic System
Name ExpressionDifferential FormMaxwell's Relation
Internal
Energy
 U   dU =   T dS - p dV
 (  T  )S   =   - (  p  )V
vinculum vinculum
V S
1
EnthalpyH U + pVdH =   T dS + V dp
 (  T  )S   =     (  V  )p
vinculum vinculum
p S
2
Free
Energy
F U - TSdF = - S dT - p dV
 (  S  )T   =     (  p  )V
vinculum vinculum
V T
3
Free
Enthalpy
G H - TSdG = - S dT + V dp
 (  S  )T   =   - (  V  )p
vinculum vinculum
p T
4

The tabulated differential relations are of the following mathematical form:

dz   =   (z/x) dx + (z/y) dy

The matching  Maxwell relations  in the last-column simply state that

2z / yx  =  2z / xy

Such trivial mathematical statements aren't physically obvious...  For example, from the  equation of state  of a gas  (i.e., the relation between its volume, temperature and pressure)  the last two give the  isothermal derivatives  of entropy with respect to pressure or volume.  This can be integrated to give an expression of entropy involving parameters which are functions of temperature alone  (example of a  Van der Waals fluid  below).

Free entropy & Massieu function (1869)   by   François Massieu (1832-1896; X1851)


(2006-09-19)     Calorimetric Coefficients.  Adiabatic Coefficient  (g).
Heat capacities, compressibilities, thermal expansibilities, etc.

Thermal capacity is defined as the ratio of the heat received to the associated increase in temperature.  For an  hydrostatic system,  that quantity comes in two flavors:  isobaric  (constant pressure) and  isochoric  (constant volume) :

Cp   =   T (  S  )p   =   (  H  )p
vinculum vinculum
T T
CV   =   T (  S  )V   =   (  U  )V
vinculum vinculum
T T

The difference between those two happens to be a quantity which can easily be derived from the  equation of state  (the relation linking  p, V and T):

Relation of Mayer   ( for  molar  heat capacities )
Cp -  CV   =    T (  p  )V (  V  )p  
vinculum vinculum
T T
=    R     [ for a mole of an ideal gas ]
Proof :
dS   =  (  S  )V   dT   +   (  S  )T  dV
vinculum vinculum
T V
 (  S  )p  =  (  S  )V      +   (  S  )T   (  V  )p
vinculum vinculum vinculum vinculum
T T V T
  Cp   =   CV     +   (  p  )V   (  V  )p
vinculum vinculum vinculum vinculum
T T T T

That last step uses the  third relation of Maxwell.   QED

By definition, the  adiabatic coefficient  is the ratio   g  =  Cp / Cv  (equal to  1+2/j  where  j  is 3, 5 or 6 for a classical perfect gas obeying Joule's law).

The adiabatic coefficient may also be expressed as other ratios of noteworthy quantities...

 Come back later, we're
 still working on this one...


(2013-01-15)     Relating  isothermal  and  isentropic  derivatives.
The  latter  must be used to compute the  speed of sound.

Sound is a typical  isentropic  phenomenon  (i.e.,  for reasonnably small intensities, sound is reversible and adiabatic).  When quick vibrations are used to probe something, what we're feeling are  isentropic  derivatives.

On the other hand,  slow  measurements at room temperature allow thermal equilibria with the room at the beginning and the end of the observed transformation.  In that case, we are measuring  isothermal  coefficients.

Consider, for example, the  stiffness  K  of a fluid or a solid  (more precisely called  bulk modulus of elasticity ).  Its  inverse  is the  relative reduction  in volume caused by an  increase  in pressure.  It comes in two flavors:

Isothermal :
1     =   -  1  (  V  )T
vinculum vinculum vinculum
KT V p
 
Adiabatic :
1     =   -  1  (  V  )S
vinculum vinculum vinculum
KS V p
I'm ignoring the name  (compressibility)  and the symbol  (k)  for the reciprocal of  stiffness.  The other two well-established elasticity coefficients expressed in the same units have unused reciprocals.

We'll need the  volumetric  thermal expansion coefficient, defined by :

b    =     1  (  V  )p
vinculum vinculum
V T

Here goes nothing  (Maxwell's fourth relation is used for the third line and the fourth line is obtained by expanding the leading factor of the last term).

dV   =  (  V  )S   dp   +   (  V  )p  dS
vinculum vinculum
p S
 (  V  )T  =  (  V  )S      +   (  V  )p   (  S  )T
vinculum vinculum vinculum vinculum
p p S p
  -V   =   -V     -   (  V  )p   (  V  )p
vinculum vinculum vinculum vinculum
KT KS S T
  1   =   1     +   (  T  )p   (  V  )p   b
vinculum vinculum vinculum vinculum
KT KS S T

    1     =     1   +   T b 2    
vinculum vinculum vinculum
KT KS Cp / V

The quantity   Cp / V   (which appears as the denominator of the last term)  is the molar heat capacity per molar volume; it's also equal to the mass density  r  multiplied into the  specific heat capacity  c (note lowercase).

All told, that term is the inverse of a quantity  W  homogeneous to a pressure, an elasticity coefficient, or an energy density  (more than 10 years ago, I proposed the term  thermal wring  as a pretext for using the symbol  W,  which isn't overloaded in this context):

    W    =     Cp     =     r cp     =     KS     =     g KT     =     KS KT    
vinculum vinculum vinculum vinculum vinculum
V T b 2 T b 2 g - 1 g - 1 KS - KT
 


(2013-01-17)     The thermal Grüneisen parameter   z
Adiabatic derivative of   Log T   with respect to   Log 1/V   (or  Log r).
Many authors use  g  (or gth )  to denote this Grüneisen parameter.  I beg to differ  (to prevent confusion with the adiabatic coefficient).

      z     =     -  V   (  T  )S   =     r   (  T  )S    
vinculum vinculum vinculum vinculum
T V T ¶r

z  is the adiabatic ratio of the logarithmic differential of temperature to that of either density or volume  (same thing but opposite signs).

The relation to the adiabatic coefficient  g  =  Cp / Cv  =  Ks / KT   is simply:

g   =   1  +  z b T

For condensed states of matter  (liquids or solids)  the volumetric coefficient of thermal expansion  (b)  is quite small and the above adiabatic coefficient remains very close to unity.  The Grüneisen parameter is more meaningful  (the adiabatic coefficient is traditionally used in the study of gases).

Pressure derivatives of shear and bulk moduli from the thermal Grüneisen parameter and volume-pressure data
A.M. Hofmeister & H.K. Mao,  Geochimica et Cosmochimica Acta, 67, 7, pp. 1207-1227  (2003).
 
Grüneisen parameters and isothermal equations of state
L. Vocadlo.  J.P. Poirier  &  G.D. Price,  American Mineralogist, 85   (2000).
 
The Grüneisen parameter:  Computer calculations via lattice dynamics
N.L. Vocadlo  &  Geoffrey D. Price,  Physics of the Earth and Planetary Interiors, 82, pp. 261-270  (1994).
 
Grüneisen parameters  by  Eric Weisstein
Wikipedia :   Grüneisen parameter   |   Eduard Grüneisen (1877-1949)


(2006-09-18)     Deriving entropy from an  equation of state :
A closed formula for the entropy of a  Van der Waals  fluid.

For one mole of a  Van der Waals gas, we have:

( p + a / V2 )  ( V-b )   =   RT

( p  -  a / V2  +  2 a b / V 3 )  dV  +  ( V-b ) dp   =   R dT

Let's use the  third relation of Maxwell :

 (  S  )T   =   (  p  )V   =    R
vinculum vinculum vinculum
V T V - b

Therefore,   S   =   f (T)  +  R Log (V-b)     for some function  f

To be more definite, we resort to calorimetric considerations, namely:

 (  S  )V   =     CV     =   f ' (T)
vinculum vinculum
T T

This shows that CV is a function of temperature alone.  So, we may as well evaluate it for large molar volumes  (very low pressure)  and find that:

CV   =   j/2 R

That relation comes from the fact that, at very low pressure, the energy of interaction between molecules is negligible.  Therefore, by the  theorem of equipartition of energy,  the entire energy of a gas is the energy which gets equally distributed among the  j  active  degrees of freedom  of each molecule, including the 3 translational degrees of freedom which are used to  define  temperature and  0, 2 or 3  rotational degrees of freedom  (we assume the temperature is low enough for vibrations modes of the molecules to have negligible effects; see below).  All told:  j = 3  for a monoatomic gas, j = 5  for a diatomic gas, j = 6  otherwise.

S   =   S0  +   j/2 R Log (T)  +  R Log (V-b)

There's no way to reconcile this expression with Nernst's third law to make the entropy vanish at zero temperature.  That's because the domain of validity of the Van der Waals equation of state does not extend all the way down to zero temperature (there would presumably be a transition to a solid phase at low temperature, which is not accounted for by the model).  So, we may as well accept the  classical  view, which defines entropy only up to an additive constant and choose the following expression  (the statistical definition of entropy, ultimately based on quantum considerations, leaves no such leeway).

S   =   R  Log [  T j /2  (V-b)  ]

Therefore, the  isentropic  equation of a Van der Waals fluid generalizes  one  of the formulations valid for a perfect gas  (with b = 0)  namely:

T j /2  (V-b)   =   constant

Unlike  CV,  Cp = g CV  is not constant for a Van der Waals fluid, since:

Cp -  CV   =    T (  p  )V (  V  )p  
vinculum vinculum
T T


(2012-07-03)   The  empirical  Dulong-Petit law   (1819) :
The heat capacity  per mole  is nearly the same  (3R)  for all crystals.

In 1819, Dulong and Petit  (respectively the third and the second holder of the chair of physics at  Polytechnique  in Paris, France)  jointly observed that the heat capacity of metallic crystals is essentially proportional to their number of atoms.  They found it to be nearly  25 J/K/mol  for every solid metal they investigated  (this would have failed at very low temperatures).

In 1907, Albert Einstein gave a simplified quantum model, which explained qualitatively why the Dulong-Petit law fails at low temperature.  His model also linked the empirical values of the  molar  heat capacity  (at high temperatures)  to the ideal gas constant R :

3 R   =   24.943386(23)  J/K/mol

In 1912, Peter Debye devised an even better model  (equating the solid's vibrational modes with propagating  phonons )  which is  also  good at low temperatures.  Its limited accuracy at intermediate temperatures is entirely due to the simplifying assumption that all phonons travel at the same speed.  When applied to a gas of  photons,  that statement is true and the model then describes blackbody radiation perfectly, explaining Planck's law !

Departure from the Law of Dulong and Petit  by  Rod Nave  (Hyperphysics).
 
Dulong-Petit law (1819)   |   Pierre Louis Dulong (1785-1838; X1801)   |   Alexis Petit (1791-1820; X1807)


wangshu (2010-12-19)   Vibrational modes of the  CO molecule.
What do the following energy levels contribute to the heat capacity of carbon dioxide at 400 K?     En  =  (n+1) hn   where  n = 20 THz

The ratio   hn / kT  =  2.4   indicates that the quanta involved are similar to the average energy of a thermal photon  (2.7 kT).

 Come back later, we're
 still working on this one...

Specific Heat Capacity of Carbon Dioxide from 175 K to 6000 K


(2005-06-25)     Latent Heat (L) and Clapeyron's Relation
As entropy varies in a change of phase, some heat must be transferred.

The British chemist Joseph Black (1728-1799) is credited with the 1754 discovery of  fixed air  (carbon dioxide) which helped disprove the erroneous  phlogiston  theory of combustion.  James Watt (1736-1819) was once his pupil and his assistant.  Around 1761, Black observed that a  phase transition  (e.g., from solid to liquid)  must be accompanied by a transfer of  heat,  which is now called  latent heat.  In 1764, he first measured the latent heat of steam.

The latent heat  L  is best described as the difference  DH  in the enthalpy (H=U+pV) of the two phases, which accurately represents heat transferred under constant pressure  (as this voids the second term in  dH = TdS + Vdp).

Under constant pressure, phase transistion  occurs at constant temperature.  So, the  free enthalpy  (G=H-TS)  remains constant  (as  dG = -SdT + Vdp).

Consider now how this  free enthalpy  G  varies along the curve which gives the pressure  p  as a function of the temperature  T  when the two phases 1 and 2 coexist.  Since  G  is the same on either side of this curve, we have:

dG   =   -S1 dT  +  V1 dp
dG   =   -S2 dT  +  V2 dp

Therefore,  dp/dT  is the ratio  DS/DV  of the change in entropy to the change in volume entailed by the phase transition.  Since  TDS = DH, we obtain:

  Emile Clapeyron 
 1799-1864
The Clausius-Clapeyron Relation :
T dp/dT   =   DH / DV   =   L / DV

That relation is one of the nicest results of  classical  thermodynamics.


(2006-09-23)     Joule-Thomson Coefficient and Inversion Temperature.
A flow expansion of a real gas may cool it enough to liquefy it.

Joule Expansion  &  Inner Pressure

Expanding dS along dT and dV, the expression dU = T dS - p dV becomes:

dU   =   T (  S  )V   dT   +  [ T (  S  )T   - p ] dV
vinculum vinculum
T V
  = CV  dT   +  [ T (  p  )V   - p ] dV
vinculum
T

This gives the following expression (vanishing for a perfect gas) of the so-called Joule coefficient which tells how the temperature of a fluid varies when it undergoes a  Joule expansion, where the internal energy (U) remains constant.  An example of a Joule expansion is the removal of a separation between the gas and an empty chamber.

 (  T  )U    =   - 1   [ T (  p  )V   - p ]
vinculum vinculum vinculum
V Cv T

 Johannes Diderik van der Waals 
 (1837-1923) earned a Nobel prize in 1910 The above square bracket is often called the  internal pressure  or  inner pressure.  It's normally a positive quantity which repays study.  Let's see what it amounts to in the case of a Van der Waals fluid;

 (  U  )T    =      [ T (  p  )T   - p ]   =  RT  -  p   =  a
vinculum vinculum vinculum vinculum
V T V-b V2

By integration, this yields:  U = U0(T) - a / V.  The latent heat of liquefaction (L) is obtained in term of the molar volumes of the gaseous and liquid phases (VG,VL) either as DH = DU+pDV  or as  TDS  (using the above expression for S):

L   =   p (VG-VL) + a (1/VL-1/VG)   =   RT Log [ (VG-b) / (VL-b) ]

Joule-Thomson (Joule-Kelvin) Expansion Flow Process

 Joule-Kelvin 
Liquefier

The  Joule-Kelvin coefficient  (m)  pertains to an isenthalpic expansion.  Its value is obtained as above (from an expression of dH instead of dU):

m   =   (  T  )H    =   1   [ T (  V  )p   - V ]   =    V   ( T a - 1 )
vinculum vinculum vinculum vinculum
p Cp T Cp

m vanishes for perfect gases but allows an expansion flow process which can cool many real gases enough to liquefy them, as long as the initial temperature is below the so-called  inversion temperature, which makes  m  positive.

More precisely, the inversion temperature is a function of pressure.  In the (T,p) diagram, there is a domain where isenthalpic decompression causes cooling.  The boundary of that domain is called the  inversion curve.  In the example of a  Van der Waals fluid, the equation of the inversion curve is obtained as follows:

0   =   T (  V  )p  -  V   =    R T   -  V
vinculum vinculum
T p  -  a / V2  +  2 a b / V3

This gives a relation which we may write next to the equation of state:

R T    =     p V  -  a / V  +  2 a b / V2
R T  +  p b=p V  +  a / V  -   a b / V2

By eliminating  V  between those two equations, we obtain a single relation which is best expressed in units of the critical point  (pc = 1, Tc = 1):

vinculum
T±    =    15 / 4   +   p / 24   ±   Ö   9 - p

If  T  is above  T+  (or below  T- )  then decompression won't cool the gas.

At fairly low pressures, the  inversion temperature  is approximately:

Ti   =   6.75 Tc

The ratio observed for most actual gases is lower than 6.75:  Although it's  7.7  for helium,  it's only  6.1  for hydrogen,  5.2  for neon,  4.9  for oxygen or nitrogen,  and  4.8  for argon.

A Joule-Thomson cryogenic apparatus has no moving parts at low temperature  (note that the cold but unliquefied part of the gas is returned in thermal contact with the high-pressure intake gas, to pre-cool it before the expansion valve).  William Thomson, 
 Baron Kelvin (1824-1907)

The effect was described in 1852 by William Thomson (before he became Lord Kelvin).  So was the basic design, with the cooling countercurrent.  Several such cryogenic devices can be "cascaded" so that one liquefied gas is used to lower the intake temperature of the next apparatus...  Liquid oxygen was obtained this way in 1877, by Louis Paul Cailletet (France) and Raoul Pierre Pictet (Switzerland).  Hydrogen was first liquefied in 1898, by Sir James Dewar (1842-1923).  Finally,  helium  was liquefied in 1908, by the Dutch physicist  Heike Kamerlingh Onnes  (1853-1926; Nobel 1913).

In 1895, the German engineer Carl von Linde (1842-1934) designed an air liquefaction machine based on this throttling process, which is now named after him  (Linde's method).


(2005-06-25)     Relativistic Nature of Thermodynamic Quantities
Heat is not the same "form of energy" as  Hamiltonian energy.

We have introduced relativistic thermodynamics elsewhere in the case of a pointlike system whose  rest mass  may vary.  The fundamental relativistic distinction between the  total Hamiltonian energy (E)  and the  internal energy (U)  is also advertised above.

Now, it should be clear from its statistical definition that entropy is a relativistic invariant, since the probability of a well-defined spacetime event does not depend on the speed of whoever observes it.  Mercifully, all authors agree on this one...

They haven't always agreed on the following (correct) formula for the temperature  T  of a body moving at speed  v  whose temperature is  T0  in its rest frame.

Mosengeil's Formula  (1906)
Vinculum
    T     =     T0   Ö  1 - v2/c2    

The invariance of the entropy  S  means that a quantity of heat  (dQ = T dS)  transforms like the temperature  T.  So do all the  thermodynamic potentials, including  internal energy  (U)  free energy  (F = U-TS)  Helmholtz' enthalpy  (H = U+pV)  and Gibbs' free enthalpy  (G = H-TS)...

Vinculum
dQ     =     dQ0   Ö  1 - v2/c2

This was wrongly ignored by Eddington (and several  later authors) who assumed that heat and thermodynamical potentials ought to transform like mechanical energy, because they are measured in the same units.  In 1952, Einstein himself recanted his 1907 support of the above...  Such bizarre episodes are discussed at length in the 1967 dissertation of  Abdelmalek Guessous, under Louis de Broglie.

One of several ways to justify the above expression for the temperature of a moving body is to remark that the frequency of a typical photon from a moving blackbody is proportional to its temperature.  Thus, if it can be defined at all, temperature must transform like a frequency.  This viewpoint was first expounded in 1906 by  Kurd von Mosengeil  and adopted at once by Planck, Hasenöhrl and Einstein (who would feel a misguided urge to recant, 45 years later).

Kurd Friedrich Rudolf von Mosengeil (1884-1906)  was among the most promising of the numerous doctorands who worked under Max Planck.  He died in a climbing accident shortly after completing his dissertation, which Planck saw through the press in 1907  (Theory of Stationary Radiation in a Uniformly Moving Cavity).

In 1911, F. Jüttner retrieved the same formula for a moving gas, using a relativistic variant of an earlier argument of Helmholtz.  He derived the  relativistic speed distribution function  recently confirmed numerically  (2008-04-23)  by  Constantin Rasinariu  in the case of a 2-dimensional gas.  (In his 1964 paper entitled  Wave-mechanical approach to relativistic thermodynamics, L. Gold gave a quantum version of Jüttner's argument.)  Mosengeil's (correct) formula was also featured in the textbook published by  Max von Laue  in 1924.

By that time, Eddington had already published his own 1923 textbook, containing the aforementioned erroneous idea which other people would later have independently  (apparently, everybody overlooked that part of Eddington's famous book until A. Gamba quoted it in 1965).  The ensuing mess is still with us.  ( 2012-03-10 )

In 1967, under the supervision of  Louis de BroglieAbdelmalek Guessous completed a full-blown attack, using Boltzmann's statistical mechanics  (reviewed below).  This left no doubt that  thermodynamical  temperature must indeed transform as stated above  (in modern physics, other flavors of temperature are not welcome).

Equating heat with a form of energy was once a major breakthrough,  but the fundamental relativistic distinction between heat and Hamiltonian energy noted by most pioneers  (including Einstein in his youth)  was butchered by others  (including Einstein in his old age)  before its ultimate vindication...

A few introductions to Relativistic Thermodynamics :

In 1968, Pierre-V. Grosjean  called the waves of controversies about Mosengeil's formula the  temperature quarrel...  In spite or because of its long history, that dispute is regularly revived  by authors who keep discarding one fundamental subtlety of thermodynamics:  Heat  doesn't  transform like a  Hamiltonian  energy  (which is the time-component of an energy-momentum 4-vector)  but like a LagrangianMany  essays are thus  going astray,  including:

Schewe's article prompted a  Physics Forums  discussion on 2007-10-27.  Related threads:  2007-06-29, 2008-01-11, 2008-04-14.  Other threads include:  2007-10-24, 2007-12-01, etc.

I stand firmly by the statement that  if  temperature can be defined at all, it must obey Mosengeil's formula.  The following articles argue against the premise of that conditional proposition, at least for one type of thermometer:

Review :

 Abdelmalek 
 Guessous (c. 2010)
Abdelmalek Guessous

 
On April 14, 1967, Guessous presented his doctoral dissertation  (Recherches sur la thermodynamique relativiste).  He establishes the relativistic  thermodynamical temperature  as the reciprocal of the integrating factor of the quantity of heat, using the above  statistical definition of entropy.  This superior definition is shown, indeed, to be compatible with Mosengeil's formula.  The actual text isn't easy to skip through, because the author keeps waving the formulas he is arguing  against  (mainly the Ott-Arzeliès formulation, inaugurated by Eddington and ruled out experimentally by P. Ammiraju in 1984).
 
In 1970, Guessous published an expanded version of the thesis as a book entitled "Thermodynamique relativiste",  prefaced by his famous adviser  Louis de Broglie (Nobel 1929).  That work is still quoted by some scholars, like Simon Lévesque, from Cégep de Trois-Rivières  (2007-01-13)  although the subject is no longer fashionable.
 
The original dissertation of Abdelmalek Guessous appears verbatim as the first five chapters of the book  (93 out of 305 pages).  Unfortunately, Guessous avoids contradicting  (formally, at least)  what was established by his adviser for pointlike systems.  This impairs some of the gems found in Chapter VI and beyond, because the author retains his early notations even as he shows them to be dubious.  For example, the paramount definition of internal energy as a thermodynamical potential  (transforming like a Lagrangian)  doesn't appear until Chapter VI where it's dubbed  U',  since  U  was used throughout the thesis for the Hamiltonian energy  E  (not clearly identified as such).  More importantly, Guessous runs into the correct definition of the inertial mass  (namely, the momentum to velocity ratio)  but keeps calling it "renormalized mass"  (denoted M' )  while awkwardly retaining the symbol  M  as a mere name for  E/c2  (denoted  U/c  by him)  which downplays the importance of the aforementioned true inertial mass  m = M'.  So, Guessous  missed  (albeit barely so)  the revolutionary expression of the inertia of energy for  N  interacting particles at a nonzero temperature  T,  presented next in the full glory of traditional notations consistent with the rest of this page.


(2008-10-07)     Inertia of Energy at Nonzero Temperature
The Hamiltonian energy  E  is  not  proportional to the inertial mass  M.

Here's the great formula which I obtained  many  years ago by pushing to their logical conclusion some of the arguments presented in the 1969 addenda to the 1967 doctoral dissertation of Abdelmalek Guessous.  I've kept this result of my younger self  in pectore  for far too long  (I first read Guessous' work in 1973).

E    =     M c 2  -  N k T

We define the inertial mass  (M)  of a relativistic system of  N  point-masses as the ratio of its total momentum  p  to the velocity  v  of its center-of-mass:

p   =   M v

It's not obvious that the dynamical momentum  p  is actually  proportional  to the velocity  v  so that  M  turns out to be simply a scalar quantity !

The description of a moving object of nonzero size always takes place at  constant time  in the frame  K  of the observer.  The events which are part of such a description are simultaneous in  K  but are usually  not simultaneous  in the  rest frame  (K)  of the object.  That viewpoint has been a basic tenet of Special Relativity ever since Einstein showed in excruciating details how it explains the Lorentz-Fitzgerald contraction, which makes the volume  V  of a moving solid appear smaller than its volume at rest  V:

Vinculum
V     =     V0   Ö  1 - v2/c2

 Come back later, we're
 still working on this one...


(2008-10-13)     Angular Momentum
Local temperature is higher on the outer parts of a rotating body.

 Come back later, we're
 still working on this one...

Relativistic Thermodynamics with Angular Momentum by Tadas K. Nakamura  & Ryuuchi Okamoto


(2005-06-20)     Stefan's Law.  The Stefan-Boltzmann Law.
Each unit of area at the surface of a black body radiates a total power proportional to the fourth power of its thermodynamic temperature.

 Joseph Stefan 
 1835-1893
Jozef Stefan

This law was discovered experimentally in 1879, by the Slovene-Austrian physicist Joseph (Jozef) Stefan (1835-1893).  It would be justified theoretically in 1884, by Stefan's most famous student:  Ludwig Boltzmann (1844-1906).

The energy density (in J/m3 or Pa) of the thermal radiation inside an oven of thermodynamic temperature T (in K) is given by the following relation:

[ 4 s / c ]   T 4     =     [ 7.566 10-16 Pa/K4 ]   T 4

On the other hand, each unit of area at the surface of a black body radiates away a power proportional to the fourth power of its temperature T.  The coefficient of proportionality is  Stefan's constant  (which is also known as the  Stefan-Boltzmann constant).  Namely:

s   =   ( 2 p 5 k 4 /  ( 15 h 3 c 3 )   =   5.6704 10-8 W/m2/K4

Those two statements are related.  The latter can be derived from the former using the following argument, based on  geometrical optics,  which merely assumes that radiation escapes at a speed equal to  Einstein's constant  (c).

One of the best physical approximations to an element of the surface of a "black" body is a small opening in the wall of a large cavity ("oven").  Indeed, any light entering such an opening will never be reflected directly.  Whatever comes in is "absorbed", whatever comes out bears no relation whatsoever to any feature of what was recently absorbed...  The thing is black in this precise physical sense.

 Come back later, we're
 still working on this one...


(2005-06-19)     A Putative "Fourth Law" about Maximal Temperature
Several arguments would place an upper bound on temperature...

Several arguments have been proposed which would put a theoretical maximum to the thermodynamic temperature scale.  This has been [abusively] touted as a "fourth law" of thermodynamics.  Some arguments are obsolete, others are still debated within the latest context of the standard model of particle physics:

In 1973, D.C. Kelly argued that no temperature could ever exceed a limit of a trillion kelvins or so, because when particles are heated up, very high kinetic energies will be used to create new particle-antiparticle pairs rather than further contribute to an increase in the velocities of existing particles. Thus, adding energy will increase the total number of particles rather than the temperature.

This quantum argument is predated by a semi-classical guess, featuring a rough quantitative agreement:  In 1952, French physicist Yves Rocard  (father of Michel Rocard, who was France's prime minister from 1988 to 1991)  had argued that the density of electromagnetic energy ought not to exceed by much its value at the surface of a "classical electron"  (a uniformly charged sphere with a radius of about 2.81794 fm).  Stefan's law would then imply an upper limit for temperature on the order of what has since been dubbed "Rocard's temperature", namely:

3.4423 1010 K

One process seems capable of generating temperatures well above Rocard's temperature:  the explosion of a black hole via Hawking radiation.

Rocard's temperature would be that of a black hole of about 8 1011 kg, which is much too small to be created by the gravitational collapse of a star. Such a black hole could only be a "primordial" black hole, resulting from the hypothetical collapse of "original" irregularities  (shortly after the big bang).  Yet, the discussion below shows that a black hole whose temperature is Rocard's temperature would radiate away its energy for a very long time: about 64 million times the age of the present Universe...  It gets hotter as its gets smaller and older.

As the derivation we gave for Stefan's Law was based on geometrical optics, it does not apply in the immediate vicinity of a black hole  (space curvature is important and wavelengths need not be much shorter than the sizes involved).  A careful analysis would show that a Schwarzschild black hole absorbs photons as if it was a massless black sphere (around which space is "flat") with a radius equal to  a = 3Ö3 MG/c2  (about 2.6 times the Schwarzschild radius).  Thus, it emits like a black body of that size  (obeying Stefan's law).  Its power output is:

P   =   ( 4pa 2 ) s T 4   =   108 p s M 2 G 2 T 4 / c 4

Using for  T  the  temperature of a Schwarzschild black hole  of mass M:

P   =   9     [  h    c6  ]
vinculum vinculum vinculum
20480 p  M2 2p  G2

As this entails a mass loss inversely proportional to the square of the mass,  the  cube  of the black hole's mass decreases at a constant rate of  27 / 20480p  (in natural units).  The black hole will thus evaporate completely after a time proportional to the cube of its mass, the coefficient of proportionality being about  5.96 10-11 s  per cubic kilogram.  A black hole of 2 million tonnes (2 109 kg) would therefore have a lifetime about equal to the age of the Universe  (15 billion years).

Hawking thus showed that his first hunch was not quite right:  a black hole's area may decrease steadily because of the radiation which does carry entropy away.  The only absolute law is that, in this process like in any other, the total entropy of the Universe can only increase.  There is little doubt that Hawking's computations are valid down to masses as small as a fraction of a gram.  However, it must be invalid for masses of the order of the Planck mass (about 0.02 mg), as the computed temperature would otherwise be such that a "typical" photon would carry away an energy  kT  equivalent to the black hole's total energy.

This allows the possibility of temperatures more than 15 orders of magnitudes higher than Rocard's temperature.  The "natural" unit of temperature is 21 orders of magnitude above Rocard's temperature.

In his wonderful 1977 book  The First Three Minutes, 1979 Nobel laureate Steven Weinberg gives credit to R. Hagedorn, of the CERN laboratory in Geneva, for coming up with the idea of a maximum temperature in particle physics when an unlimited number of hadron species is allowed.  Weinberg quotes work on the subject by a number of theorists including Kerson Huang (of MIT) and himself, and states the "surprisingly low" maximum temperature of  2 000 000 000 000 K  for the limit based on this idea...

However, in an afterword to the 1993 edition of his book, Weinberg points out that the "asymptotically free" theory of strong interactions made the idea obsolete:  a much hotter Universe would "simply" behave as a gas of quarks, leptons, and photons  (until unknown territory is found in the vicinity of Planck's temperature).

In spite of these and other difficulties, there may be a maximum temperature well below Planck's temperature which is not attainable by any means, including black hole explosions:  One guess is that newly created particles could form a hot shell around the black hole which could radiate energy back into the black hole.  The black hole would thus lose energy at a lesser rate, and would appear cooler and/or larger to a distant observer.  The "fourth law" is not dead yet...


(2005-06-20)     Hawking Radiation and the Entropy of Black Holes
Like all perfect absorbers, black holes radiate with blackbody spectra.

The much-celebrated story of this fundamental discovery starts with the original remark by Stephen W. Hawking (in November 1970) that the surface area of a black hole can never decrease.  This law suggested that surface area is to a black hole what entropy is to any other physical objects.

Jacob Bekenstein was then a postgraduate student working at Princeton under John Wheeler.  He was the first to take this physical analogy seriously, before all the mathematical evidence was in.  Following Wheeler, Bekenstein remarked that black holes swallow the entropy of whatever falls into them.  If the second law of thermodynamics is to hold in a Universe containing black holes, some entropy  must  be assigned to black holes.  Bekenstein suggested that the entropy of a black hole was, in fact, proportional to its surface area...

At first, Hawking was upset by Bekenstein's "misuse" of his discovery, because it seemed "obvious" that anything having an entropy would have a temperature, and that anything having a temperature would radiate away some of its energy.  Since black holes were thought to be unable to let anything escape (including radiation) they could not have a temperature or an entropy.  So it was thought for a while.  But in 1973, Hawking himself made calculations confirming Bekenstein's hunch.

The entropy  (S)  of a black hole is proportional
to the area
  (A)  of its event horizon.
S   =     k 2p c 3    ( ¼ A )
Vinculum
h G

What Stephen Hawking found is that quantum effects allow black holes to radiate (and force them to do so).  One of several explanatory pictures is based on the steady creation and anihilation of particle/antiparticle pairs in the vacuum close to a black hole...  Occasionally, a newly-born particle falls into the black hole before recombining with its sister, which will fly away  as if  it had been directly emitted by the black hole.  The expenditure of work to separate both particles is credited to the black hole and it exceeds the mass-energy of the "absorbed" particle by an amount equal to the mass-energy of the "emitted" one.

For a massive enough black hole, Stephen Hawking found the corresponding radiation spectrum to be that of a perfect  blackbody  having a temperature proportional to the  surface gravity  g  of the black hole  (which is constant over the entire horizon of a  stationary  black hole).  In 1976, Bill Unruh generalized this proportionality between any gravitional field  g  (or acceleration) and the temperature  T  of an associated  heat bath.

Temperature  T  is proportional to  (surface)  gravity  g.
kT= g h / (4p2c)         [ Any coherent units ]
T=g / 2p         [ In  natural units ]

In the case of the simplest black hole  (established by Karl Schwarzschild as early as 1917)  g  is  c2/2R,  where  R  is the "Schwarzschild radius" equal to  2MG/c2  for a black hole of mass  M.  In SI units,  kT  is about 1.694 / M.

Temperature of a Schwarzschild black hole :
kT= c 3 (h/2p) / (4pG M)         [ Any coherent units ]
T=1 / M         [ In  natural units ]

Rationalized Natural System of Units
c=1         [ Einstein's constant ]
h=2p         [ Planck's constant ]
G=1 / 4p         [ gravitational constant ]
mo=1         [ magnetic constant ]
k=1         [ Boltzmann's constant ]


(2005-06-13)     Statistical Approach :  The Partition Function (Z).
Z is a sum over all possible quantum states:   Z(b) = å exp (-b E)
E is the energy of each state.  b = 1/kT  is related to temperature.

A simplified didactic model: N independent spins

Consider a large number (N) of  paramagnetic ions  whose locations in a crystal are sufficiently distant so interactions between them can be neglected.  The whole thing is submitted to a uniform magnetic induction B.  Using the simplifying assumption that each ion behaves like an electron, its magnetic moment can only be measured to be aligned with the external field or directly opposite to it...  Thus, each eigenvalue  E  of the [magnetic] energy of the ions can be expressed in terms of  N  two-valued quantum numbers  si = ±1

 
E   =   m B  
 
N
å
i =1
 
  si
 

m is a constant (it would be equal to Bohr's magneton for electrons).  As the partition function introduced above involves exponentials of such sums, which are products of elementary factors, the entire sum boils down to:

Z(b)   =   å exp (-b E)   =   [ exp (b m B) + exp (-b m B) ] N

 Come back later, we're
 still working on this one...

Partition function

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