Fundamental relations and equations of state

A fundamental relation contains all the information about a system--all the thermodynamic information about a thermodynamic system--and involves independent extensive variables. For instance, entropy S expressed as a function of internal energy U, system volume V, and conserved particle number N (same type of particles)  is a fundamental relation, expressed symbolically as S(U,V,N).  Or, equivalently, the internal energy U can be expressed as a function of entropy, volume, and particle number, U(S,V,N), and this is also a fundamental relation.

In contrast, the more commonly used equations of state are not fundamental relations, and involve functions of state that are not independent of each other, such as temperature T,  pressure P, and chemical potential μ, that are partial derivatives of fundamental relations. A complete set of equations of state, however, is informationally equivalent to a fundamental relation. The three equations of state 

T(S,V,N) = ∂U/∂S,

 -P(S,V,N) = ∂U/∂V,

and μ(S,V,N) = ∂U/∂N,

for example, contain all the information of the energetic fundamental relation, U(S,V,N). For an interesting case where N isn't involved (because it's not conserved) see my July 22 post on the Stefan-Boltzmann law as one of the equations of state for black-body radiation.

(This discussion should make you think of similar descriptions used in quantum theory, such as "the wave function contains all the information about the system," the need for "a complete set of commuting observables," and the choice of  energy or entropy fundamental relations being somewhat like the choice of  Schrödinger or Heisenberg 'pictures'.)

The connections between fundamental relations and equations of state are discussed in Callen's book in the first three sections of Chapter 3, "Some Formal Relationships and Sample Systems." In case you don't have access to that, I'm posting (below) some meticulous notes on this subject put together for a Statistical Physics class by Professor Mark Loewe in the mid-1990s, when he was teaching at Southwest Texas State University in San Marcos (now Texas State University-San Marcos). He used Callen's book, but he also handed out notes such as these that he wrote himself.  He is concise and thorough, as you can see. (Well, yeh, sorry the notes are hard to see. Slide 'em over to your desktop, maybe.) There's more info in the notes than you would likely ever want or need, but the accompanying descriptions are worth reading. One little thing Mark wasn't thorough about: in the corrections and clarifications, he doesn't say which problem Callen gives an incorrect answer to. Maybe I have it in my class notes, and will post it if I find it. 

Just how fundamental relations and equations of state are related to the first law of thermodynamics in its most basic form,

,

is something worth thinking about and looking up. Does the first law contain all the thermodynamic (macroscopic) information about a system? Oh yeah, and we have to talk about isolated systems versus systems attached to a heat reservoir. The 1st law equation above is for the change in internal energy for Q (energy as heat) supplied to the system, and W the work done by the system on its surroundings. It's a good starting point for imagining either an isolated system that has energy added to it (temporarily connected to, then disconnected from, some kind of reservoir) or a system connected to a heat reservoir and not yet in thermal equilibrium with the reservoir.

Just to keep things in perspective before going on to specific examples in later posts, I'll quote Callen from the top of page 26, "The single, all-encompassing problem of thermodynamics is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system." 

Mark Loewe's notes:

Herb Callen problems 3.6-1, 3.6-2, and 3.6-3

(Revised 5 Aug 2021.  See my previous post for notes and comments on Callen's Section 3.6.)

3.6-1   The universe is considered by cosmologists to be an expanding electromagnetic cavity containing radiation that now is at a temperature of 2.7 K. What will be the temperature of the radiation when the volume of the universe is twice its current value?  Assume the expansion to be isentropic (this being a nonobvious prediction of cosmological model calculations).

So the entropy will be the same when the volume V₁ becomes V₂  and when  T₁ becomes T₂ :

S(U, V)   =  constant =  (4/3) (bV₂)^1/4 (U₂ )^3/4   =   (4/3) (bV₁ )^1/4 (U₁ )^3/4 , 

where  U₂ = bV₂ (T₂ ) ⁴   and  U₁ = b V₁ (T₁ ) ⁴ .  

Canceling the 4/3 and b factors on each side of the above entropy equation and putting in temperature T in place of the energy U gives (exponent on V's  becomes 1/4 + 3/4 = 1, and exponent of T's becomes 4 x 3/4 = 3)

V₂ (T₂ )³  =   V₁ (T₁ )³  

(T₂ /T₁ ) ³  =  V₁/V₂  =

(T₂ /T₁ ) ³  =  1/2    

T₂  =  (1/2)^1/3 T₁    =     0.79 (2.7 K)  =   2.1 K .

Question for further research: in how many years will this doubling in size occur? The accepted answer to this question is not the same now as it was in 1985 when Callen’s book was published, due to the discovery in 1998 of the acceleration of the expansion (dark energy). 

 

3.6-2 Assuming the electromagnetic radiation filling the universe to be in equilibrium at  T = 2.7 K,  what is the pressure associated with this radiation?  Express the answer in both pascals and atmospheres.

From Stefan-Boltz law, we have U/V = bT⁴ , and the pressure equation for isotropic radiation is P_R = U/3V.  So P_R = (b/3)T⁴

= (7.56x10^-16 Joule · meter^-3 · Kelvin^-4))(2.7 K)⁴/3 

= 134 x 10^-16  Newton/meter²

P_R =  1.34 x 10^-14 Pa

P_R =  1.32 x 10^-19 atm.

 

3.6-3  The density of matter (primarily hydrogen atoms) in intergalactic space is such that its contribution to the pressure is of the order of 10^-23 Pa. 

(a) What is the approximate density of matter (in atoms/meter³) in intergalactic space?

(b) What is the ratio of the kinetic energy of matter to the energy of radiation in intergalactic space? (Recall Problems 3.6-1 and 3.6-2.)  

(c)  What is the ratio of the total matter energy (i.e., the sum of the kinetic energy and the relativistic energy mc²) to the energy of radiation in intergalactic space?

 

(a)  We want to calculate N/V from P_M, and, assuming (maybe incorrectly) the hydrogen gas in intergalactic space is in equilibrium with the CMB radiation, we use the ideal gas law, P_MV = NkT, where T = 2.7 K.  Rearranging gives

 N/V  =  P_M/kT  =  10^-23 Pa/[(1.38 x 10^-23 J/K) · 2.7K]

=  (1/3.7) atoms/meter³ 

=  0.27 atoms/meter³ 

^{The answer is actually about one per cubic centimeter, or one million per cubic meter, alas. That means the temp of the very diffuse hydrogen gas is about 4 million degrees Kelvin.  Here's the first paragraph from a good discussion of the subject:}

There are some missing details that are often skipped over in popular science articles/documentaries, which can make this confusing. First: the temperature of space, 2.7 K or -270 C, refers to the remaining radiation from the Big Bang, not to the temperature of any matter. If you were in space and somehow prevented any matter from touching you, you would still absorb microwave radiation equivalent to being surrounded by matter with a temperature of 2.7 K (since all matter with temperatures above 0K emits radiation). Of course, with all the stars and such, you also receive a lot of other radiation, so you wouldn't necessarily cool down to 2.7 K. That temperature refers specifically, and only, to the background microwave radiation left over from the early history of the universe.

^--------

   (b)  For our intergalactic atomic hydrogen gas, the only energy is kinetic energy (well, ignoring the spin flipping the electron can undergo relative to the spin of the proton, resulting in the famous 21 cm spectral line).  (Also ignoring, until part (c), the relativistic mass-energy.)  So we have (KE)_H = (3/2) NkT = (3/2)P_MV for the kinetic energy of matter, which we want to divide by the U_rad =  bVT⁴  = 3P_RV of radiation:

 

(KE)_H / U_rad  =  (3/2) P_MV / 3P_RV

=  P_M / 2P_R

=  10^-23 Pa / 2(1.34 x 10^-14 Pa)

=   3.7 x 10^-10 

 

(c) The relativistic energy is U_rel = Mc², where M is the mass of all the hydrogen atoms in intergalactic space. We could do this calculation by estimating the size of the universe which we’d multiply by the density of hydrogen atoms. But volume cancels out in a ratio of energy density calculations, and energy density is the determining factor for the total energy of radiation, of hydrogen atoms’ KE, and of relativistic energy. And we've calculated number density of H atoms, so we should use it. We divide by total volume, V, of the universe to get density:

U_rel/V = (Mc² /V) =  m(N/V)c²

= (1.67 x 10^-27 kg for one H atom) (0.27 H atoms per meter³)(3 x 10⁸ m/s)²

=  9 x 10¹⁶ · 0.45 x 10^-27 joule/meter³

=  4 x 10^-11 joule/meter³

Now I need to divide this by U_rad/V for radiation, and add the result to the answer to (b). I didn’t do the U_rad/V calculation separately for radiation energy density in part (b)—but we know U/3V = P for the radiation pressure, so we take 3P as our radiation energy density

U_rad/V = 3P = 3 x 1.34 x 10^-14 Pa = 4 x 10^-14 joule/meter³

so

U_rel/U_rad  =  1000.

The KE/U_rad  ratio of 3.7 x 10^-10 is negligible in comparison with the factor-of-1000 ratio of relativistic-to-radiation energy. No point in adding them! But we’d expect the relativistic energy to be much greater than the total KE of atoms in a gas at temperature 2.7 K.

Questions for further research:

At what temperature does the KE of the H atoms equal their mass-energy?

At what temp does the radiation energy equal the H atoms mass-energy?

Does the mass-energy of all the loose H atoms in the universe have any significance beyond the fact that it has to be included when we account for all the energy released (created) in the Big Bang?