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?

The Stefan-Boltz law as a thermo equation of state

In this discussion, the Stefan-Boltzmann law will be written in terms of the energy contained in a "trapped" electromagnetic field, rather than being written in terms of the power of an "escaping" electromagnetic field (as I discussed for sunlight in a recent post).  An energy-versus-temperature equation is one example of a thermodynamic equation of state. The well-known energy-versus-temp relation for a monatomic ideal gas, for instance, is U=3NkT/2. I'll look at many more examples of equations of state, and at how they are related to the "fundamental relations" of thermodynamics in a future post.

Our main man Herb Callen has a good discussion (Section 3-6 in his 2nd edition text) on the thermodynamics of radiation in an empty box: 

If the walls of any 'empty' vessel are maintained at a temperature T, it is found that the vessel is, in fact, a repository of electromagnetic energy. The quantum theorist might consider the vessel as containing photons, the engineer might view the vessel as a resonant cavity supporting electromagnetic modes, whereas the classical thermodynamicist might eschew any such mechanistic models.

The empty vessel could be almost anything. A metal breadbox, a wooden breadbox, a closed cardboard box, a spherical Christmas tree ornament, or Santa Claus ornament, as long as it's hollow. Yes, it's generally going to have air in it, which is a point that doesn't get discussed much. That's not to say it will be inflated with air like a tennis ball, football or basketball.  But it could be, and there's an electromagnetic field in there with that compressed air. We can just let these "vessels" be at room temperature. Otherwise we have the problem of keeping the walls uniformly at the same high or low temperature. 

Even if a box has walls that are at different temperatures, there will be some sort of radiation spectrum in the box.  Let's say one wall is heated so it's emitting most of the radiation in the cavity, presumably with a different emission spectrum near its surface than the spectra at the surfaces of the other walls.  But is this even possible? Can the spectrum of the radiation in the box be different on different sides of the box?

Well, what we would have is the usual linear superposition (addition) of the amplitudes of the all the waves/photons in the box.,* resulting in constructive and destructive interference, which results in an equilibrium energy-versus-frequency spectrum.  If we have a rectangular or square box, we can ask what the role of the radiation is in changing the temperature distributions in the five walls of our box that are not directly heated. This changing of the temps is a temporary, or transient, behavior. An equilibrium temperature distribution will come into being after the transient behavior is over, due to heat conduction in the walls and radiation from the heated wall of the box to the other walls of the box.

But the  radiation in the box won't have a black-body spectrum unless the walls of the empty vessel are maintained, as Callen says, at the same temperature, which gives equality of absorption and emission rates for every part of the walls. At least this is the desired black-body cavity experimental set-up. The other thing we need for the usual, real, set-up is a little peek-a-boo hole in the box to observe the spectrum of the radiation. 

But we aren't going to try to observe it here. We just want to ask, for a given temperature how much electromagnetic energy is in the vessel? The answer is given by the Stefan-Boltzmann law in yet another form, different from the three ways I wrote it in my February post. Instead of flux (power per unit area) or intensity, the total energy of the radiation is given as proportional to Kelvin-temperature-to-the-fourth-power. Callen writes this as

U = bVT⁴ 

where V is the volume of the empty vessel and b = 4σ/c = 7.56x10^-16 J/m³K⁴, where σ is the Stefan-Boltzmann constant and c is the speed of light. (See my 12 November 2020 post, where the integral of the Planck spectral energy density function at temperature T is shown symbolically and the factor of  c/4 is used.  Here, we're getting rid of that factor. If you divide both sides of the above equation by V, the result is energy density.) Sooooooo, the above equation really comes from integrating the Planck spectral energy density formula over all frequencies while holding temperature constant.  

What, then, is the difference between the views of the quantum theorist, the engineer, and the thermodynamicist in Callen's description above? The photons of the quantum theorist are after all normal modes of the electromagnetic field (see page 97 of these notes on the EM field), and a normal mode is a standing wave that has one of the resonant frequencies of the box. The best brief reference on this subject is Rodney Loudon's two-page discussion, "Introduction: The Photon," at the beginning of his book The Quantum Theory of Light, 3rd ed., and the five references therein.

The quantum theorist is not so much interested anymore in cavity radiation by itself as he or she is in cavity quantum electrodynamics, when there are a controlled number of atoms in a very small cavity (and no air, unless the "atoms" in the cavity are a few oxygen and nitrogen molecules) . An electrical engineer who wants to use the "vessel" as a resonant chamber for electromagnetic radiation is just like an acoustical engineer who wants to use an acoustic chamber to enhance certain frequencies and suppress others. (Here we can think about the usual "box of gas" as having acoustical modes of vibration.) Microwave ovens possess electromagnetic resonant cavities to enhance the microwave cooking frequencies, while musical instruments, for example, enhance the desired harmonics of sound waves.

But I'm getting off the subject here. The subject, which I haven't fully explained yet, is the equations of state of blackbody radiation, and how they can be used to find the entropy of blackbody radiation in terms of U and V. This is a "fundamental relation" in thermodynamics, and would be something the thermodynamicist is interested in, particularly the fact that N--whether he or she thinks of it as the number of photons in the box or number of normal modes--isn't an independent variable along with U and V. (See the quote from Callen below, and note he uses an exclamation mark.)

Besides the above equation of state for the total energy of trapped blackbody radiation in terms of temperature, there's a well-known relation between radiation pressure P, total energy U, and volume V,

P = U/3V

which you can copy and paste into an internet search box to find out more about. "It will be noted," says Callen, "that these empirical equations of state are functions of U and V, but not of N. This observation calls our attention to the fact that in the 'empty' cavity there exist no conserved particles to be counted by a parameter N. The electromagnetic radiation within the cavity is governed by a fundamental equation of the form S = S(U, V) in which there are only two rather than three independent extensive parameters!"

The relevant equation for finding the entropic fundamental relation is 

ST  =  U  +  PV ,

or 

S  =  U / T  +  PV / T.

From the two equations of state, we have 

1/T   = b^1/4 (V/U)^1/4

and 

P/T   = (1/3) b^1/4 (V/U)^1/4

so that the entropy fundamental equation is 

S(U, V)   = (4/3) (bV)^1/4 U^3/4

One of the things I'm interested in is how this expression for the entropy of black-body radiation in terms of U and V compares with the logarithmic expression found by Planck, but I'll come back to that later. First, I'll solve the three problems in this section of Callen's book (Section 3.6), then I'll post some detailed typed handouts from a Statistical Physics class I took in 1995 at Texas State University that used Callen's book as the text (thank you,  Mark Loewe!).

----------------

*Professor Hazeltine said in a junior-level electrodynamics class I took at the UofTexas at Austin 30 years ago that "photons don't interact." I raised my hand and said, "What about interference and diffraction?" Hazeltine explained that "interacting" has to do with things that exert forces on each other, and photons don't exert forces on each other. I asked, "Could we still say they interact linearly then?" I don't recall his response, but he stuck with the "photons don't interact" paradigm. It's not a good paradigm, because photons, being bosons, are gregarious and will indeed interact in a certain way (a sort of attraction) that fermions don't (they sort of repulse each other). We don't call mutual boson interactions and mutual fermion interactions by the name "forces," but maybe we should, or maybe we should modify our idea of forces.

Trinity bomb test described as ammo dump explosion

 (From p.1 of the El Paso Herald-Post, Monday afternoon, July 16, 1945)

Army Ammunition Explosion Rocks Southwest Area

Great Blast Near San Marcial Lights El Paso

            An ammunition magazine containing a considerable amount of high explosives and pyrotechnics exploded at 5:30 a.m. in the New Mexico desert near San Marcial on a remote section of the Alamogordo Air Base reservation.

            No one was hurt.

            The blast was seen and felt throughout an area extending from El Paso to Silver City to Gallup, Socorro and Albuquerque.

            Many persons saw a flash light up the sky, like daylight, and felt earth tremors.  They thought an earthquake had struck.

            William O. Eareckson, commanding officer of the Alamogordo Air Base, released the following statement:

            “Several inquiries have been received concerning a heavy explosion which occurred on the Alamogordo Air Base reservation this morning.

            “A remotely located ammunition magazine containing a considerable amount of high explosives and pyrotechnics exploded.

            “There was no loss of life or injury to anyone, and the property damage outside the explosives magazine itself was negligible.

            “Weather conditions affecting the content of gas shells exploded by the blast may make it desirable for the Army to evacuate temporarily a few civilians from their homes.”

            El Paso men going to work at 5:30 a.m. said the flash illuminated Mt. Franklin.  E.R. Carpenter, Louie Ratliff and Jack Coulehan, riding down Alabama avenue, said the whole sky was ablaze with light.  Mr. Carpenter, mechanical superintendent of the Newspaper Printing Corporation, said many persons called the newspaper’s composing room to report seeing the flash and hearing the explosion.  Callers asked if a meteor had fallen.

Big Light in Sky

            L.R. Lessell, Gila forest headquarters supervisor, said rangers reported the shock was felt throughout the Mogollon mountains.

            Rangers at Chloride reported the blast lighted the sky brightly in the area of San Marcial.  The blast was followed by a terrific explosion, like a detonation, rangers said.

            Forest rangers, believing that an earthquake had struck, checked with  Alfred E. Moore at the Smithsonian Observatory on Burro mountain.  The observatory is near Tyrone and 800 feet high.

            Mr. Moore confirmed the flash, saying it was distinctly visible at the observatory, but the shock was unlike earthquake vibrations.  The observatory head has experienced earthquakes in South America and Mexico.

Silver City Shaken

Silver City residents reported three distinct blasts were felt there.  The shock cracked plate glass windows in downtown buildings.

            “The blasts sounded like heavy claps of thunder,” Don Lusk of Silver City, said.  “Houses shook.  People were roused out of sleep from the noise and tremors.” 

            Mrs. H.E. Wieselman of 901 North Ochoa street saw the explosion as she crossed the Arizona-New Mexico state line.  She was enroute to El Paso from California.

            “We had just left Safford, and it was still dark,” Mrs. Wieselman said.  “Suddenly the tops of high mountains by which we were passing were lighted up by a reddish, orange light.

            “The surrounding countryside was illuminated like daylight for about three seconds. 

            “Then it was dark again.

            “The experience scared me.  It was just like the sun had come up and gone down again.”

Front Seat at Sky Show

            Ed Lane, Santa Fe railroad engineer, was at Belen, N.M., when the blast occurred.

            He said he had a front seat to the greatest fireworks show he had ever seen.  The blast was in the direction of San Marcial and seemed to be only a few miles from Belen.
            “I was coming to El Paso,” Mr. Lane said.  “My engine was standing still.  All at once it seemed as if the sun had suddenly appeared in the sky out of darkness.  There was a tremendous white flash.  This was followed by a great red glare and high in the sky were three tremendous smoke rings.  The highest was many hundreds of feet high.  They swirled and twisted as if being agitated by a great force.  The glare lasted about three minutes and then everything was dark again with dawn breaking in the east.”

I copied this article sometime in the early 1990s by hand (pencil and paper) from the original July 16, 1945 El Paso Herald-Post in the bound newspaper archives of the Barker Center for Texas History at the University of Texas at Austin.  In May1994, I used it as an appendix to a paper I wrote ("Zero Patience for Zero Hour") for a History of the Atomic Bomb class taught by Bruce Hunt. Here are some other front-page Herald-Post headlines from July 16, 1945:

450 Super-Forts Heap 2500 Tons of Fire on Japan

Woman Happy as She Gets Back Lost $1480

Ice Cream Firm Owner Freezes to Death in Plant

Capitalizing on Prejudice, Nationalist Party Drives for Control of Government

B-29 Crew Throws Man to Safety from 9000 Feet