Photon Gas

Max Planck

Aaah! Natural Light! Get of off me, get it off me!

Barney Gumble

 

The ideal classical particle gas is sometimes too simple. The independence of the internal energy on the volume often leads to confusion when discussing more realistic processes. In particular the idea that the specific heat at constant volume can depend on the volume for almost everything but an ideal classical particle gas is a source of endless head-scratching for the average student. Therefore it is useful to introduce a second system, slightly less simple, but for which nevertheless all properties are still amenable to easy exact analysis [1].

That system is a box filled with light particles, or photons. Photons have the peculiar property that they can be created and destroyed, unlike real particles at ordinary temperatures, so the number of photons in the box is not fixed. You may very well be familiar with it. The inner ear is a rather good approximation to a box with photons, and if your mother ever stuck a thermometer in it, what she did was measure the temperature of the infrared radiation near the tympanic membrane, which is in turn a good approximation to your body temperature. This so–called black body radiation also played an important role in the development of Planck’s quantum ideas.

Every object emits electromagnetic radiation of which the distribution is related to its temperature. Even the universe itself does so. The cosmic background radation has a temperature of 2.725 K, and is thought to be a remnant of the Big Bang. Adiabatic expansion – it must be adiabatic, there is no heat bath around the universe – causes cooling, and although the universe began incredibly hot, it is now actually rather cold, and still cooling down [2]. A recent picture of the fluctuations in the background radiation is given in the top figure on the left. These fluctuations are surprisingly small, less than ~0.0002 K from the average, so small that regular expansion cannot explain it, and the inflationary model had to be invented. The spectrum itself is shown in the bottom figure, and is indistinguishable from Planck’s prediction.

The internal energy of the photons in a box with volume \( V\) and temperature \(T\) is given by

\[ U=bVT^4\quad\mbox{with}\quad b = \frac{8\pi^5k_B^4}{15c^3h^3}=7.56577\times10^{-16}\,{\rm J\,K^{-4}m^{-3}} \]

At 300 K the energy of the photons in a volume of 1 m3  is about 6.13 μJ, which is completely negligible compared to the energy of  250 kJ/m3 of an ordinary gas at standard ambient temperature and pressure. In our daily lives we do not have to worry about it.

The specific heat at constant volume, \(C_V\), for this gas is

\[ C_V =\left(\frac{\partial U}{\partial T}\right)_V = 4bVT^3 \]

so that  \( \Delta U \ne C_V\Delta T\), and furthermore \(C_V\) depends on the volume: the bigger the volume, the more heat I have to add in order to raise the temperature by 1 degree while keeping the volume constant.

The number of photons in the box is not fixed, they can be created and destroyed: the wall gives photons and  it takes them away. There is, however, an average number of photons, given by

\[ <N> = \frac{16\zeta(3)\pi k_B^3}{(hc)^3}VT^3\]

where \(\zeta(z)\) is the Riemann zeta function and the value of \(\zeta(3)\approx1.202\) [3].

The “ideal photon gas law”, the equation of state for the photon gas, is given by:

\[ p = \frac{1}{3}bT^4 \]

The pressure of the photon gas does not depend on the volume. This makes the isotherms in a \(p, V\) diagram rather boring curves [1]. In an ordinary gas the pressure is caused by collisions of particles with the wall. Most collisions are elastic, and in an elastic collision  the part of the momentum perpendicular to the wall is reversed. For all processes in nature momentum must be conserved, and the wall needs to take care of that. The change in momentum is the force on the wall, according to Newton’s second law of motion.

Although photons have no rest mass, they do carry momentum: \(\vec p =\hbar\vec k\) where \(\vec k\) is a vector of length \(c/\lambda\) in the direction the photon moves. If a photon is absorbed its momentum vanishes and if a photon is created its momentum must come from the wall. Momentum conservation tells us once again that a force on the wall  is the result.

The photon gas has another interesting property the classical particle gas does not have: an internal pressure. Small changes in the internal energy expressed as a function of temperature and volume, \(U(T,V)\), can be written as

\[ dU = \left(\frac{\partial U}{\partial T}\right)_VdT + \left(\frac{\partial U}{\partial V}\right)_TdV\equiv C_VdT +\pi_T dV \]

In this equation \(\pi_T\) is called the internal pressure, equal to zero for the ideal classical particle gas. In such a gas particles do not feel each other, so they don’t care how far apart they are or if I move them closer together. For the photon gas, however,

\[ \pi_T = bT^4\]

Photons don’t feel each other either, but if I increase the volume while keeping the temperature constant I need to make more of them and if I decrease the volume I need to destroy them, which is the cause of this internal pressure.

Now I can finally come to the question that needs to be answered before I can work out the Carnot cycle for a photon gas: how does the temperature change if I change the volume while the system is isolated? In other words, what is an adiabat for the photon gas? To that end I combine the first Law of Thermodynamics with the above expression for \(dU\):

\[ dU = dq + dw = -pdV = C_VdT + \pi_T dV\]

to relate small changes in the volume to small changes in the temperature under adiabatic conditions (\(dq=0)\). Some straightforward algebra [4] gives

\[ -3\frac{dT}{T} = \frac{dV}{V} \]

which shows that along an adiabat \(VT^3\) must be constant. In other words: if the volume decreases by a factor of 2, the temperature increases by a factor of 1.3. When the radius of the universe has doubled once more, the background temperature will be  only 1.36 K.


[1] I use this system elsewhere to discuss the Carnot cycle, to get insight into some of the errors Carnot made.

[2] The so–called heat death as as the final fate of the universe is therefore somewhat of a misnomer. In the end it is going to be very cold and all temperature differences will vanish which makes it very hard to do useful things. This may take a while though: see F. Dyson, Time without end – Physics and biology in an open universe, Rev. Mod. Phys., 51, (1979), 447-460

[3] A very famous function with the conjectured property that all its non–trivial zeroes have real part \(\frac{1}{2}\), which can gross you a million dollars and eternal fame among mathematicians if you prove it.

[4] Most students hated me for making statements like this, but now I am fortunate enough not to have to care about that anymore.

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