By Joshua N. Winn, Princeton University
According to the ideal gas law, pressure is proportional to number density and temperature. The equation is the same for both one-dimensional gas and for three-dimensional. Knowing the properties of the ideal gas gives us an opportunity to compare them to the properties of a gas of photons, which are not intuitive.
Average Energy
The first thing to know is that photons, unlike particles, don’t collide with each other. They sail right through each other. The only things photons interact with are charged particles. So, in order to randomize the positions and energies of photons, you need to have charged particles around, which are themselves in thermodynamic equilibrium. So, let’s assume we are filling a box with charged particles, that are colliding all the time, producing momentary accelerations, and thereby producing and absorbing photons.
Let’s start our comparison with the average energy per particle. For an ideal gas, it’s 3/2 kT. For the photons, it turns out to be 2.7 kT. That’s not so weird. They’re both proportional to temperature; there is just a different numerical constant in front.
Number Density
Things get weird, though, with the number density. It goes without saying that for an ideal gas in a closed box, n is constant, the particles don’t just spontaneously pop out of nowhere, or vanish. Even if we add energy, speeding up the particles, their number stays the same.
But photons do pop just out of nowhere whenever a charged particle accelerates. The particle flings away some of its own energy in the form of photons. Likewise, a photon vanishes when its energy is absorbed by a charged particle. So, for photons, we shouldn’t expect n to be a constant. If we inject more energy into the gas, speeding up the particles, the magnitude of their accelerations will rise, and they’ll produce more photons.
It turns out that the number density of photons rises as temperature to the 3rd power. The number density n is the cube of 3.9 kT over hc.
Energy Density and Pressure
Next, let’s compare energy density. For the ideal gas, the energy density u equals 3/2 nkT, so it’s proportional to nT. For photons, n varies as T-cubed, and so if energy density varies as nT, we might expect the energy density of photons to vary as T to the 4th power. And it does. The constant of proportionality is traditionally written 4 sigma over c, where sigma is the Stefan-Boltzmann constant.
Then comes pressure. For the gas, pressure equals nkT, the ideal gas law. For photons, again, n itself goes like T-cubed, so we might expect pressure to go like T to the 4th power, and it does. In this case, the proportionality constant is 4-sigma over 3c.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
Stefan-Boltzmann Law
Finally, let’s consider the flux. The power per unit area that would emerge from a tiny hole in the box. For the gas, it’s n times the average of v-epsilon, which was proportional to T to the 3/2 power. For photons, the number density n scales with T-cubed, v is always c, and epsilon is proportional to kT, so we might guess that flux is proportional to T to the 4th power, and it is.
That’s a very important result: the flux of electromagnetic radiation from a body at temperature T is proportional to T to the 4th power. That’s important enough to deserve its own name: it’s the Stefan-Boltzmann law. The constant of proportionality is sigma; that’s the one that also appeared in the equations for energy density and pressure. Sigma isn’t a new fundamental constant; it’s a certain combination of h, c, and k, but it occurs so frequently that the abbreviation is helpful. Numerically, sigma is 5.7 times 10 to the minus 8 Watts per square meter per degree Kelvin to the 4th power.
Distribution of Energies
The last comparison I want to make between particles and photons is in their distribution of energies. For the case of the gas, the average energy is 3/2 kT. If we pick a particle at random, we expect its energy to be about 3/2 kT, but not exactly. Sometimes it’ll be a little higher, sometimes lower; it depends on its recent history of collisions.
Likewise, the speed of a given particle is always fluctuating. A fundamental rule that emerges from classical statistical physics is that the probability to find a particle in a state with energy of epsilon is proportional to e to minus epsilon over kT. It’s an exponential function, and it’s called the Boltzmann factor. It means that the energy will almost always be on the order of kT. Much larger energies are vanishingly rare, because of that exponential fall-off. The particles tend to share the energy equally. There’s very little chance that one particle is going to end up with a disproportionate share of the total energy.
Maxwell-Boltzmann Distribution
From that basic rule, it’s possible—although not easy—to derive the probability distribution for the energy, or the speed, of a particle in a gas. What makes it difficult is that there are a lot of different states with the same energy. If we change the direction of a particle’s velocity, but not the speed, the particle is in a different state, but it has the same energy. That means, to calculate the probability of having a certain speed, we need to multiply the Boltzmann factor by the number of possible states with that speed, and so there’s a lot of bookkeeping associated with counting all of those states.
But if you do go through all that, you can derive the Maxwell-Boltzmann distribution: that’s the probability distribution for particle speed, in an ideal gas. It’s the product of the Boltzmann factor, and a factor of v-squared which comes from all that state-counting. The horizontal axis is speed, in meters per second, and the vertical axis is the relative probability that you’ll find a particle to have that speed or, another way to think of it, it’s the fraction of particles that have that speed, at any given time. The function depends on the particle mass, and the temperature.
Common Questions about Particles and Photons
The only things photons interact with are charged particles
The flux of electromagnetic radiation from a body at temperature T is proportional to T to the 4th power. This is the Stefan-Boltzmann law.
The Maxwell-Boltzmann distribution is the probability distribution for particle speed, in an ideal gas.
