# Bernouilli (ideal gas) method on Photongas? Components of light?

1. Aug 25, 2010

### nonequilibrium

Hello,

A friend of mine (we're both first-years) has just done his herexam of Thermodynamics & Intr. to Statistical Mechanics. He had to get a relationship between pressure P and energy density E/V for a photongas. As a simplification $$E = \sum |p| c$$ was given. He couldn't do it with the ensemble method, so he decided to try and use the Bernouilli method. Apparently he got it right at the exam, although I don't see the mathematical validity/physical sense of his method:

So basically we use Bernouilli's method for an ideal gas. (I didn't think this was possible at first, but apparently the professor said he was right) Imagine a box with N photons bouncing around without interacting. We focus on one photon and we look at its force (and thus its pressure) on the right wall (of course for one photon this force is statistical and not physical):
$$F_1 = \frac{\Delta p_x}{\Delta t} = \frac{2 |p_x|}{2 L/v_x} = \frac{|p_x| v_x}{L}$$ with L the length of the box and v_x the component for the direction in question (so we're averaging the momentum change for one bounce on the right wall over the time it takes to get back to where it was, at least with respect to the x-component).
Now to get the total and thus physical force, we have $$F = \sum F_i = \sum \frac{|p_x| v_x}{L}$$ (I'm taking the liberty of not specifying the i-dependence as it's pretty obvious).

Now I would personally be stuck at this moment with the product of p_x and v_x, yet apparently he assumed v_x = c... He does admit he can't really justify it (at the moment it seemed obvious), but apparently it is allowed? Or did the professor skip the derivation and just look at the result, which did happen to be right (which implies there might be a reason for v_x = c).

Also, as a side-Q he got "isn't the energy normally dependent on v²?" and as an answer he gave it might have to do with the constancy of light to which my professor said "indeed". How does this all tie together?

PS: Of course once you have F, you can solve the initial question by P = F/A and V = AL