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**[SOLVED] Stat. Mech: energy-temperature relation of a perfect classical gas**

Note: This is really a problem I gave myself in an attempt to make myself understand thermodynamics better. If the problem itself is flawed (which is a possibility,) then please explain to me why and how so.

## Homework Statement

Consider the system consisting of a single particle of infinitesimal volume and finite mass m confined to a cube. (Without using the equipartition theorem,) verify that the relationship between the system temperature T = (dE/dS) and the speed |v| of the particle can be written in the form

(1/2)m|v|^2 = (3/2)kT

as predicted by the equipartition theorem.

## Homework Equations

T = (dE/dS)

E = 1/2 m|v|^2

S = k ln W

## The Attempt at a Solution

The strategy is to express both energy E and entropy S as functions of the particle's speed |v|, and then differentiate and divide:

T = dE/dS = (dE/d|v|)/(dS/d|v|)

Obviously, since E = 1/2 m|v|^2, dE/d|v| = m|v|.

Also, S = k ln W, dS/d|v| = k (1/W) (dW/d|v|).

The tricky part is how to approximate W as a function of |v|, and I think that this is the part I don't really understand. W is the number of states the particle can have. Since particle motion is quantized at some level:

W = (# possible positions) * (# possible velocities)

The only information that we have on the particle is that it is confined to a cubic box (let its volume be V) and that its speed is |v|. The number of possible positions this particle could occupy is proportional to the volume, and the number of momentum-states with speed |v| varies as |v|^2, corrresponding to the numer of lattice points that exist within the spherical shell of radius |v|+d|v|; hence,

W = MV * N|v|^2 = A|v|^2, and

dS/d|v| = k (1/W) (dW/d|v|) = k * (1/A|v|^2) * (2A|v|) = k * 2/|v|.

Dividing, we have

T = dE/dS = (dE/d|v|)/(dS/d|v|) = m|v|/(2k/|v|) = m|v|^2/(2k); rearranging gives

(1/2)m|v|^2 = kT,

which is not what we wanted.

I am aware that if my approximation for W were to have the form W = Av^3 instead of W = Av^2, then everything would work, but I don't understand how to rationalize that. What is wrong with the way I am approximating the number of possible states W?

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