- #1

fordhamdining

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## Homework Statement

I'm given that the energy of a particle in a rectangular box is the following:

[tex]E =\frac{\hbar \pi^2}{2m}(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2})[/tex]

I'm to show that if the length of the box is increased adiabatically and quasistatically from L_x to 8L_x, the work done by the particle is 3/4 of the initial mean energy.

## Homework Equations

The mean pressure exerted on the walls is

[tex]p=\frac{2E}{3V}[/tex]

where V is the volume of the box. The force on the wall parallel to the x-axis is given by

**[tex]**\frac{-\partial E}{\partial L_x} = \hbar \pi^2n_x^2/mL_x^3[/tex][/B]

## The Attempt at a Solution

I tried calculating the work by integrating pdV from 1 to 8 (since increasing one dimension by a factor of 8 increases the volume by the same) but I did not get the 3/4 I needed. There was a factor of ln(8) that could not possibly simplify to 3/4. Then I figured since the expansion is only happening in the x-direction, I can integrate F_xdL_x from 1 to 8 but I still didn't get the right answer. Am I missing something?