What is the Correct Hamiltonian for an Isolated Gas and Piston System?

[luisgml_2000]

Homework Statement

A system is contained within the walls of a box and a movable piston. An object weighing W is placed on the piston. If you regard the gas and the piston as an isolated system, use the microcanonical ensemble to deduce the equation of state for p (pressure). Take a look at the picture.

Homework Equations

$$\mathcal{H}=\frac{1}{2m}\sum p_i^2 + Wx$$

$$\Omega=V^{n}\int_{\mathcal{H} <E}dx d^{3N}p$$

$$S=klog\Omega$$

$$p/V=\frac{\partial S}{\partial V}$$

The Attempt at a Solution

The only thing I need to solve this problem is to write down the hamiltonian of this system. I'd like to know if the expression that I have written is correct; in that case, I don't know how to evaluate the integral for the number of microstates.

thanks!

 

[luisgml_2000]

I'm sure that the hamiltonian of this system is as written above. Do you have any ideas for solving the integral for $$\Omega[\tex]?$$