Construct a partition function for the system

[elduderino]

Homework Statement

Consider a system of N noninteracting particles in a container of cross-sectional area A. Bottom of the container is rigid. The top consists of an airtight, frictionless piston of mass M. Neglect the potential energy of the molecules of gas.

Construct the partition function Q of the (N+1) particle system (N particles of mass m+ piston)

Calculate the fluctuations in the volume of the system?

Homework Equations

$$Z= \frac{1}{N!h^{3N}}\int e^{-\beta H(p,q)}d^3pd^3q$$

The Attempt at a Solution

System is in equilibrium for theory to be applicable, hence piston is at rest at some height y.

$$H(p,q) = \sum_i \frac{p_i^2}{2m} +mgy$$

6N+1 dimensional phase space

$$Z= \frac{1}{N!h^{3N}}\int e^{-\beta (\sum_i \frac{p_i^2}{2m} +mgy )}d^3pd^3q dy$$

 

[elduderino]

Addition: I did the integration for Z but the answer is not coming right. How do I know the answer? It SHOULD come out to be the same as the Gibb's potential for an ideal gas. ie A=-kTlnZ.