# Construct a partition function for the system

• elduderino
In summary, the conversation discusses a system of N noninteracting particles in a container with a rigid bottom and a frictionless piston on top. The partition function Q is constructed for the system with N particles and one piston, and the fluctuations in the system's volume are calculated. The system is assumed to be in equilibrium and the phase space is 6N+1 dimensional. The integration for Z is done, but the answer does not match the Gibb's potential for an ideal gas.
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

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

Last edited:
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.

## 1. What is a partition function?

A partition function is a mathematical expression used in statistical mechanics to calculate the thermodynamic properties of a system. It takes into account the energy levels and degeneracies of a system's particles to determine the probability of a given state occurring.

## 2. How do you construct a partition function for a system?

To construct a partition function, one must first determine the energy levels and degeneracies of the system's particles. Then, the partition function is calculated by summing over all possible states of the system, with each state weighted by the Boltzmann factor e^(-E/kT), where E is the energy of the state, k is the Boltzmann constant, and T is the temperature of the system.

## 3. What is the significance of the partition function in statistical mechanics?

The partition function is essential in statistical mechanics as it allows us to calculate the thermodynamic properties of a system, such as the internal energy, entropy, and free energy. It serves as a bridge between the microscopic and macroscopic properties of a system.

## 4. How does the partition function change with temperature?

The partition function is dependent on temperature as it is used to calculate the probability of a given state occurring at a certain temperature. As temperature increases, the Boltzmann factor decreases, resulting in a decrease in the overall value of the partition function.

## 5. Can the partition function be used for all types of systems?

Yes, the partition function can be used for both classical and quantum mechanical systems. However, for quantum systems, the partition function must be modified to account for the quantum nature of the particles.

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