Thermo attributes of ideal gas in 3D harmonic potential

AI Thread Summary
The discussion focuses on solving a homework problem involving a classical system of N distinguishable, non-interacting particles in a 3D harmonic potential. Participants are tasked with finding the partition function and Helmholtz free energy, as well as entropy, internal energy, and total heat capacity at constant volume. The appropriate equation for the partition function is provided, emphasizing the integration over both position and momentum variables. Clarifications are made about using Helmholtz free energy instead of enthalpy and how to handle the momentum integration. The conversation highlights the need to break down the exponential terms into manageable components for further calculations.
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Homework Statement


A classical system of N distinguishable, non-interacting particles of mass m is placed in a 3D harmonic potential,

U(r) = c \frac{x^2 + y^2 + z^2}{2 V^{2/3}}

where V is a volume and c is a constant with units of energy.

(a) Find the partition function and the Helmholtz free energy of the system.

(b) Find the entropy, the internal energy and the total heat capacity at a constant volume for the system.

Homework Equations


Z = \Sigma \exp{-E_i / kT}

H = U - TS

f(E)= A \exp{-E/kT}


The Attempt at a Solution



Unfortunately, I'm not sure where to start on this one. Anybody able to give me a tip in the right direction?
 
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Your system is a classical one, so the appropriate equation for the partition function is

Z = \int d\mathbf{r}_1 d\mathbf{r}_2 \dots d\mathbf{r}_N d\mathbf{p}_1 d\mathbf{p}_2 \dots d\mathbf{p}_N \exp\left[-\frac{E(\mathbf{r},\mathbf{p})}{k_BT}\right]

The energy is

E(\mathbf{r},\mathbf{p}) = \sum_{i=1}^{N}\left[\frac{\mathbf{p}_i^2}{2m} + U(\mathbf{r}_i)\right].

This should get you started with that part.

For the free energy, instead of using F = U - TS, it's easy to use the equation

F = -k_BT \ln Z

(it's more common to use F than H to denote the Helmholtz free energy, as H is typically used for the enthalpy).
 
Mute said:
Your system is a classical one, so the appropriate equation for the partition function is

Z = \int d\mathbf{r}_1 d\mathbf{r}_2 \dots d\mathbf{r}_N d\mathbf{p}_1 d\mathbf{p}_2 \dots d\mathbf{p}_N \exp\left[-\frac{E(\mathbf{r},\mathbf{p})}{k_BT}\right]

The energy is

E(\mathbf{r},\mathbf{p}) = \sum_{i=1}^{N}\left[\frac{\mathbf{p}_i^2}{2m} + U(\mathbf{r}_i)\right].

(it's more common to use F than H to denote the Helmholtz free energy, as H is typically used for the enthalpy).

Still stuck. How would the integration over the momentums work? That is, what would the \mathbf{p_i}^2 be?
 
\mathbf{p}_i^2 = p_{i,x}^2 + p_{i,y}^2 + p_{i,z}^2

Hence, you end up with a sum of squares of terms in the exponential that can all be split into products of exponentials.
 
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