# Thermodynamics: Work Under Isothermal Condition

1. Oct 23, 2014

### cox24

1. The problem statement, all variables and given/known data

One mole of 2-D ideal gas is confined in an isotropic cone potential:

$U = \lambda |r|$

where $\lambda$ is a positive parameter and $r$ is the displacement vector (2-dimensional) from the origin. The mass of each molecule is $m$.

(1) Determine the Helmholtz free energy $A$ of this gas confined in the potential at temperature $T$.

(2) If you want to vary $\lambda → \lambda + \delta \lambda$, you must do some work $\delta W$. Compute $\delta W$, following Einstein.

(3) Show, as thermodynamics tells us, that $\delta W$ in (2) agrees with the variation of $\delta A$ of the Helmholtz free energy $A$ due to the same parameter change in (2).

2. Relevant equations

$A = -k_{B}TlogZ$

$Z = \sum e^{\frac{-H}{k_{B}T}}$

H = Hamiltonian
$k_{B}$=Boltzmann constant

Einstein: $\delta W = < \delta H >$

3. The attempt at a solution

(1) For an ideal gas, the system hamiltonian (with the potential term added) is:

$H = \sum \frac{p_{i}^2}{2m} + \lambda |r|$

so,

$Z = \sum e^{\frac{-(p_{x}^2 + p_{y}^{2})}{2mk_{B}T} - \frac{\lambda |r|}{k_{B}T}}$

How is this sum computed for the canonical partition function of a 2-D ideal gas with the hamiltonian included?

Also, for (2),

$\delta W = < \delta \lambda H > = \delta \lambda < |r| >$

this answer is wrong, but I really don't see what I'm missing here.

Any help would be appreciated, thanks.

2. Oct 28, 2014