Thermodynamics: Work Under Isothermal Condition

[cox24]

Homework Statement

[/B]
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).

Homework Equations

$A = -k_{B}TlogZ$

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

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

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

The Attempt at a Solution

[/B]
(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.

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?