A simple model for piezo electric effect

  • #1
I'm told I have a cylinder with a simple piston and a trapped ideal gas. The area of the piston is A, and the position of the piston with respect to the bottom of the cylinder is denoted by [tex]l[/tex]. The cylinder itself is electrically isolating, while the bottom and the piston surface are connected to a battery which maintains the voltage [tex]\Phi[/tex]. Thus the arrangement is a capacitor, and we have both mechanical and electrical work on the system:

[tex]\delta W=fdl+\Phi dQ[/tex]​

With [tex]\delta[/tex] denoting an inexact differential, and [tex]f=P_{ex}A[/tex] (with [tex]P_{ex}[/tex] being the external pressure). Q is the charge of the piston surface. This problem will only concern reversible processes. The number of molecules, N, in the gas is constant. [tex]k_B=\frac{R}{N_A}[/tex] is the Boltzman constant.

I am asked to use the fundamental equation of thermodynamics to establish the relevant thermodynamic potential [tex]\mathcal{A}[/tex] for the system with control variables [tex](T,l,\Phi )[/tex] , and give the equations of state.

I am honestly rather confused about this problem. Further down the page, there is a specific potential given, which leads me to think that I'm looking for something on the form

[tex]\mathcal{A}(T,l,\Phi )=-\frac{\epsilon A}{2l}\Phi ^2-\mathcal{A}_0(T,V)[/tex]​

but I doubt that is specific enough. Any sort of guidance would be appreciated.
 

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