A simple model for piezo electric effect

[johnmadsen88]

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 $$l$$. The cylinder itself is electrically isolating, while the bottom and the piston surface are connected to a battery which maintains the voltage $$\Phi$$. Thus the arrangement is a capacitor, and we have both mechanical and electrical work on the system:

$$\delta W=fdl+\Phi dQ$$​

With $$\delta$$ denoting an inexact differential, and $$f=P_{ex}A$$ (with $$P_{ex}$$ 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. $$k_B=\frac{R}{N_A}$$ is the Boltzman constant.

I am asked to use the fundamental equation of thermodynamics to establish the relevant thermodynamic potential $$\mathcal{A}$$ for the system with control variables $$(T,l,\Phi )$$ , 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

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

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

 

[NateD]

The fundamental equation of thermodynamics states that the change in the internal energy of a system is equal to the difference between the total heat supplied to the system and the total work done by the system. This can be written as: dU=\delta Q-\delta W Using this equation, we can derive an expression for the relevant thermodynamic potential as: \mathcal{A}(T,l,\Phi )=U(T,V)-\int f dl-\int \Phi dQ The equation of state for this system is then given by: P_{ex}=\frac{\partial \mathcal{A}}{\partial V}=\frac{\partial U}{\partial V}-f-\Phi \frac{\partial Q}{\partial V} The specific form of this equation will depend on the form of the internal energy of the system.