A rather specific thermodynamic potential

[johnmadsen88]

Today, during class, our professor went through a simple example about Piezoelectricity. We have a cylinder with a trapped ideal gas and a piston which is part of a capacitor together with the bottom of the cylinder. The voltage across the capacitor is [tex]\Phi[/tex], and the charge is [tex]Q[/tex]. The distance between the piston and the bottom of the cylinder is [tex]l[/tex]. A potential for such a system could be

[tex]\mathcal{A}(T,l,\Phi )=-\frac{\epsilon A}{2l}\Phi ^2-Nk_BT\ln{Al}+\frac{3}{2}Nk_BT-\frac{3}{2}Nk_BT\ln{T}[/tex]​

with [tex]k_B[/tex] and [tex]N[/tex] denoting the Boltzman constant and the number of molecules, respectively. [tex]\epsilon[/tex] is the dielectric constant. He then said that one can (I assume, by Legendre transformation) express this using another potential [tex]\mathcal{B}[/tex] which is relevant for variables [tex](T,Q,l)[/tex]:

[tex]\mathcal{B}(T,Q,l)=-\frac{1}{2\epsilon A}Q^2-Nk_BT\ln{Al}+\frac{3}{2}Nk_BT-\frac{3}{2}Nk_BT\ln{T}[/tex]​

I was wondering if someone could give this a more physical twist. In other words, I'd like it if someone explained where these contributions come from and what characterizes them. If I have been too unclear, please tell me, so I can reword my writing. Thanks.

PS: I cannot currently see my TeX code properly when previewing (either my computer or the site is having problems), but I hope I haven't made errors.

 

[LightHero]

The potential \mathcal{A} is related to the energy of the system, which includes the mechanical and electrical energies. The first term, -\frac{\epsilon A}{2l}\Phi ^2, is the electrical energy of the capacitor due to the voltage \Phi across it, where A is the area of the capacitor plates and \epsilon is the dielectric constant of the material between the plates. The second term, -Nk_BT\ln{Al}, is the thermal energy associated with the ideal gas, where T is the temperature of the gas and N is the number of molecules in the cylinder. The third term, \frac{3}{2}Nk_BT, is the kinetic energy of the molecules, and the fourth term, -\frac{3}{2}Nk_BT\ln{T}, is the entropy contribution due to the thermal fluctuations of the molecules. The potential \mathcal{B} is related to the free energy of the system, which is a combination of the energy and entropy contributions. The first term, -\frac{1}{2\epsilon A}Q^2, is the electrical energy of the capacitor due to the charge Q stored on it. The second and third terms are the same as in \mathcal{A}, while the fourth term is the entropy contribution associated with the thermal fluctuations of the molecules.