Maxwell Relations (not equations)

[eddo]

I just recently learned the Maxwell Relations in Thermodynamics. We aren't really doing anything with them, just went through the derivations.
In deriving them, we started with the equation of state:

TdS=dU+PdV

where T is temperature, S entropy, U internal energy, P pressure, V volume. We would than pick two of the four variables S,P,V,T, and with some math derive one of the maxwell relations. This was done using differentials, ie, if
df(x,y)=Adx+Bdy, than A=df/dx (partial derivative) and B=df/dy (p.d.), also, as long as the function isn't pathological, dA/dy=dB/dx (p.d.).

Here's my question, each of the four maxwell relations is found by chosing 2 of the four variables, and there are four maxwell relations. But there are 6 possible ways to chose 2 of the 4 variables, so how come there aren't 6 Maxwell Relations? Is there simply no way to manipulate the resulting equations to make it work for the other 2 pairs of variables, or is there some other reason for this? Thank you.

 

[kanato]

You have to start with some differential relationship. Usually used are the thermodynamic differentials:
$$dU = T dS - P dV$$
$$dA = -S dT - P dV$$
$$dH = T dS + V dP$$
$$dG = -S dT - P dV$$

For the variables T, S, P, V, there are 6 possible choices for pairs: TS, TP, TV, SP, SV, PV. Notice the two that you can't get are TS and PV. It's because these pairs are thermodynamic conjugate pairs, and always appear together in these differentials.

If you consider the dimensional analysis of the above equations, the LHS is always an energy, and the right hand side is always products involving temperature and entropy, which is energy, and producst of pressure and volume, which is energy. In order to get T and S in different spots, ie. T dV + S dP, you have to have products on the RHS which are not energy, and don't make sense to be added together.

 

[dextercioby]

H is enthalpy.Okay.A must be Helmholtz potential.U've considered closed system (no change of # of particles)...
To conclude,the # of Maxwell relations is very large and it's even greater,if you consider the Massieu functions $\Phi$ and $\Psi$.

Daniel.