[xender]

Is "Summation over i of (Nu_i*Mu_i) = 0" an equilibrium critieria for a reaction happening under constant P and V, and if so, how do you prove it?

(Nu_i = Stoichiometric Coefficient of Chemical i, Mu_i = Chemical Potential of Chemical i)

The example system in the problem is a rigid, sealed, batch reactor, at some temperature, initially containing only reactant A. Reaction is A <-> 2B. P is held constant by a coolant/steam jacket on a control loop that keeps P perfectly constant. Reaction is gas phase, and components can be treated as ideal gases.

As far as I can tell, the traditional approaches to equilibrium criteria won't work because the constraints are not independent, and thus I can't write a potential function in terms of only P, V, and moles.

I've been directed that the solution goes down the path:

d(PV) = 0, d(NRT) = d(NT) = 0
Thus: N_initial*T_initial = N_final*T_final
N_final = N_a_final + N_b_final = (N_a_initial - X)+(N_b_initial + 2*X) = N_a_initial +X
So N_final/N_initial = 1 + X/N_a_initial = 1 + Conversion
T_final = T_initial / (1+Conversion)
The conversion at equilibrium will be a function of T. If one makes some hypotheticals about the relationship between conversion and temperature you can plot out a line for conversion, and another for the temperature, and seeing that there is one intersection you would establish that there is a single equilibrium state.

However, I don't see how this proves the initial equilibrium criteria in terms of chemical potentials.