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In thermodynamics you work with certain potentials, which are a function of the thermodynamic variables, i.e.:

U(S,T,V,N,P)

Now for U one has the identity:

dU= TdS+SdT-pdV+VdP etc etc.

From these one figure out relations like:

T = dU/dS at fixed V,T,P...

It is this thing about the thermodynamic variables being fixed that has always confused me. In general are the thermodynamic variables S,P,T,V,N not correlated? How am I to understand then the derivative if I am keeping the other thermodynamic variables fixed. Consider for instance including particles of different kinds:

U = ... + μ1N1 + μ2N2

Now we have that:

μ1 = dU/dN1 at fixed T,S,V, N2

But how can I keep N2 fixed if I am in a resevoir, where adding a particle to one phase with N1 particles, actually takes away a particle from the other i.e. dN1=-dN2

Similarly, if I change for instance V, don't I change S or P etc etc.