dU = T dS - P dV + mu_1 dN1 + mu_2 dN2 + ...+ mu_r dNr

We want to generalize the fundamental thermodynamic relation:

dU = T dS - P dV (1)

which is valid for systems with fixed particle numbers.

Any differentiable function f(x1,x2,x3,...,xr) of r independent variables x1, x2,...,xr will have a differential df that can be expressed in terms of its partial derivatives as:

df = (df/dx1) dx1 + (df/dx2) dx2 + (df/dx3) dx3 + .....+(df/dxr) dxr (2)

where (df/dxi) denotes the partial derivative of f w.r.t. xi where all the xj for j not equal to i are kept constant. Consider then the internal energy U as a function of not just S and V but also the numbers of different types of particles N1, N2,...,Nr. Then (2) implies that:

dU = (dU/dS) dS + (dU/dV) dV + (dU/dN1) dN1 + (dU/dN2) dN2 + ...+(dU/dNr) dNr (3)

where the derivatives are partial derivatives where all the other variables in the set {S,V,N1,N2,...,Nr} are kept fixed. The aprtial derivatives can be related to physical quantities of the system. The partial derivative (dU/dS) is taken at constant volume and constant particle numbers. Now we know that if we keep all the particle numbers constant, then the infinitesimal change in U is given by (1). The coefficient of dS in (1) is T, which by (2) implies that the partial derivative of U w.r.t. S equals T, and this partial derivative is taken at constant V but also all the particle numbers are kept constant. Of course, when using (1) it is taken for granted that the particle numbers are kept constant, so one doesn't write that down explicitely. In (3) we can thus put:

(dU/dS) = T

Similarly, (dU/dV) is taken at constant S and constant particle numbers, therefore it is also the coefficient of dV in (1) which is -P:

(dU/dV) = -P

The partial derivatives of U w.r.t. the particle numbers define the chemical potentials. (dU/dNi) is taken at constant Nj for all j not equal to i and at constant S and V. This is, by definition, mu_i:

(dU/dNi) = mu_i

Substituting the expressions for the partial derivatives in (3) gives:

dU = T dS - P dV + mu_1 dN1 + mu_2 dN2 + ...+ mu_r dNr

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