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

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Derive equation dU = Tds - PdV + ε_i μ_i dN_I, where the chemical potential for the ith type of particle is
μ_i = (∂U / ∂N_I)_S,V,Nk

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Solution Summary

We prove the extended fundamental thermodynamic relation from the definition of the chemical potential.

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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 ...

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