first law of thermodynamics

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For a reversible process,

dS = \left(\frac{\partial S}{\partial U}\right)_{V,N} dU + \left(\frac{\partial S}{\partial V}\right)_{U,N} dV + \left(\frac{\partial S}{\partial N}\right)_{U,V} dN\,,

and with the definitions in place,

dS = \frac{1}{T} dU + \frac{P}{T} dV - \frac{\mu}{T} dN\,,

or

dU = T dS - P dV + \mu dN\,.

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