Second law of thermodynamics

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The second law of thermodynamics for a reversible change (that is to say that the system evolves through a succession of equilibrium states) can be written as

\left.dQ\right.=TdS

Thus for a closed system, having a fixed number of atoms/molecules, one has

\left.dU\right.=TdS -PdV

where U is the internal energy. For an open system:

\left.dU\right.=TdS -PdV + \mu dN

where \mu is the chemical potential.

For U(S,V) one has the following total differential

dU=\left(\frac{\partial U}{\partial S}\right)_V dS + \left(\frac{\partial U}{\partial V}\right)_S dV

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