Second law of thermodynamics

For a reversible change

$dQ=TdS$

Thus for a closed system ($n$ fixed):

$dU=TdS-PdV$

For an open system:

$dU=TdS-PdV+\mu dN$

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

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