Second law of thermodynamics

For a reversible change

$dQ=TdS$

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

$dU=TdS-PdV$

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

$dU=TdS-PdV+\mu dN$

where $\mu$ is the chemical potential.

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

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