Second law of thermodynamics

For a reversible change

$\left.dQ\right.=TdS$

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

$\left.dU\right.=TdS -PdV$

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

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

where $μ$ is the chemical potential.

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

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

[edit] See also

Nicolas Léonard Sadi Carnot "Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance" (1824), printed in Annales scientifiques de l'École Normale Supérieure Sér. 2, 1 pp. 393-457 (1872)