Gibbs Equation

Valid for pure substances

d U = T d S - P d V

Temperature definition:

T = \frac{d U}{d S} |_{V}

First law has general form: $d U = δ Q - δ W$

Change in energy independent of path
$\Rightarrow$ Let's choose a reversible path

Work term

Heat term

Definition of Temperature:

T = \frac{\partial U}{\partial S} |_{V, N}

Gibbs Equation:

d u = T d s - P d v

$T = \frac{\partial u}{\partial s} |_{v}$
$P = - \frac{\partial u}{\partial v} |_{s}$

d h = T d s + v d P

$T = \frac{\partial h}{\partial s} |_{P}$
$v = \frac{\partial h}{\partial P} |_{s}$

d f = - s d T - P d v

$s = - \frac{\partial f}{\partial T} |_{v}$
$P = - \frac{\partial f}{\partial v} |_{T}$

d g = - s d T + v d P

$s = - \frac{\partial g}{\partial T} |_{P}$
$v = \frac{\partial g}{\partial P} |_{T}$

d G = d H - T d S \leq 0

d G = d H - T d S \leq 0

Important

At Equilibrium $d G_{T, P} = 0$