# SL: Entropy and Spontaneity/Equilibrium

### Entropy and Spontaneity/Equilibrium

Positive Entropy change of the universe $\to$ spontaneous process

For equilibrium: $dS_{univ} \to$ "maximum"

$dS_{univ} = dS_{sys} + dS_{surr} = dS_{sys} + \frac{\delta Q_{surr}}{T_{surr}} = dS_{sys} - \frac{\delta Q_{sys}}{T_{surr}} \ge 0$

• Closed System: Constant T,V
• First Law: $dU = \delta Q_{rev}+ \delta W_{rev}$ $\to$ $dU = \delta Q_{rev}$
• Second Law: $dS_{sys} - \frac{Q_{sys}}{T_{surr}} = dS_{sys} - \frac{dU}{T_{surr}} \ge 0$
• (constant T) $T_{surr} = T_{sys}$

$T_{sys}dS_{sys} - dU \ge 0$

(constant T) $d(T_{sys}S_{sys}) = T_{sys}dS_{sys} + S_{sys}dT_{sys} = T_{sys}dS_{sys}$

For equilibrium: $d(T_{sys}S_{sys} - U) \ge 0$

• ##### Definition:

The Helmholtz Free Energy, $A$, is defined as $A \equiv U-TS$.

For equilibrium: $dA \le 0$ ("minimum")

• Closed System: Constant T,P
• First Law: $dU = \delta Q_{rev}+ \delta W_{rev}$ $\to$ $dU = \delta Q_{rev} - PdV$
• Second Law: $dS_{sys} - \frac{Q_{sys}}{T_{surr}} = dS_{sys} - \frac{dU+PdV}{T_{surr}} \ge 0$
• (constant T) $T_{surr} = T_{sys}$

$T_{sys}dS_{sys} - (dU+PdV) \ge 0$

(constant T) $d(T_{sys}S_{sys}) = T_{sys}dS_{sys} + S_{sys}dT_{sys} = T_{sys}dS_{sys}$

(constant P) $d(PV) = PdV + VdP = PdV$, so $(dU+PdV) = dH$

For equilibrium: $d(T_{sys}S_{sys} - H) \ge 0$

• ##### Definition:

The Gibbs Free Energy, $G$, is defined as $G \equiv H-TS$.

For equilibrium: $dG \le 0$ ("minimum")