Equilibrium I

• Products of complete combustion (PCC) are useful for simple calculations.
  • Can be considered an upper bound on the extent of reaction.
  • Useful for defining heating values and providing a combustion basis.
• At high temperatures products dissociate to form $H_2$, $CO$, $H$, $O$, $OH$, etc.

Criteria for equilibrium

• Consider a fixed mass at equilibrium

• $dU = dQ + dW$

  • $dW = -PdV$
  • at equilibrium differential changes are reversible, so $dQ = TdS$.

• $dU = TdS - PdV$

• Assume const $T$ and $P$, then we can move $T$ and $P$ inside the derivatives:

  • $dU = d(TS) - d(PV)$.

• Collect terms:

  • $d(U+PV-TS) = d(H-TS)= dG = 0$.

• At equilibrium

  $$dG_{T,P} = 0$$

• Hence $G$ is a critical point.

• Note equilibrium is a state not a process.

  • The $T,P$ in $dG_{T,P}=0$ does not mean that that the equilibrium applies to a constant $T$ and $P$ process.
  • Rather, if you look in directions of constant $T$ and $P$, then the $G$ curve will be minimum at the equilibrium point.

Other equivalent equilibrium conditions

• Above, we had $$dU = TdS - PdV$$
  • For constant $T$, $P$, we have $dG_{T,P}=0$.
  • For constant $S$, $V$, we have $dU_{S,V}=0$.
  • For constant $S$, $P$, we have $dU=-d(PV)\rightarrow dH_{S,P}=0$.
  • For constant $T$, $V$, we have $dU=d(TS)\rightarrow dA_{T,V}=0$, where $A$ is Helmholz free energy.
• All of these four expressions are true of an equilibrium state at a given temperature and pressure.
  • Similarly, all of these four expressions are true at a given $S$ and $V$, etc.
  • Again, this is because equilibrium is a state, not a process.