Chemical & Phase Equilibrium

What does equilibrium minimise?

An isolated system heads toward maximum entropy (Lesson 6). But a flame, a chemical reactor or a glass of iced water is rarely isolated: it is held at a fixed temperature and pressure by its surroundings. For such a system, combining the first and second laws gives a new criterion. With $T$ and $p$ fixed,

$dG_{T,p} \le 0,$

where $G = H - TS$ is the Gibbs functionGibbs function GG = H − TS. At fixed temperature and pressure, a system reaches equilibrium where G is a minimum.. Every spontaneous change at constant temperature and pressure lowers $G$, and equilibrium is the state of minimum $G$. The two terms pull in opposite directions: a low enthalpy favours strong chemical bonds (CO₂ and water), a high entropy favours many, scattered molecules (CO, O₂, H, O). Temperature sets the exchange rate between them. At room temperature enthalpy wins outright. In a flame at 2500 K, entropy claims a share.

The equilibrium constant

For a reaction between ideal gases, $\nu_A A + \nu_B B \rightleftharpoons \nu_C C + \nu_D D$, setting $dG = 0$ at the minimum gives the equilibrium constantEquilibrium constant KK(T) = exp(−ΔG°/RT): fixes the balance between reactants and products at equilibrium. It depends on temperature only.:

$K(T) = \frac{y_C^{\nu_C}\,y_D^{\nu_D}}{y_A^{\nu_A}\,y_B^{\nu_B}}\left(\frac{p}{p_\text{ref}}\right)^{\nu_C + \nu_D - \nu_A - \nu_B}, \qquad \ln K = -\frac{\Delta G^\circ}{\bar R T}.$

Here $\Delta G^\circ$ is the change in Gibbs function for the reaction with each gas pure at 1 atm. $K$ depends only on temperature, and it is tabulated — Moran and Shapiro's Table A-27 is the standard list:

Two rules tell you how the equilibrium shifts:

• Temperature. For reactions that absorb heat, like every dissociation, $K$ rises steeply with temperature. The van 't Hoff equation makes it precise: $d\ln K/dT = \Delta H/\bar R T^2$.
• Pressure. Raising the pressure pushes the equilibrium toward the side with fewer moles of gas. Dissociation increases the number of moles, so high pressure suppresses it. That is Le Chatelier's principle, falling straight out of the $p^{\Delta\nu}$ factor.

Hot flames don't finish burning

Put these ideas inside a flame. At 2200 K, CO₂ and H₂O are partly dissociated into CO, H₂ and O₂. Dissociation absorbs energy, so the equilibrium flame temperature is lower than the complete-combustion one, and the products contain CO and H₂ even with enough oxygen to burn them. As the gas cools downstream, they mostly recombine, releasing their energy — the heat is delayed, not lost.

And the nitrogen, Lesson 13's inert ballast, is not quite inert. At flame temperatures, the equilibrium $\tfrac12\mathrm{N_2} + \tfrac12\mathrm{O_2} \rightleftharpoons \mathrm{NO}$ shifts appreciably to the right:

That explains a puzzle about NOₓ emissions. At exhaust temperatures, equilibrium would leave almost no NO. But the reactions that make and destroy NO are slow; once the gas leaves the flame and cools, they freeze, and the NO formed in the hottest zone survives into the atmosphere, where it causes smog and acid rain. That is why low-NOₓ gas-turbine combustors burn lean, to keep the peak temperature down, and why their residence time at high temperature is kept short. Equilibrium thermodynamics tells you the destination; chemical kinetics tells you whether there is time to get there.

Phase equilibrium

The same principle governs how a substance shares itself between phases. Each component in each phase has a chemical potential, $\mu_i = (\partial G/\partial n_i)_{T,p}$: the change in $G$ per mole added. Matter flows from high chemical potential to low until they are equal — exactly as heat flows from high temperature to low until temperatures are equal.

For a pure substance with liquid and vapour, equilibrium means $g_f = g_g$: the Gibbs function is the same in both phases. That is what fixed the vapour pressure in Maxwell's equal-area construction (Lesson 11), and differentiating it along the saturation line gives the Clapeyron equation. It also explains why water evaporates from a glass in dry air, why salt lowers the freezing point of water, and how a distillation column separates crude oil into petrol, kerosene and diesel.

For mixtures with several components and phases, the Gibbs phase rule counts the freedom left over:

$F = 2 + N - P,$

where $N$ is the number of components and $P$ the number of phases. Pure water ($N = 1$) as liquid and vapour ($P = 2$) has $F = 1$: fix the temperature and the pressure is fixed too — the saturation curve. At the triple point, with ice, liquid and vapour together, $F = 0$: a single, unique state, 0.01 °C and 611.7 Pa, which is why it defines the Kelvin scale.

That completes the theory. Next: put it all to work. Can you design a power plant that beats the competition?