Chemical equilibrium

Stoichiometry tells you how much product a reaction could make if it went to completion. Many reactions never do. They run until the forward and reverse reactions balance, and then the composition stops changing. Ammonia synthesis, methanol production, the dissolving of minerals in boiler water, the acidity of process streams and the chemistry of blood all settle into this state, called chemical equilibrium.

Engineers care because equilibrium sets the limit on what a process can achieve, no matter how good the reactor. This chapter explains what equilibrium is, how the equilibrium constant describes it, how to calculate equilibrium compositions, and how changing concentration, pressure or temperature shifts it. It then applies the same ideas to acids, bases, pH, buffers and solubility.

Dynamic equilibrium

Consider dinitrogen tetroxide, a colourless gas that splits into brown nitrogen dioxide: N₂O₄ ⇌ 2 NO₂. Seal some N₂O₄ in a flask and it starts to decompose. As NO₂ builds up, the reverse reaction (two NO₂ molecules combining) speeds up. Eventually the two rates are equal and the color stops changing.

Equilibrium is dynamic: molecules are still reacting in both directions, just at equal rates. You reach the same equilibrium whether you start from pure N₂O₄, pure NO₂ or any mixture, as long as the total amount of material and the temperature are the same.

The equilibrium constant

For a general reaction \(a\text{A} + b\text{B} \rightleftharpoons c\text{C} + d\text{D}\), the concentrations at equilibrium always satisfy

where square brackets mean molar concentration at equilibrium. This is the law of mass action. \(K_c\) depends only on the reaction and the temperature. Pure solids and pure liquids (including water as a solvent) are left out of the expression, because their concentrations do not change.

For gases, the same idea is often written with partial pressures (in bar or atm) as \(K_p\). The two are related by \(K_p = K_c(RT)^{\Delta n}\), where \(\Delta n\) is the change in moles of gas (products minus reactants) and \(R = 0.08206\) L·atm/(mol·K).

The size of \(K\) tells you where the equilibrium lies:

• \(K \gg 1\) (say above \(10^3\)): products dominate; the reaction goes nearly to completion.
• \(K \ll 1\) (below \(10^{-3}\)): reactants dominate; very little product forms.
• \(K\) near 1: significant amounts of both.

Two manipulation rules are useful. Reversing a reaction inverts \(K\). Multiplying the equation by a factor \(n\) raises \(K\) to the power \(n\).

The reaction quotient

The reaction quotient \(Q\) has the same form as \(K\) but uses the current concentrations, which may not be at equilibrium. Comparing \(Q\) with \(K\) tells you which way the reaction will go:

• \(Q < K\): too few products; the reaction runs forward.
• \(Q > K\): too many products; the reaction runs in reverse.
• \(Q = K\): at equilibrium.

Calculating equilibrium compositions

The standard tool is an ICE table: Initial concentrations, Change (in terms of an unknown \(x\), using the stoichiometry), and Equilibrium concentrations. Substitute the equilibrium row into the \(K\) expression and solve for \(x\).

Le Chatelier's principle

If a system at equilibrium is disturbed, it shifts in the direction that partly counteracts the disturbance. This qualitative rule, Le Chatelier's principle, follows from comparing \(Q\) with \(K\), and it is how chemical engineers push reactions toward the products they want.

• Concentration: adding a reactant (or removing a product) makes \(Q < K\), so the reaction runs forward. Ammonia plants condense NH₃ out of the loop and recycle the unreacted gas for exactly this reason.
• Pressure: compressing a gas mixture favors the side with fewer moles of gas. For N₂ + 3 H₂ ⇌ 2 NH₃, four moles of gas become two, so high pressure (150 to 300 bar in practice) favors ammonia. Reactions with equal moles of gas on both sides are unaffected.
• Temperature: unlike the others, temperature changes \(K\) itself. Raising the temperature favors the endothermic direction. The Haber process is exothermic (\(\Delta H = -92\) kJ), so higher temperature lowers the equilibrium yield.
• Catalysts: speed up both directions equally. They help a system reach equilibrium faster but do not change where it ends up.

How much does temperature change K?

The van 't Hoff equation gives the change in \(K\) between two temperatures, assuming \(\Delta H\) stays roughly constant:

The engineer's compromise

The Haber process shows the trade-off between equilibrium and rate. Low temperature gives a high equilibrium yield of ammonia, but the reaction is then far too slow. Industrial converters run at about 400 to 500 °C with an iron catalyst: the equilibrium yield per pass is lower, but equilibrium is reached quickly, and recycling the unreacted gas makes up the difference. Designing that compromise is a classic chemical engineering problem.

Acids, bases and pH

Acid–base reactions are equilibria too, and they matter everywhere in engineering: water treatment, corrosion control, boiler chemistry, mining effluents, concrete durability and food processing. In water, an acid donates H⁺ (as hydronium, H₃O⁺) and a base accepts it.

Water itself ionizes slightly: H₂O ⇌ H⁺ + OH⁻, with \(K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14}\) at 25 °C. In pure water both are \(10^{-7}\) M. Because these numbers span many orders of magnitude, we use a logarithmic scale:

pH 7 is neutral; below 7 is acidic and above 7 is basic. Each unit is a factor of ten: pH 3 is ten times more acidic than pH 4 and a thousand times more than pH 6.

Strong and weak acids

A strong acid such as HCl, HNO₃ or H₂SO₄ (first proton) ionizes completely, so [H⁺] equals the acid concentration: 0.010 M HCl has pH 2.00. A weak acid ionizes only partly, and its equilibrium constant \(K_a\) tells you how much. Acetic acid (vinegar) has \(K_a = 1.8 \times 10^{-5}\).

Buffers

A buffer is a mixture of a weak acid and its conjugate base (or a weak base and its conjugate acid). It resists changes in pH because it can neutralize both added acid and added base. The pH of a buffer is given by the Henderson–Hasselbalch equation:

Natural buffers keep blood at pH 7.4, and the carbonate buffer in lakes protects them from acid rain. Engineers add buffering chemicals to control pH in fermentation, electroplating baths and drinking-water treatment.

Solubility equilibria

A sparingly soluble salt in contact with its solution reaches an equilibrium described by the solubility product \(K_{sp}\). For calcium carbonate, CaCO₃(s) ⇌ Ca²⁺ + CO₃²⁻ and \(K_{sp} = [\text{Ca}^{2+}][\text{CO}_3^{2-}]\). The solid is left out because it is a pure solid.

Solubility equilibria explain scale. When hard water is heated, dissolved CO₂ escapes and CaCO₃ becomes less soluble, so it precipitates as scale on kettle elements, boiler tubes and heat exchangers. A layer only a millimeter thick can cut heat transfer noticeably and raise fuel use, so plants soften feed water, add scale inhibitors or control pH to stay below saturation.