13.5Equilibrium Calculations

Before it becomes a quadratic, an ICE-table equilibrium expression is a rational equation, Kc = x²/(1.00 − x), and clearing the denominator to reach polynomial form is the rational-equation move — students who try to cross-multiply incorrectly (or drop the −x term) get the wrong quadratic to begin with.

The PCl5 ICE table collapses to x² + 0.0211x − 0.0211 = 0, solved with the quadratic formula from Appendix B; the two roots are computed and the negative one (x = −0.156) is explicitly discarded because a concentration cannot be negative — an equilibrium calculation is unworkable without being able to run and interpret the quadratic formula.

The small-x approximation — replacing (0.15 − x) with 0.15 because x ≪ 0.15 — is an approximation criterion that only makes sense next to the exact quadratic solution it is standing in for; the section explicitly says if the resulting x fails the ≪ test, redo the calculation without the approximation, i.e. fall back to algebra:2.6.

ΔG = ΔG° + RT ln Q and its equilibrium special case ΔG° = −RT ln K, K = e^(−ΔG°/RT), require solving an equation that mixes a variable inside and outside a natural-log/exponential — exactly an exponential-and-logarithmic-equation solve, used here to compute Ksp for AgCl from tabulated free energies of formation.