Calculating Equilibrium Concentrations

Choose a change that has meaning

For \(X_2(g)\rightleftharpoons2X(g)\), let x be the molar concentration of X₂ that dissociates. Then [X₂] falls by x while [X] rises by 2x. The coefficient belongs in the change row as well as in the equilibrium expression. Here the container volume and temperature remain constant.

Organize an original starting mixture

Because the system begins without X, the initial net direction is forward. Physical limits require \(0\leq x\leq0.100\ \mathrm M\).

Solve rather than guess

At the stated temperature, let \(K_c=0.0400\). Substitution gives

The positive root is x = 0.0270 M to three significant figures. Thus [X₂] = 0.0730 M and [X] = 0.0540 M. The negative root would make [X] negative and must be rejected on physical grounds, not just because “we always take plus.”

Perform two independent checks

Substitution gives \(0.0540^2/0.0730\approx0.0400\). Atom conservation gives \(2[X_2]+[X]=0.200\ \mathrm M\), matching the initial total concentration of X atoms. The total concentration of molecules increases, so checking [X₂] + [X] for constancy would be the wrong conservation test.

Test the tempting approximation

If we assumed x was negligible beside 0.100, we would obtain \(x\approx\sqrt{0.0400(0.100)/4}=0.0316\ \mathrm M\). That is 31.6% of the initial concentration, far beyond the common 5% classroom check. The approximation therefore fails. Small K alone is not enough; its size relative to the starting concentration also matters.

Interpret experimental concentration traces

In a suitable spectroscopic model, the X signal should increase while the X₂ signal decreases. Their molar changes must obey the two-to-one relationship after calibration. Raw absorbance changes need not have that ratio because the species may have different molar absorptivities. Check the measurement model before diagnosing a stoichiometric inconsistency.

When the mixture starts with products too

Calculate Q first. If Q exceeds K, choose reverse changes or allow a signed extent and interpret its sign consistently. Do not automatically write every reactant change as negative. The table records what the mixture does, not what the reaction arrow seems to suggest at first glance.