## Q. 13.4

Judging Whether a Mixture Is at Equilibrium

The following pictures represent mixtures of A molecules (red spheres) and B molecules (blue spheres), which interconvert according to the equation A $\rightleftharpoons$ B. If mixture (1) is at equilibrium, which of the other mixtures are also at equilibrium? Explain.

STRATEGY

The equilibrium constant for the reaction is given by $K_c$ = [B]/[A], where the concentrations are equilibrium concentrations in units of mol/L. Since the equilibrium-constant expression has the same number of concentration terms in the numerator and denominator, the volume cancels and $K_c$ = (moles of B)/(moles of A). Because the number of moles is directly proportional to the number of molecules, $K_c$ = (molecules of B)/(molecules of A) in the equilibrium mixture, mixture (1). To determine whether the other mixtures are at equilibrium, count the number of molecules and compare the B/A ratio in mixtures (2)–(4) with the B/A ratio in the equilibrium mixture.

## Verified Solution

For mixture (1), $K_{\mathrm{c}}=[\mathrm{B}] /[\mathrm{A}]=2 / 6=1 / 3$

For mixture (2), $[\mathrm{B}] /[\mathrm{A}]=4 / 4=1 \neq K_c .$

For mixture (3), $[\mathrm{B}] /[\mathrm{A}]=3 / 9=1 / 3=K_{\mathrm{c}}$

For mixture (4), $[\mathrm{B}] /[\mathrm{A}]=9 / 3=3 \neq K_{\mathrm{c}}$

Mixture (3) is at equilibrium, but mixtures (2) and (4) are not at equilibrium because their equilibrium-constant expressions [B]/[A] do not equal $K_c$.