Question 4 of 5: Atmospheric Stability Regimes and Copper–Hydroxide Complexation

Problem 4: Atmospheric Stability Regimes and Copper–Hydroxide Complexation (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Meaning of Atmospheric Stability

Atmospheric stability describes the atmosphere's resistance to, or encouragement of, vertical (convective) motion of an air parcel displaced from its original level. It is governed by the comparison, developed in Problem 2(c), between the rate at which a displaced parcel cools adiabatically as it rises or sinks (the dry, or moist once saturated, adiabatic lapse rate) and the rate at which the surrounding (environmental) air's temperature actually changes with height. A stable atmosphere resists and damps vertical motion — a displaced parcel is forced back toward its starting level — while an unstable atmosphere amplifies it — a displaced parcel keeps accelerating away from its starting level. Stability is the single most important atmospheric variable for air-quality engineering because it directly controls how effectively pollutants released near the ground are diluted vertically: stable air traps and concentrates emissions (as in an inversion), while unstable air disperses them rapidly through deep convective mixing.

(b) Neutral, Stable, and Unstable Conditions

(i) The atmosphere is neutral when the environmental lapse rate exactly equals the dry adiabatic lapse rate ($\Gamma_{env} = \Gamma_d \approx 0.98\ {}^{\circ}\text{C}/100\ \text{m}$): a displaced parcel always finds itself at the same temperature (and density) as its new surroundings, so it experiences no net buoyant restoring or accelerating force and simply remains wherever it is moved — vertical motion neither is damped nor amplified.

(ii) The atmosphere is stable when the environmental lapse rate is less than the dry adiabatic lapse rate ($\Gamma_{env} < \Gamma_d$), including the special case of a temperature inversion ($\Gamma_{env}<0$, temperature increasing with height). A parcel forced upward cools faster than its surroundings, ends up colder and denser, and experiences a net downward (restoring) buoyant force back toward its origin — vertical mixing is suppressed, exactly as computed for the atmosphere in Problem 2(c)(ii).

(iii) The atmosphere is unstable when the environmental lapse rate exceeds the dry adiabatic lapse rate ($\Gamma_{env} > \Gamma_d$), i.e. the surrounding air cools with height faster than a rising parcel cools itself. The displaced parcel remains warmer and less dense than its surroundings at every level, so the buoyant force continues to accelerate it further from its starting point — strong solar heating of the ground on a clear, calm day is the classic cause, producing vigorous convective (thermal) mixing and rapid, efficient dispersion of ground-level pollutant releases.

(c) Cu²⁺ / CuOH⁺ Equilibrium Ratio

Given. Cu²⁺ + OH⁻ → CuOH⁺, log K = 6.3; [OH⁻] = 10⁻⁴ mol/L.

Find. The equilibrium ratio [Cu²⁺]/[CuOH⁺].

Approach. Write the mass-action expression for the complexation equilibrium and solve directly for the requested species ratio.

1. Write the equilibrium constant expression. $$K = \frac{[\text{CuOH}^+]}{[\text{Cu}^{2+}][\text{OH}^-]}, \qquad K = 10^{\log K} = 10^{6.3} = 1.995\times10^{6}$$
2. Rearrange for the requested ratio and substitute. $$\frac{[\text{Cu}^{2+}]}{[\text{CuOH}^+]} = \frac{1}{K\,[\text{OH}^-]} = \frac{1}{(1.995\times10^{6})(10^{-4})} = \frac{1}{199.5}$$ $$\frac{[\text{Cu}^{2+}]}{[\text{CuOH}^+]} = \boxed{5.01\times10^{-3}\ \ (\text{i.e. about 1:200})}$$