Chemical Equilibrium Mixed Problems

Introduction

Mixed equilibrium problems test whether the right question is asked at each stage. A disturbance can change the current quotient, the direction of adjustment, and the final composition without changing K. Reaction extent and conservation then determine the new concentrations. Keeping those stages separate prevents qualitative shift rules from replacing the actual calculation.

Core explanation

Start with a balanced reaction, for example A(g) + B(g) ⇌ C(g), and write its dimensionless activity quotient Q = aC/(aAaB). At a stated temperature K has one value for that reaction as written. Current activities give Q. If Q < K, net forward reaction is favored; if Q > K, net reverse reaction is favored; at Q = K, the mixture is at equilibrium. Changing temperature can change K; changing composition or volume at fixed temperature changes Q immediately and then the reaction composition adjusts.

To calculate a final state, introduce a reaction extent ξ. For A + B ⇌ C, changes in mole amounts are −ξ, −ξ, +ξ for forward progress. Concentrations or partial pressures follow from these changed amounts and the final volume or total pressure. Substitute them into Q = K and solve. A mathematical root is physically acceptable only if every final mole amount is nonnegative and any phase assumptions are met. The initial direction from Q versus K often indicates the expected sign of ξ and helps reject the wrong root.

Stress cases require particular care. If a reactant is added, Q can fall, but the final reactant concentration may still be higher than before addition despite partial consumption. If a product is removed, Q falls and forward reaction tends to replace some product; it rarely restores the original amount exactly. A pressure change matters according to gaseous stoichiometry and the manner of change. Adding an inert gas at constant volume does not change reacting species partial pressures in an ideal-gas model, while expansion at constant temperature does. Avoid an unqualified “pressure always shifts equilibrium” rule.

Combining reactions also changes constants systematically: reverse a reaction and invert K; multiply reaction coefficients by a factor and raise K to that factor; add reactions and multiply their constants. This algebra presumes consistent reaction definitions and standard states. Units in concentration-style numerical K expressions can be misleading, so use the convention stated in the problem.

Step-by-step reasoning

1. Balance the target reaction and write Q with proper exponents. 2. Compare current Q with K to determine initial direction. 3. Use a signed reaction extent for all species changes. 4. Solve Qfinal = K and reject physically impossible roots.

Visual explanation

Draw a timeline: initial mixture, immediate disturbance with changed Q, reaction adjustment with extent ξ, and final state with Q = K.

Real-world analogy

A balance scale may be pushed by adding weight to one side. The immediate imbalance predicts direction, but the final load on each side depends on how much can move between them.

Real-world example

Industrial gas reactions are managed by adjusting feed composition, pressure, and temperature. A plant must distinguish an immediate quotient change from a new equilibrium composition before estimating yield.

Why?

Why does Q versus K predict direction without giving final amounts? It says which way free energy initially favors reaction; conservation and the magnitude of K determine how far the mixture moves.

Common misconception

“A shift right means every reactant ends below its original concentration.” If reactant was added, some is consumed but its final amount may still exceed its pre-addition amount.

Worked example

For A ⇌ B, take K = [B]/[A] = 4.0 at fixed temperature. Initially [A] = 0.60 M and [B] = 0.20 M, so Q = 0.333 < 4.0 and forward change is favored. Let x convert from A to B. Then (0.20+x)/(0.60−x) = 4.0. Solving gives 0.20+x = 2.40−4x, so x = 0.44 M. Final concentrations are [A] = 0.16 M and [B] = 0.64 M; their ratio is 4.0 and both are nonnegative.

Quick check

1. What direction is favored when Q > K for the reaction as written? Answer: Net reverse reaction, lowering Q toward K.

Exam focus

Identify whether the question asks direction, new K, or full final composition. Use Q for direction and an extent table for composition; temperature alone can alter K.

Advanced insight

The sign of ΔrG = RT ln(Q/K) gives the same direction test. It connects the quotient comparison to thermodynamics rather than to a memorized shift slogan.

Summary

Mixed equilibrium reasoning separates immediate quotient change from final equilibration. A balanced equation, Q-versus-K test, reaction extent, and physical-root check produce a consistent answer.

Practice questions

1. If Q = 2 and K = 5, which direction is initially favored? Answer: Forward, because Q < K. 2. What happens to K when a reaction equation is reversed? Answer: The new constant is 1/K. 3. Why reject a calculated extent that makes a concentration negative? Answer: It violates the nonnegative amount constraint and cannot represent a physical equilibrium state.