Equilibrium constant manipulation — worked example Summary & Study Notes

🧪 Problem overview

Given two equilibrium reactions and their constants, we must combine and manipulate them to obtain the equilibrium constant for a target reaction.

Given: $2A + B \rightleftharpoons 3C,; K_1 = 2.60$

$2C + D \rightleftharpoons 2B,; K_2 = 0.0250$

Target reaction: $4C \rightleftharpoons 4A + D$

🔁 Key rules for manipulating equilibrium constants

• If a reaction is reversed, the new equilibrium constant is the reciprocal: $K_{rev} = \frac{1}{K}$.
• If a reaction is multiplied by a factor $n$, the equilibrium constant is raised to that power: $K_{new} = K^{n}$.
• If reactions are added, the overall equilibrium constant is the product of the individual $K$ values.

🧭 Strategy to get the target reaction

We want $4C$ on the left and $4A + D$ on the right. Use the given reactions as building blocks and apply the rules above to cancel intermediate species ($B$ and extra $C$) and produce the desired stoichiometry.

1. Reverse reaction 1 so that $C$ appears on the left: $3C \rightleftharpoons 2A + B,; K = \frac{1}{K_1}$

2. Multiply the reversed reaction 1 by 2 to reach coefficients that can combine cleanly with reaction 2: $6C \rightleftharpoons 4A + 2B,; K = \left(\frac{1}{K_1}\right)^2$

3. Reverse reaction 2 so that $B$ appears on the left and can cancel with the $2B$ produced above: $2B \rightleftharpoons 2C + D,; K = \frac{1}{K_2}$

4. Add the two modified reactions: $6C + 2B \rightleftharpoons 4A + 2B + 2C + D$

Cancel $2B$ on both sides and subtract $2C$ from the left to obtain the target: $4C \rightleftharpoons 4A + D$

🔬 Combine equilibrium constants

Multiply the $K$ values for the two modified steps: $K_{total} = \left(\frac{1}{K_1}\right)^2 \times \frac{1}{K_2}$

Substitute numbers: $K_{total} = \left(\frac{1}{2.60}\right)^2 \times \frac{1}{0.0250}$

Compute stepwise: $\frac{1}{2.60} = 0.384615\ldots$

$\left(0.384615\ldots\right)^2 = 0.147928\ldots$

$\frac{1}{0.0250} = 40$

$K_{total} = 0.147928\ldots \times 40 = 5.91712\ldots$

✅ Final answer and sig figs

Rounded to three significant figures (matching the precision of the given constants): $K = 5.92$

✳️ Quick summary

• Use reciprocity for reversed reactions and exponentiation for multiplied reactions.
• Add reactions and multiply their $K$ values.
• Carefully cancel intermediate species to reach the target stoichiometry.
• Final result for $4C \rightleftharpoons 4A + D$ is $K = 5.92$ (three sig figs).