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K1, K2, and K3 represent the equilibrium constants for the following reactions, respectively: (I) \(N_2O_4 \rightleftharpoons 2NO_2\) (II) \(2NO_2 \rightleftharpoons N_2 + 2O_2\) (III) \(NO_2 \rightleftharpoons \tfrac{1}{2}N_2 + O_2\) Which of the following correctly describes the relationship between these constants?

1. K1 equals the reciprocal of K2 and also equals the reciprocal of K3
2. K1 equals the reciprocal of K2 and is equal to the square of K3
3. K1 is the square root of K2 and is equal to K3
4. K1 equals the reciprocal of K2 and is the same as K3

Correct answer: K1 equals the reciprocal of K2 and is equal to the square of K3

Solution

Reaction (II) is the reverse of reaction (I) after appropriate scaling, so its equilibrium constant is the reciprocal of \(K_1\). Reaction (III) is half of reaction (II), so its constant is the square root relation; equivalently, \(K_1 = K_3^2\).

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