When the following reaction is at equilibrium, which of these relationships is always true?

2 O3(g) 3 O2(g)

A. [O3]2 = [O2]3

B. Kc[O2]2 = [O3]3

C. [O3] = [O2]
D. Kc[O3]2 = [O2]3

E. Kc[O2]3 = [O3]2

Kc = (O2)^3/(O3)^2. Look at how this can be rearranged to find the right choice.

To determine which relationship is always true at equilibrium for the given reaction, we need to consider the balanced chemical equation and the expression for the equilibrium constant (Kc).

The balanced chemical equation is:
2 O3(g) → 3 O2(g)

The expression for the equilibrium constant (Kc) is:
Kc = [O2]3 / [O3]2

Now, let's evaluate each option to see which one is consistent with the balanced equation and the expression for the equilibrium constant.

A. [O3]2 = [O2]3
This option is not consistent with the balanced equation (the exponents are reversed), so it is not always true at equilibrium.

B. Kc[O2]2 = [O3]3
This option is not consistent with the expression for the equilibrium constant (the exponents are reversed), so it is not always true at equilibrium.

C. [O3] = [O2]
This option is not consistent with the balanced equation (the exponents are different), so it is not always true at equilibrium.

D. Kc[O3]2 = [O2]3
This option is consistent with the balanced equation and the expression for the equilibrium constant. The exponents are correctly matched, so it is always true at equilibrium.

E. Kc[O2]3 = [O3]2
This option is not consistent with the balanced equation (the exponents are reversed), so it is not always true at equilibrium.

Therefore, the correct answer is D. Kc[O3]2 = [O2]3.

To determine the correct relationship, we need to understand the concept of equilibrium in a chemical reaction. In a chemical equilibrium, the rate of the forward reaction is equal to the rate of the reverse reaction. At equilibrium, the concentrations of the reactants and products remain constant.

To find the relationship between the concentrations of O3 and O2 at equilibrium, we need to examine the balanced equation:

2 O3(g) ⇌ 3 O2(g)

The coefficients in the balanced equation represent the stoichiometry of the reaction. Let's analyze the options given:

A. [O3]2 = [O2]3
This option suggests that the concentration of O3 squared is equal to the concentration of O2 cubed. However, based on the balanced equation, the coefficients are not the same, so this relationship is not correct.

B. Kc[O2]2 = [O3]3
This option suggests that the equilibrium constant (Kc) multiplied by the concentration of O2 squared is equal to the concentration of O3 cubed. Though this equation includes the correct stoichiometric coefficients, the actual coefficient for O3 in the equation is 2 (not 3). Therefore, this relationship is not correct.

C. [O3] = [O2]
This option suggests that the concentration of O3 is equal to the concentration of O2. However, based on the balanced equation, the coefficient for O3 is 2, while the coefficient for O2 is 3. Therefore, this relationship is not correct.

D. Kc[O3]2 = [O2]3
This option suggests that the equilibrium constant (Kc) multiplied by the concentration of O3 squared is equal to the concentration of O2 cubed. This relationship incorporates the correct stoichiometric coefficients for the equation, as the coefficient for O3 is indeed 2, and the coefficient for O2 is 3. Consequently, this relationship is correct.

E. Kc[O2]3 = [O3]2
This option suggests that the equilibrium constant (Kc) multiplied by the concentration of O2 cubed is equal to the concentration of O3 squared. This relationship does not match the stoichiometric coefficients given in the balanced equation, so it is not correct.

Based on the analysis, the correct relationship between the concentrations of O3 and O2 at equilibrium is:

D. Kc[O3]2 = [O2]3