Question

Consider the following reaction at \(250^{\circ} \mathrm{C}\) : $$ \mathrm{A}(s)+2 \mathrm{~B}(g) \rightleftharpoons \mathrm{C}(s)+2 \mathrm{D}(g) $$ (a) Write an equilibrium constant expression for the reaction. Call the equilibrium constant \(K_{1}\). (b) Write an equilibrium constant expression for the formation of one mole of \(\mathrm{B}(g)\) and call the equilibrium constant \(K_{2}\) (c) Relate \(K_{1}\) and \(K_{2}\).

Short answer

Answer: The relationship between the equilibrium constants \(K_1\) and \(K_2\) is \(K_1 = K_2^2\).

Step-by-step solution

(Step 1: Write the equilibrium constant expression for the given reaction)

For the given reaction: $$ \mathrm{A}(s)+2 \mathrm{~B}(g) \rightleftharpoons \mathrm{C}(s)+2 \mathrm{D}(g) $$ We can write the equilibrium constant, \(K_{1}\), as the ratio of the product concentrations of their equilibrium concentrations to the reactant concentrations raised to their stoichiometric coefficients. Since A and C are in solid-state, their concentrations remain constant and do not appear in the expression. $$ K_{1}=\frac{ [\mathrm{D} ]^2 } { [\mathrm{B}]^2 } $$

(Step 2: Write the equilibrium constant expression for the formation of one mole of B(g))

We can write the hypothetical reaction for the formation of one mole of B(g) as follows: $$ 0.5\mathrm{A}(s)+\mathrm{B}(g) \rightleftharpoons 0.5\mathrm{C}(s)+\mathrm{D}(g) $$ Now, write the equilibrium constant, \(K_2\), using the same principle as in Step 1: $$ K_2=\frac{ [\mathrm{D}] } { [\mathrm{B}] } $$

(Step 3: Relate \(K_1\) and \(K_2\))

To relate \(K_1\) and \(K_2\), we can rearrange the expression for \(K_2\) to have \([\mathrm{D}]\) on one side: $$ [\mathrm{D}] = K_2[\mathrm{B}] $$ Now, square both sides of the equation: $$ [\mathrm{D}]^2= K_2^2[\mathrm{B}]^2 $$ Substitute the expression for \([\mathrm{D}]^2\) from above into the expression for \(K_1\): $$ K_1=\frac{K_2^2[\mathrm{B}]^2} { [\mathrm{B}]^2 } $$ Now, cancel the \([\mathrm{B}]^2\) term on both numerator and denominator: $$ K_1=K_2^2 $$ Thus, the relationship between \(K_1\) and \(K_2\) is \(K_1 = K_2^2\).