A sample of \(\mathrm{H}_{2} \mathrm{S}\) gas is placed in an evacuated, sealed container and heated until the following decomposition reaction occurs at \(1000 \mathrm{K} :\) \(2 \mathrm{H}_{2} \mathrm{S}(g) \rightarrow 2 \mathrm{H}_{2}(g)+\mathrm{S}_{2}(g) \qquad K_{\mathrm{c}}=1.0 \times 10^{-6}\) (A) \(K_{\mathrm{c}}=\frac{\left[\mathrm{H}_{2}\right]^{2}\left[\mathrm{S}_{2}\right]}{\left[\mathrm{H}_{2} \mathrm{S}\right]^{2}}\) (B) \(K_{\mathrm{c}}=\frac{\left[\mathrm{H}_{2} \mathrm{S}\right]^{2}}{\left[\mathrm{H}_{2}\right]^{2}\left[\mathrm{S}_{2}\right]}\) (C) \(K_{\mathrm{c}}=\frac{2\left[\mathrm{H}_{2}\right]\left[\mathrm{S}_{2}\right]}{2\left[\mathrm{H}_{2} \mathrm{S}\right]}\) (D) \(K_{\mathrm{c}}=\frac{2\left[\mathrm{H}_{2} \mathrm{S}\right]}{2\left[\mathrm{H}_{2}\right]\left[\mathrm{S}_{2}\right]}\)

Step by step solution

01

Understand the law of mass action for the reaction

The law of mass action stipulates that for a reaction of the form \(aA + bB \rightarrow cC + dD\), the equilibrium constant expression can be defined as \[K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}\], where the brackets represent the molar concentrations of the respective substances at equilibrium and a, b, c, d are the stoichiometric coefficients. In this case, the reaction is \(2 \mathrm{H}_{2} \mathrm{S}(g) \rightarrow 2 \mathrm{H}_{2}(g)+\mathrm{S}_{2}(g)\). Therefore, a and b should be 2, whereas c and d should also be 2.

02

Identify the given options for \(K_{c}\) expressions

Here, the alternatives for the \(K_c\) expression are (A) \(\frac{\left[\mathrm{H}_{2}\right]^{2}\left[\mathrm{S}_{2}\right]}{\left[\mathrm{H}_{2} \mathrm{S}\right]^{2}}\), (B) \(\frac{\left[\mathrm{H}_{2} \mathrm{S}\right]^{2}}{\left[\mathrm{H}_{2}\right]^{2}\left[\mathrm{S}_{2}\right]}\), (C) \(\frac{2\left[\mathrm{H}_{2}\right]\left[\mathrm{S}_{2}\right]}{2\left[\mathrm{H}_{2} \mathrm{S}\right]}\), and (D) \(\frac{2\left[\mathrm{H}_{2} \mathrm{S}\right]}{2\left[\mathrm{H}_{2}\right]\left[\mathrm{S}_{2}\right]}\)

03

Choose the correct expression for \(K_{c}\)

By analyzing the four options given, it can be found that only Option (A) is consistent with the law of mass action. The expression \(K_{c} = \frac{\left[\mathrm{H}_{2}\right]^{2}\left[\mathrm{S}_{2}\right]}{\left[\mathrm{H}_{2} \mathrm{S}\right]^{2}}\) includes the correct stoichiometric coefficients in the reaction and correctly represents the molar concentration ratios at equilibrium. Therefore, the correct option is (A).

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