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A mixture of $\pu{1 g}$ of $\ce{H2(g)}$ and $\pu{1.06 g}$ $\ce{H2S(g)}$ in a $\pu{0.5 L}$ flask come to equilibrium at $\pu{1670 K}.$ At equilibrium, there are $\pu{8E-6 mol}$ of $\ce{S2(g)}$ present. Determine $K_p.$

I set up an ICE table:

$$ \begin{array}{ccc} \ce{& &2 H2 &+ &S2 &<=> &2 H2S} \\ \text{Initial} & &\pu{1 M} && 0 && \pu{0.062 M} \\ \text{Change} & &2x && x && 2x \\ \text{Equilibrium / Final} & &? && \pu{1.6E-5 M} && ? \end{array} $$

From the $\ce{S2}$ column we see that $x = \pu{1.6E-5 M}$, so I calculated the equilibrium constant as follows:

$$K_p = \frac{(0.062 + 2\times\pu{1.6E-5})^2}{(\pu{1.6E-5})(1-2\times\pu{1.6E-5})^2} = 240$$

What is wrong with my solution?

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The problem lies in that our information of the system is in terms of concentration, so you actually calculated $K_c$.

$$\ce{2H2 +S2<=>2H2S}$$ $$\begin{array}{|l|l|l|} \ce{H2 & S2 & H2S} \\ \hline 0.992\ \mathrm{M}& 0\ \mathrm{M}& 0.062\ \mathrm{M} \\ 2(1.6\times 10^{-5}) & (1.6\times 10^{-5}) & -2(1.6\times 10^{-5}) \\ 0.992+2(1.6\times 10^{-5}) & 1.6\times10^{-5}\ \mathrm{M} & 0.062-2(1.6\times 10^{-5})\end{array}$$

$$K_c=\frac{(0.062\ \mathrm{M}-3.2\times 10^{-5}\ \mathrm{M})^2}{(1.6\times10^{-5}\ \mathrm{M})\cdot(0.992\ \mathrm{M}+3.2\times 10^{-5}\ \mathrm{M})^2}=245\ \mathrm{M^{-1}}$$

which is numerically the same as the value you obtained, except that I used a more precise molecular mass for $\ce{H2}$.

Using the relation $$K_p=K_c(RT)^{\Delta n},$$ where $$\Delta n=(\sum \text{coefficient of gaseous products})-(\sum\text{coefficient of gaseous reactants}),$$ we can calculate $K_p$: $$K_p=(245\ \mathrm{M^{-1}})\cdot(8.3144598\ \mathrm{J\ K^{-1}\ mol^{-1}}\cdot 1670\ \mathrm{K})^{-1}=0.0176\ \mathrm{kPa^{-1}}$$

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