In the Haber process, nitrogen gas reacts with hydrogen gas to form gaseous ammonia (see Problem 14.76). (a) Write the equilibrium expression for the reaction. (b) The equilibrium constant at a certain temperature is \(1.5 \times 10^{-2}\). If the equilibrium concentrations of hydrogen and nitrogen are both \(0.20 \mathrm{M}\) at this temperature, what is the equilibrium concentration of ammonia?

In the Haber process, nitrogen gas (\(\mathrm{N_2}\)) reacts with hydrogen gas (\(\mathrm{H_2}\)) to form ammonia (\(\mathrm{NH_3}\)). The balanced equation for this reaction is: \[ \mathrm{N_2(g)} + 3\mathrm{H_2(g)} \rightleftharpoons 2\mathrm{NH_3(g)} \]

The equilibrium expression for the reaction is the product of concentrations of products raised to their respective coefficients in the balanced equation divided by the product of concentrations of reactants raised to their respective coefficients. In this case, the equilibrium expression is: \[K_\mathrm{eq} = \dfrac{[\mathrm{NH_3}]^2}{[\mathrm{N_2}][\mathrm{H_2}]^3}\]

We can now use the given values of equilibrium constant \((K_\mathrm{eq}\)) and the equilibrium concentrations of nitrogen and hydrogen to calculate the equilibrium concentration of ammonia. The equilibrium constant is given as \(1.5 \times 10^{-2}\), and the equilibrium concentrations of hydrogen and nitrogen are both \(0.20 \mathrm{M}\). We can rearrange the equilibrium expression to solve for \([\mathrm{NH_3}]\): \[[\mathrm{NH_3}]^2 = K_\mathrm{eq} [\mathrm{N_2}][\mathrm{H_2}]^3\] Now, plug the given values and concentrations into the equation: \[[\mathrm{NH_3}]^2 = (1.5 \times 10^{-2})(0.20)((0.20)^3)\] Now, solve for \([\mathrm{NH_3}]\): \[[\mathrm{NH_3}] = \sqrt{(1.5 \times 10^{-2})(0.20)((0.20)^3)}\] \[[\mathrm{NH_3}] = \sqrt{0.00012}\] \[[\mathrm{NH_3}] \approx 0.011\ \mathrm{M}\]

Thus, the equilibrium concentration of ammonia at this temperature is approximately \(0.011\ \mathrm{M}\).