Nitrogen gas \(\left(\mathrm{N}_{2}\right)\) reacts with hydrogen gas \(\left(\mathrm{H}_{2}\right)\) to form ammonia \(\left(\mathrm{NH}_{3}\right) .\) At \(200^{\circ} \mathrm{C}\) in a closed container, 1.00 atm of nitrogen gas is mixed with 2.00 \(\mathrm{atm}\) of hydrogen gas. At equilibrium, the total pressure is 2.00 atm. Calculate the partial pressure of hydrogen gas at equilibrium, and calculate the \(K_{\mathrm{p}}\) value for this reaction.

The balanced equation for the given reaction is: \[N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)\]

At the beginning of the reaction, we have the following partial pressures: - \(P_{N_2} = 1.00 \, atm\) - \(P_{H_2} = 2.00 \, atm\) - \(P_{NH_3} = 0 \, atm\), since no ammonia has been formed yet.

Let x be the change in the partial pressure of nitrogen gas as the reaction proceeds. The change in partial pressures for each component can be summarized as follows: - \(N_2\): \(1.00 - x\, atm\) - \(H_2\): \(2.00 - 3x \, atm\), since the stoichiometry shows that for 1 mole of \(N_2\), 3 moles of \(H_2\) react - \(NH_3\): \(2x\,atm\), since for every 1 mole of \(N_2\), we form 2 moles of \(NH_3\)

Since the total pressure at equilibrium is given as 2.00 atm, we can write the equation as follows: \[P_{N_2} + P_{H_2} + P_{NH_3}=(1.00-x)+(2.00-3x)+(2x)=2.00\] Solving for x: \[3.00-x=2.00\] \[x=1.00\,atm\] Now we can determine the equilibrium partial pressure of hydrogen gas: \[P_{H_2} = 2.00 - 3x = 2.00 - 3(1.00) = -1.00\,atm\] However, since a negative value for pressure is not physically significant, we can conclude that all the hydrogen gas has reacted and got consumed, while a small fraction of nitrogen gas is still present. So, we can revise the partial pressures of the components at equilibrium to: - \(P_{N_2} = 1.00 - 0.67 = 0.33 \, atm\) - \(P_{H_2} = 0 \, atm\) - \(P_{NH_3} = 2 \times 0.67 = 1.34 \, atm\)

Using the balanced equation and the equilibrium partial pressures, we can compute the \(K_p\) value as follows: \[K_p = \frac{P_{NH_3}^2}{P_{N_2}\cdot P_{H_2}^3} = \frac{(1.34)^2}{(0.33)\cdot(0)^3}\] Since \(P_{H_2}^3 = 0\), the formula is undefined. This indicates that we have made an error, or we have an incomplete reaction description. In real systems, it would be highly unlikely for hydrogen gas's pressure to be exactly zero. We would usually have an infinitesimal or tiny quantity present that couldn't be accurately measured. This situation should prompt us to revisiting the problem conditions or considering uncertainties. However, based on the given problem statement, we have to admit that computing \(K_p\) is not possible.