Problem 79 - Chemistry: The Central Science

Question

Solid \(\mathrm{NH}_{4} \mathrm{SH}\) is introduced into an evacuated flask at \(24^{\circ} \mathrm{C}\). The following reaction takes place: $$ \mathrm{NH}_{4} \mathrm{SH}(s) \rightleftharpoons \mathrm{NH}_{3}(g)+\mathrm{H}_{2} \mathrm{~S}(g) $$ At equilibrium, the total pressure (for \(\mathrm{NH}_{3}\) and \(\mathrm{H}_{2} \mathrm{~S}\) taken together) is \(0.614 \mathrm{~atm}\). What is \(K_{p}\) for this equilibrium at \(24^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

The equilibrium constant \(K_{p}\) for the reaction at \(24^{\circ} \mathrm{C}\) is \(0.094249\).

Step 1: Find the individual partial pressures of NH3 and H2S at equilibrium

Since the total pressure is given as \(0.614 \mathrm{~atm}\), and we assumed that \(P_{NH_3} = P_{H_2S}\), we can distribute the total pressure equally between the two gases: $$P_{NH_3} = P_{H_2S} = \frac{0.614 \mathrm{~atm}}{2} = 0.307 \mathrm{~atm}$$

Step 2: Determine the value of standard pressure

In this step, we will calculate the value of the standard pressure (\(P^\circ\)) used in the calculation of \(K_{p}\). Since the temperature is \(24^{\circ} \mathrm{C}\), we know that $$P^\circ = 1 \mathrm{~atm}$$

Step 3: Calculate the equilibrium constant Kp

Now that we have the partial pressures of \(\mathrm{NH}_{3}\) and \(\mathrm{H}_{2} \mathrm{~S}\), and the standard pressure, we can calculate the equilibrium constant \(K_{p}\) by using the formula: $$K_{p} = \frac{P_{NH_3}P_{H_2S}}{P^\circ} = \frac{(0.307 \mathrm{~atm})(0.307 \mathrm{~atm})}{1 \mathrm{~atm}}$$ Now, we'll calculate the value: $$K_{p} = \frac{0.094249 \mathrm{~atm}^2}{1 \mathrm{~atm}} = 0.094249$$ Thus, the equilibrium constant \(K_{p}\) for this reaction at \(24^{\circ} \mathrm{C}\) is \(0.094249\).

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. This balance means that the concentrations of the reactants and products remain constant over time, not that they are equal to each other. It's a dynamic process - molecules are constantly moving between reactants and products, but because the rates are the same, there is no net change.

Imagine a dance floor where dancers represent molecules. Equilibrium is like having an equal number of dancers switching between two dance styles at any given moment, maintaining a steady rhythm in the dance party. Similarly, at the equilibrium point in a chemical reaction, there is a constant 'dance' of molecules shifting back and forth, but the overall 'dance' of the reaction remains unchanged.

Partial Pressures

Partial pressures refer to the pressure exerted by a single gas in a mixture of gases. Each gas in a mixture behaves independently and contributes to the total pressure of the system. This is based on Dalton's Law of Partial Pressures, which states that the total pressure in a mixture of non-reacting gases is equal to the sum of the individual pressures that each gas would exert if it were alone in the container.

Think of it as a team sport where each player contributes to the score. In a basketball game, each player’s points are like the partial pressures, and the final score is like the total pressure. The performance of each player adds up to the final outcome, just as in a gas mixture, each gas's partial pressure adds to the total pressure. In the case of the chemical reaction we're looking at, the partial pressures of NH3 and H2S contribute to the overall pressure inside the flask.

Equilibrium Constant Kp

The equilibrium constant for reactions involving gases, symbolized as Kp, is a number that expresses the relationship between the partial pressures of the products and reactants at equilibrium. It’s calculated using the equation: $$K_{p} = \frac{(P_{products})^n}{(P_{reactants})^m}$$ where n and m are the stoichiometric coefficients of the balanced equation, and P represents the partial pressures of the gases.

When a reaction reaches equilibrium at a given temperature, the Kp value remains constant. It indicates the extent to which a reaction proceeds; a high Kp value means the products are favored at equilibrium, while a low Kp suggests the reactants are favored. In our example, the balanced chemical equation has equal stoichiometric coefficients for NH3 and H2S, thus the powers of their partial pressures are both to the first power when calculating Kp. The calculated Kp value provides insight into the reaction's behavior at the specific temperature of 24°C.