Question

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

Short answer

The equilibrium constant $K_p$ for the given reaction at $24^{\circ} \mathrm{C}$ is approximately 0.094.

Step-by-step solution

Calculate partial pressures

The total pressure of the system at equilibrium is given as P_total = 0.614 atm. To calculate the partial pressures of NH3(g) and H2S(g), we make use of the fact that both are in equal amounts. If x atm is the pressure of NH3(g), then the pressure of H2S(g) will also be x atm. Therefore, the total pressure can be expressed as: P_total = P_NH3 + P_H2S => 0.614 atm = x + x => x = 0.307 atm So, P_NH3 = P_H2S = 0.307 atm.

Calculate the equilibrium constant Kp

Now that we know the partial pressures of NH3(g) and H2S(g) at equilibrium, we can use them to find the equilibrium constant (Kp). For the given reaction, the Kp can be expressed as: Kp = P_NH3 * P_H2S Since the partial pressures of both NH3(g) and H2S(g) are found to be equal, we can plug in the values and find Kp: Kp = (0.307 atm) * (0.307 atm) => Kp = 0.094249 So, the equilibrium constant Kp for the given reaction at 24°C is approximately 0.094.

Key concepts

Equilibrium Constant

In the realm of chemical reactions, knowing how far a reaction proceeds helps us understand the behavior of chemicals in equilibrium. The equilibrium constant, denoted as \( K \), is a key concept here. It represents the ratio of concentrations of products to reactants at equilibrium. Specifically, when dealing with reactions involving gases, we often use the equilibrium constant in terms of pressure, represented as \( K_p \).

• \( K_p \) is valuable because it uses partial pressures instead of concentrations.
• It remains constant for a given reaction at a specific temperature.
• This constancy allows chemists to predict the position of equilibrium.

When \( K_p \) is much greater than 1, it indicates that products are favored at equilibrium. Conversely, a \( K_p \) much less than 1 suggests the reactants are favored. Understanding \( K_p \) helps in predicting how different conditions such as pressure and temperature changes can affect the direction of the reaction.

Partial Pressure

In studies of chemical equilibrium, especially for gaseous reactions, the concept of partial pressure is crucial. Partial pressure refers to the pressure exerted by a single gas in a mixture of gases. Each gas behaves as if it occupies the entire volume by itself, contributing to the total pressure observed in the system.

• The sum of all partial pressures equals the total pressure of the system.
• Mathematically, it is expressed as \( P_{ ext{total}} = P_{1} + P_{2} + \ldots \).

In our exercise, the total pressure was 0.614 atm, split equally between ammonia \( \text{NH}_3 \) and hydrogen sulfide \( \text{H}_2\text{S} \). This is a typical scenario when each product is generated in equal molar amounts.

Reaction Quotient

The reaction quotient, \( Q \), provides a snapshot of the state of a reaction regardless of whether it has reached equilibrium. It resembles the expression for an equilibrium constant, but it uses the current concentrations or pressures instead of those at equilibrium.

• The formula for \( Q \) is the same as that for \( K \).
• \( Q \) can help determine which direction a reaction needs to proceed to reach equilibrium.

If \( Q < K \), the reaction shifts forward towards the products. Conversely, if \( Q > K \), the reaction shifts backward towards the reactants. Understanding \( Q \) is essential for manipulating chemical processes to favor product formation. It serves as a guide to understanding the dynamics of equilibrium and the ongoing adjustments within a chemical system.