Question

At \(25^{\circ} \mathrm{C}\) the reaction $$ \mathrm{NH}_{4} \mathrm{HS}(s) \rightleftharpoons \mathrm{NH}_{3}(g)+\mathrm{H}_{2} \mathrm{~S}(g) $$ has \(K_{p}=0.120\). A 5.00-L flask is charged with \(0.300 \mathrm{~g}\) of pure \(\mathrm{H}_{2} \mathrm{~S}(\mathrm{~g})\) at \(25^{\circ} \mathrm{C}\). Solid \(\mathrm{NH}_{4} \mathrm{HS}\) is then added until there is excess unreacted solid remaining. (a) What is the initial pressure of \(\mathrm{H}_{2} \mathrm{~S}(\mathrm{~g})\) in the flask? (b) Why does no reaction occur until \(\mathrm{NH}_{4} \mathrm{HS}\) is added? (c) What are the partial pressures of \(\mathrm{NH}_{3}\) and \(\mathrm{H}_{2} \mathrm{~S}\) at equilibrium? (d) What is the mole fraction of \(\mathrm{H}_{2} \mathrm{~S}\) in the gas mixture at equilibrium? (e) What is the minimum mass, in grams, of \(\mathrm{NH}_{4} \mathrm{HS}\) that must be added to the flask to achieve equilibrium?

Short answer

The initial pressure of H2S in the flask is 0.0435 atm. No reaction occurs until NH4HS is added because the reaction requires solid NH4HS as one of the reactants to proceed. The partial pressures of NH3 and H2S at equilibrium are 0.0362 atm and 0.0073 atm, respectively. The mole fraction of H2S at equilibrium is 0.167. The minimum mass of NH4HS that needs to be added is 0.523 g.

Step-by-step solution

1. Calculate the initial pressure of H2S

To calculate the initial pressure of H2S, we first need to determine the number of moles of H2S using the given mass and molar mass. Then we can apply the ideal gas law to find the pressure. We are given 0.300 g of H2S. Molar mass of H2S = 34 g/mol. Using mass/molar mass, we can find the moles of H2S: Moles of H2S = 0.300 g / 34 g/mol = 0.00882 mol Now, using the ideal gas law (PV=nRT), we can determine the initial pressure of H2S: P = nRT/V Here, n = 0.00882 mol, R = 0.0821 Latm/molK (gas constant), T = 25°C (convert to Kelvin: 25 + 273 = 298 K) and V = 5.00 L. P = (0.00882 mol)(0.0821 Latm/molK)(298 K) / (5.00 L) = 0.0435 atm So, the initial pressure of H2S in the flask is 0.0435 atm.

2. Explain why no reaction occurs until NH4HS is added

No reaction occurs until NH4HS is added because the reaction requires solid NH4HS as one of the reactants to proceed. If there is no NH4HS present in the system, the reaction cannot occur. The reactants must come into contact with one another for a reaction to take place.

3. Calculate the partial pressures of NH3 and H2S at equilibrium

Use the equilibrium equation and Kp value to solve for equilibrium partial pressures: Kp = [NH3]*[H2S] / [NH4HS] Since NH4HS is a solid, its concentration does not change and the Kp equation becomes: 0.120 = [NH3]*[H2S] Let x be the partial pressure of NH3 and H2S at equilibrium. Since the reaction consumes 1 mole of H2S for every 1 mole of NH3 produced, the change in pressure for both gases will be the same (x atm). At equilibrium, the pressure of H2S is (0.0435 - x) atm and the pressure of NH3 is x atm. Now, substitute these pressures into the Kp equation: 0.120 = x(0.0435 - x) Now, solve for x: x^2 - 0.0435x + 0.120 = 0 Using the quadratic formula, we get a positive and a negative root. We discard the negative root as pressure cannot be negative. Hence, x = 0.0362 atm Therefore, the partial pressures of NH3 and H2S at equilibrium are 0.0362 atm and 0.0435 - 0.0362 = 0.0073 atm, respectively.

4. Calculate the mole fraction of H2S at equilibrium

To find the mole fraction of H2S at equilibrium, we need to find the total moles of gas and the moles of H2S. Mole fraction is defined as the moles of a component divided by the total moles in the mixture. Total moles of gas at equilibrium = moles of NH3 + moles of H2S Total moles of gas = (0.0362 atm * 5 L) / 0.0821 Latm/molK(298 K) + (0.0073 atm * 5 L) / 0.0821 Latm/molK(298 K) Total moles = 0.00987 + 0.00198 = 0.01185 mol Mole fraction of H2S (X_H2S) = moles of H2S / total moles X_H2S = 0.00198 / 0.01185 = 0.167 Thus, the mole fraction of H2S at equilibrium is 0.167.

5. Calculate the minimum mass of NH4HS required to achieve equilibrium

To find the minimum mass of NH4HS needed, we need to determine the moles of NH4HS reacted. In this case, the change in pressure of H2S corresponds to the moles of NH4HS used in the reaction (0.0435 atm - 0.0073 atm = 0.0362 atm). Moles of NH4HS reacted = Moles of H2S change Moles of NH4HS = (0.0362 atm * 5 L) / 0.0821 Latm/molK(298 K) = 0.00987 mol Now, determine the mass of NH4HS using the molar mass (Molar mass of NH4HS = 53 g/mol): Mass of NH4HS = moles * Molar mass Mass of NH4HS = 0.00987 mol * 53 g/mol = 0.523 g So, the minimum mass of NH4HS that needs to be added is 0.523 g.

Key concepts

Equilibrium Constant

When we speak about equilibrium constant, often denoted as K_eq or K_p for gas-phase reactions, we're addressing the quantitative measure of the extent of a reaction at equilibrium.

It plays a critical role in understanding how much reactants and products will be present when the system is in a state of balance, a state where the rates of the forward and reverse reactions are equal. The expression for the equilibrium constant depends on the balanced chemical equation and is a ratio of the concentrations (for solutions) or partial pressures (for gases) of the products over the reactants, each raised to the power of their stoichiometric coefficients.

For example, in the exercise, we have the equilibrium constant K_p because the products and reactants are gases, and it is expressed in terms of their partial pressures. The numerical value of K_p provides insight into the proportions of products and reactants at equilibrium; a smaller value suggests that reactants are favored, while a larger value indicates a product-favored reaction.

Partial Pressure

The term partial pressure is used to describe the pressure that a single gas in a mixture of gases would exert if it occupied the entire volume on its own. It's a way to understand the contribution of each gas within a container. The partial pressure of each gas is directly proportional to its mole fraction in the mixture.

In our exercise, we determine the initial partial pressure of H₂S and then find the partial pressures of ammonia (NH₃) and hydrogen sulfide (H₂S) at equilibrium. These are vital because they influence how we use the equilibrium constant to measure the system's position at equilibrium. We use the ideal gas law and equilibrium expressions to calculate these partial pressures as key steps towards understanding the system.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure (P), volume (V), number of moles (n), and temperature (T) of a theoretical ideal gas. The law is commonly expressed as PV = nRT, where R is the universal gas constant.

For the chemical equilibrium problem, we use the ideal gas law to calculate the initial pressure of H₂S gas. By knowing the mass of H₂S and its molar mass, we can determine the moles and then apply the law to find pressure. This step is integral as it sets the stage for determining how much of the solid reactant needs to be added to achieve equilibrium and later for calculating mole fractions.

Mole Fraction

A mole fraction quantifies the composition of a component in a mixture by dividing the number of moles of that component by the total number of moles of all substances within the mixture. Represented by X, it is a dimensionless quantity which is very useful in the gas phase as partial pressure calculations often require it.

In our exercise, we use mole fractions to establish the proportion of H₂S in the gas mixture at equilibrium. By dividing the moles of H₂S by the total moles of gas (which includes both NH₃ and H₂S post-reaction), we are able to solve for the mole fraction of H₂S. This helps in understanding the behavior of each gas in the mixture and in further calculations concerning the system’s equilibrium.