Question

If \(0.3\) moles of hydrogen gas and \(2.0\) moles of sulphur solid are heated to \(87^{\circ} \mathrm{C}\) in a \(2.0 \mathrm{~L}\) vessel, what will be the partial pressure of \(\mathrm{H}_{2} \mathrm{~S}\) gas at equilibrium? (Given: \(R=0.081-\mathrm{atm} / \mathrm{K}-\mathrm{mol}\) ) \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{S}(\mathrm{s}) \rightleftharpoons \mathrm{H}_{2} \mathrm{~S}(\mathrm{~g}) ; K_{\mathrm{c}}=0.08\) (a) \(0.32 \mathrm{~atm}\) (b) \(0.43 \mathrm{~atm}\) (c) \(0.62 \mathrm{~atm}\) (d) \(0.48 \mathrm{~atm}\)

Short answer

The partial pressure of H2S at equilibrium is approximately 0.48 atm, which matches option (d).

Step-by-step solution

Determine the Equilibrium Concentration of H2S

According to the balanced chemical equation, 1 mole of hydrogen gas reacts with 1 mole of sulfur solid to produce 1 mole of hydrogen sulfide gas. Let's denote the change in concentration of hydrogen gas as '-x', meaning at equilibrium the concentration of hydrogen gas is 0.3 - x moles/L. Since solid sulfur doesn't affect the equilibrium expression, we focus on gas species. For hydrogen sulfide, the change will be '+x', leading to an equilibrium concentration of x moles/L. The reaction quotient Q can be compared with the equilibrium constant Kc. In this case, we can write the equilibrium constant expression as Kc = [H2S]. Given Kc = 0.08, we can equate this to 'x' as the other substances are solids or in excess.

Calculate the Equilibrium Concentration of H2S

Using the ICE table method (Initial, Change, Equilibrium), we have the initial concentrations (in moles/L): [H2] = 0.3/2, [S] is in excess so we don't include it in calculations, and [H2S] = 0. We then have the changes: [H2] changes by -x, and [H2S] changes by +x. At equilibrium, [H2S] = x, and Kc = 0.08 = x. Therefore, the equilibrium concentration of H2S is 0.08 moles/L.

Convert Moles to Pressure

To find the partial pressure of H2S, we use the ideal gas law equation, PV = nRT. We have n (the number of moles of H2S) = 0.08, R (the ideal gas constant) = 0.081 atm/K/mol, T (the temperature in K) = 87 + 273 = 360 K, and V (the volume of the vessel) = 2.0 L. Plugging these values into the ideal gas law equation, we get P = (nRT)/V.

Calculate the Partial Pressure of H2S

Using the values from Step 3, the partial pressure of H2S can be calculated as P = (0.08 moles * 0.081 atm/mol/K * 360 K) / 2.0 L = 0.11664 atm. Since this result does not match any of the answer choices, we need to recognize that not all 0.3 moles of hydrogen will react. Our initial approach assumed 0.3 moles would react completely into H2S, but we know that at equilibrium, some H2 will remain unreacted.

Re-evaluate the Equilibrium Concentration

We need to consider the value of x when calculating the equilibrium concentration. Since x equals Kc and the amount of moles of H2S at equilibrium, we have: 0.3 - x equals the concentration of H2 at equilibrium. Plugging these values back into the original equilibrium constant expression, we get Kc = x / (0.3 - x).

Solve for x

From the equilibrium expression Kc = 0.08 = x / (0.3 - x), we can set up the equation: 0.08(0.3 - x) = x. Solving for x, we get 0.024 - 0.08x = x, which simplifies to 0.024 = 1.08x, thus x = 0.024 / 1.08. The value of x represents the moles of H2S gas formed per liter at equilibrium.

Calculate the Final Partial Pressure of H2S

After solving the value of x in step 6, plug the value back into the ideal gas law to get the partial pressure. The number of moles of H2S (n) is 0.024 moles/L * 2.0 L = 0.048 moles. Use the ideal gas law again: P = (0.048 moles * 0.081 atm/mol/K * 360 K) / 2.0 L to get the final partial pressure.

Key concepts

Equilibrium Constant

Understanding equilibrium in chemical reactions is crucial for working out many chemistry problems. The equilibrium constant, represented by the symbol \( K_c \), is a measure of the extent of a reaction at equilibrium—it tells us how far the reaction proceeds before halting.

In the given exercise, we have the equilibrium constant \( K_c = 0.08 \) for the reaction between hydrogen gas (H2) and solid sulfur (S) to produce hydrogen sulfide gas (H2S). The \( K_c \) value suggests that at equilibrium, the amount of H2S produced is relatively low compared to the amount of reactants. Since reactants and products are in a dynamic balance at equilibrium, the \( K_c \) expression for this reaction is based solely on the concentration of the gaseous product (H2S), as the concentration of solid sulfur remains constant and is not included in the expression.

This number, \( K_c \), not only serves as an essential guide for calculating changes during the reaction but also helps to determine at which point the reaction will settle into a state of balance. It's essential to understand that \( K_c \) is temperature-dependent and can vary under different conditions, which emphasizes the need for keen attention to the given experimental setup.

ICE Table Method

The ICE table method stands for Initial, Change, and Equilibrium. It's an organized approach to solving equilibrium problems, helping to keep track of concentrations throughout a chemical reaction.

Initially, we list the initial concentrations of reactants and products before any reaction has occurred. Next, we note the changes in concentration as the reaction progresses towards equilibrium. Finally, we document the concentrations of all species at equilibrium.

For our exercise, we commence with an initial concentration of hydrogen gas and sulfur. The sulfur, being a solid, won't appear in the ICE table. As the reaction proceeds, hydrogen is consumed, and hydrogen sulfide gas forms. By establishing that these changes in concentration are represented by \( -x \) and \( +x \) respectively, we can deduce the concentration of all species at equilibrium.

The change part of the ICE table is where students often get confused, mistaking the equilibrium constant for an immediate solution to the reaction. However, it's essential to realize that the equilibrium constant represents the ratio of products to reactants at equilibrium, which requires us to solve for 'x' thoughtfully. This method becomes powerful when we remember that the changes are the same as the stoichiometry of the balanced equation.

Ideal Gas Law

The ideal gas law is a cornerstone of chemistry and physics, encapsulating the relationship between pressure (P), volume (V), the number of moles (n), and temperature (T) for an ideal gas. Represented by the equation \( PV=nRT \), where R is the ideal gas constant, the law approximates the behavior of real gases under a variety of circumstances but assumes no intermolecular forces and that gas particles occupy no space, which is not true in reality but often close enough to be practical.

In our problem, we use the ideal gas law to calculate the partial pressure of hydrogen sulfide at equilibrium. After determining the moles of H2S produced at equilibrium using the ICE table method and \( K_c \), we then apply the ideal gas law by inserting the relevant values for \( n \), \( R \), \( T \) (adjusted to Kelvins), and \( V \).

It's important to note that this law allows us to convert between moles and pressure—a key step in solving this type of problem. The ideal gas law proves especially useful when dealing with gases at equilibrium because it directly relates observable properties, like pressure, to the microscopic aspects of moles and temperature, completing the link between theory and practical measurement.