Question

When sulphur is heated at \(900 \mathrm{~K}, \mathrm{~S}_{\mathrm{g}}\) is converted to \(S_{2}\). What will be the equilibrium constant for the reaction if initial pressure of 1 atm falls by \(25 \%\) at equilibrium? (a) \(0.75 \mathrm{~atm}^{3}\) (b) \(2.55 \mathrm{~atm}^{3}\) (c) \(25.0 \mathrm{~atm}^{3}\) (d) \(1.33 \mathrm{~atm}^{3}\)

Short answer

The equilibrium constant for the reaction is 0.25 atm, which would match choice (a) if the units were given correctly. However, due to the correction in stoichiometry and the misunderstanding in the options, no given choice is correct.

Step-by-step solution

Write the balanced chemical equation

First, we must establish the balanced chemical equation for the reaction. It is not provided, but based on the information, the reaction is: \[ S_{g} \longrightarrow S_{2} \]

Calculate the change in pressure

The initial pressure of sulfur is given to be 1 atm. At equilibrium, this pressure falls by 25%, which means the pressure of Sg decreases by \(0.25 \times 1 \text{ atm} = 0.25 \text{ atm}\).

Calculate the equilibrium pressures

Let the equilibrium pressure of Sg be \(P\). Then the equilibrium pressure is \(P = 1 \text{ atm} - 0.25 \text{ atm} = 0.75 \text{ atm}\). Now since the stoichiometry of the reaction is 1:1, the pressure of S2 at equilibrium will also be 0.25 atm.

Write the expression for the equilibrium constant

The equilibrium constant expression (Kp) for the given reaction is: \[ K_p = \frac{(P_{S2})^2}{(P_{Sg})} \]

Calculate the equilibrium constant Kp

Substitute the equilibrium pressures into the equilibrium constant expression: \[ K_p = \frac{(0.25 \text{ atm})^2}{(0.75 \text{ atm})} = \frac{0.0625 \text{ atm}^2}{0.75 \text{ atm}}\]\[ K_p = \frac{1}{12} \text{ atm} = 0.0833 \text{ atm}^{-1} \text{(to four significant figures)} \] However, the given options suggest a mistake might have happened, either in the formulation of the equilibrium expression or in the interpretation of the changes in pressure. Please double check the steps and the initial assumptions about the equation and stoichiometry.

Review and correct the mistake

Reviewing the earlier steps, we find the stoichiometry used in the equilibrium constant expression is incorrect. The correct stoichiometry according to the balanced chemical reaction is a disproportionation in which one mole of Sg produces half a mole of S2. Thus, the correct equilibrium constant expression should consider the stoichiometry and is given by: \[ K_p = (P_{S2})^1 \]Now, we can calculate the correct equilibrium constant using the pressure of S2: \[ K_p = (0.25 \text{ atm})^1 = 0.25 \text{ atm} \]The correct answer reflects the fact that the pressure of S2 rises to 0.25 atm while Sg is 0.75 atm. Since the reaction has a 1:2 stoichiometry, there's no need to square the pressure of S2 in the expression for Kp.

Key concepts

Chemical Equilibrium

In the realm of chemistry, when we talk about chemical equilibrium, we’re referring to the state in which the concentrations of reactants and products in a chemical reaction are no longer changing over time. It implies that the forward and reverse reactions are occurring at the same rate. This balance, however, doesn't mean that the reactants and products are present in equal amounts but rather that their ratios remain constant.

To quantify the state of equilibrium, we use the equilibrium constant denoted as K or K_eq. For reactions occurring in the gas phase, this constant is often in terms of pressures (K_p) and relates the partial pressures of the gases involved. The general expression for K_p takes into account the stoichiometry of the reaction and is given by the ratio of the products' pressure raised to their stoichiometric coefficients to that of the reactants.

Understanding chemical equilibrium is crucial because it shows the extent to which a reaction proceeds and predicts how the system will respond to changes in conditions, such as pressure, temperature, or concentration.

Pressure Change and Equilibrium

The position of equilibrium can shift as a result of changes in pressure, a concept that is key in understanding how systems respond to physical changes in their surroundings. According to Le Chatelier’s Principle, if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium will move to counteract the change.

In a closed system containing gases, increasing the pressure favors the formation of fewer moles of gas. Conversely, decreasing the pressure favors the side with more moles of gas. This is easily visualized through the ideal gas law where pressure is inversely related to volume, assuming constant temperature. When the pressure of the system in our exercise decreased, it triggered a shift in equilibrium, impacting both the amount of reactants and products and the value of the equilibrium constant.

Stoichiometry

Stoichiometry is the section of chemistry that involves calculating the amounts of reactants and products in a chemical reaction. It is based on the conservation of mass and the fact that elements cannot be created or destroyed in a chemical reaction, hence amounts need to balance according to the reaction's stoichiometry. In the reaction discussed in our exercise, sulfur gas (S_g) is converted into S₂, with a stoichiometry that directly affects the equilibrium constant's calculation.

Impact of Stoichiometry on Equilibrium

The correct equilibrium constant expression must reflect the stoichiometry of the balanced equation; in this case, for every one mole of S_g used, half a mole of S₂ is produced. This is vital for students to grasp since it impacts how the partial pressures are used in calculating the equilibrium constant K_p. Incorrect stoichiometry leads to wrong predictions about the reaction's behavior, which is the mistake initially made in the solution provided.