Question

A sealed flask has \(0.541 \mathrm{~atm}\) of \(\mathrm{SO}_{3}\) at \(1000 \mathrm{~K}\). The following equilibrium is established. $$ 2 \mathrm{SO}_{3}(g) \rightleftharpoons 2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) $$ At equilibrium, the partial pressure of oxygen is measured to be 0.216 atm. Calculate \(K\) for the decomposition of \(\mathrm{SO}_{3}\) at \(1000 \mathrm{~K}\)

Short answer

Answer: The equilibrium constant (K) for the decomposition of SO₃ at 1000 K is approximately 0.095.

Step-by-step solution

Initial situation

Initially, only SO₃ is present, so the initial pressures can be represented as follows: \([SO_3] = 0.541 \mathrm{~atm}\) \([SO_2] = 0 \mathrm{~atm}\) \([O_2] = 0 \mathrm{~atm}\)

At equilibrium

At equilibrium, the given oxygen partial pressure is 0.216 atm. Let's denote the change in molarity occurring during the reaction as x. From the balanced chemical equation, for every mole of O₂ produced, two moles of SO₂ are produced and two moles of SO₃ are consumed. So when x moles of O₂ are produced, the change in partial pressures becomes: \([SO_3] = 0.541 - 2x \mathrm{~atm}\) \([SO_2] = 2x \mathrm{~atm}\) \([O_2] = 0.216 \mathrm{~atm}\)

Determine x for SO₂ and SO₃ at equilibrium

Since O₂ is produced at half the rate of SO₂, this means two moles of SO₂ are produced for every 1 mole of O₂ formed at equilibrium. $$x = \frac{[O_2]}{2} = \frac{0.216}{2} = 0.108 \mathrm{~atm}$$ Now we can find the partial pressures of SO₂ and SO₃ at equilibrium: \([SO_3] = 0.541 - 2 (0.108) = 0.325 \mathrm{~atm}\) \([SO_2] = 2 (0.108) = 0.216 \mathrm{~atm}\)

Calculate K

Now that we have the partial pressures of SO₂, SO₃, and O₂ at equilibrium, we can calculate the equilibrium constant (K) using the equilibrium expression: $$K = \frac{[SO_2]^2 \times [O_2]}{[SO_3]^2} = \frac{(0.216)^2 \times 0.216}{(0.325)^2}$$ Calculating K: $$K = \frac{0.0100656}{0.105625} = 0.0953$$ So the equilibrium constant for the decomposition of SO₃ at 1000 K is approximately 0.095.

Key concepts

Chemical Equilibrium

When a chemical reaction occurs, the reactants can transform into products and vice versa. At some point, the rate of the forward reaction (reactants forming products) becomes equal to the rate of the backward reaction (products reforming reactants), leading to a state known as chemical equilibrium. At this stage, the concentrations of the reactants and products remain constant over time, not because the reactions have stopped, but because they are proceeding at an equal and steady rate in both directions.

Understanding chemical equilibrium is crucial as it governs the yields of chemical processes, the design of chemical reactors, and even the behavior of substances in biological systems. It's important to remember that equilibrium can be influenced by changes in concentration, pressure, volume, or temperature, as described by Le Chatelier's principle. In practical terms, this principle indicates that if you alter the conditions of a reaction at equilibrium, the system will adjust to partially counteract the change.

Partial Pressure

In the context of gas-phase reactions, partial pressure is a term used to describe the pressure exerted by a single gas in a mixture of gases. It's essentially how much pressure one gas contributes to the overall pressure if it existed alone in the same volume. The sum of the partial pressures of all gases in a mixture equals the total pressure exerted by the gas mixture.

To apply this when dealing with equilibrium in gas-phase reactions, like the decomposition of SO₃, we use the partial pressures of the reactants and products instead of their concentrations. The concept of partial pressure is rooted in Dalton's Law, which states that each gas in a mixture behaves independently. This law makes it possible to write the equilibrium constant expression in terms of partial pressures for reactions involving gases.

Reaction Quotient

The reaction quotient (Q) is a measure that tells us whether a reaction at a given set of conditions is at equilibrium, and if not, in which direction it has to shift to reach equilibrium. It is calculated in the same way as the equilibrium constant (K), by taking the ratio of the concentrations (for solutions) or partial pressures (for gases) of the products raised to the power of their stoichiometric coefficients to those of the reactants raised to their respective powers.

If Q is less than K, the forward reaction is favored, and the reaction will proceed to the right to form more products. If Q is greater than K, the backward reaction is favored, and the reaction will shift to the left to form more reactants. When Q equals K, the system is at equilibrium, and no net change occurs in the concentrations of the reactants and the products.

By calculating the reaction quotient, you can predict the direction of the reaction progress. This is crucial for chemists who may be looking to optimize a reaction for the best possible yield of a desired product.