Question

Halving the volume of a gas doubles its pressure. Using the reaction quotient corresponding to \(K_{\mathrm{P}}\), explain why halving the volume shifts the following equilibrium to the left. $$ \mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g) $$

Short answer

Halving the volume of the gas mixture results in doubling the pressure. For the reaction \(N_2O_4(g) \rightleftharpoons 2NO_2(g)\), the increased pressure causes the equilibrium to shift left, favoring the formation of \(N_2O_4\) to reduce pressure, as it has a smaller number of gas molecules.

Step-by-step solution

Understand the Relationship Between Volume and Pressure

According to Boyle's Law, at a constant temperature, the volume of a gas is inversely proportional to its pressure. Thus, when the volume of a gas is halved, its pressure is doubled.

Identify the Reaction Quotient for the Equilibrium

For the given equilibrium reaction: \(\mathrm{N}_2\mathrm{O}_4(g) \rightleftharpoons 2\mathrm{NO}_2(g)\), the reaction quotient \(Q_P\) is defined as the ratio of the partial pressures of the products to the reactants raised to the power of their stoichiometric coefficients in the balanced equation.

Describe the Effect of Increased Pressure on the Reaction Quotient and the Equilibrium Shift

When volume is halved and pressure is doubled, the partial pressures of the gases involved change. For the reaction quotient \(Q_P\), the partial pressure of \(NO_2\) will have a larger effect because it is squared in the reaction quotient equation. This increase in pressure will result in a higher \(Q_P\) compared to \(K_P\), causing the reaction to shift towards the reactants to reduce pressure, which results in the formation of \(N_2O_4\), hence the equilibrium shifts to the left.

Key concepts

Boyle's Law

Boyle's Law is a fundamental principle of gas behavior that relates the pressure of a gas to its volume at constant temperature. According to this law, for a fixed amount of gas at constant temperature, the pressure exerted by the gas is inversely proportional to the volume it occupies. Mathematically, Boyle's Law is expressed as:
\[ P_1V_1 = P_2V_2 \]
where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the pressure and volume after a change. In practical terms, if you decrease the volume of a contained gas (assuming temperature remains constant), the pressure increases proportionally. As a result, when the exercise mentions halving the volume of a gas, Boyle's Law explains why its pressure doubles, assuming no change in temperature.
This principle is crucial when considering the behavior of gases under various conditions and plays a direct role in understanding how changes in volume and pressure can influence chemical reactions that involve gases.

Reaction Quotient

The reaction quotient, represented as \(Q_P\), is a measure of the relative amounts of products and reactants present during a reaction at a given instant. It is similar to the equilibrium constant, but unlike the equilibrium constant which applies at equilibrium, the reaction quotient applies to any point in time during the reaction. For gaseous reactions, the reaction quotient is based on the partial pressures of the reactants and products. The formula to find \(Q_P\) is:
\[ Q_P = \frac{{(P_{\text{product}})^{\text{coefficient of product}}}}{{(P_{\text{reactant}})^{\text{coefficient of reactant}}}} \]
In the context of our exercise involving the equilibrium between \(\text{N}_2\text{O}_4\) and \(2 \text{NO}_2\), the reaction quotient would be calculated by squaring the partial pressure of \(\text{NO}_2\) since its coefficient is 2, and dividing it by the partial pressure of \(\text{N}_2\text{O}_4\). When the system is not at equilibrium, the reaction quotient will shift the reaction towards the equilibrium state (either to the left or the right) in its attempt to achieve the equilibrium constant, \(K_P\).

Le Chatelier's Principle

Le Chatelier's Principle is a key concept in chemical equilibrium, providing insight into how a system at equilibrium responds to changes in concentration, pressure, volume, or temperature. It states that if an external change is applied to a system at equilibrium, the system will adjust itself to counteract that change and re-establish equilibrium.
For example, increasing the pressure of an equilibrium system by decreasing the volume favours the reaction that produces fewer gas molecules. This principle helps us understand why in our exercise, reducing the volume causes the equilibrium to shift to the left. The original reaction, \(\text{N}_2\text{O}_4(g) \rightleftharpoons 2 \text{NO}_2(g)\), goes from one molecule of \(\text{N}_2\text{O}_4\) to two molecules of \(\text{NO}_2\). By Le Chatelier's Principle, the system will shift towards the side with fewer gas molecules to relieve the increased pressure, which is towards the reactant, \(\text{N}_2\text{O}_4\).
Understanding Le Chatelier's Principle is vital for predicting how different changes will affect a chemical reaction and can be applied to optimize industrial chemical processes, such as in the synthesis of ammonia in the Haber process.