Question

At a certain temperature, the equilibrium constant for the following reaction is 0.0472 . $$ \mathrm{NO}(g)+\mathrm{SO}_{3}(g) \rightleftharpoons \mathrm{SO}_{2}(g)+\mathrm{NO}_{2}(g) $$ All gases are at an initial pressure of \(0.862 \mathrm{~atm} .\) (a) Calculate the partial pressure of each gas at equilibrium. (b) Compare the initial total pressure with the total pressure of the gases at equilibrium. Would that relation be true of all gaseous systems?

Short answer

Short answer: (a) The partial pressures of the gases at equilibrium are: $$ P_{NO} = 0.579 \, \mathrm{atm} $$ $$ P_{SO3} = 0.579 \, \mathrm{atm} $$ $$ P_{SO2} = 0.283 \, \mathrm{atm} $$ $$ P_{NO2} = 0.283 \, \mathrm{atm} $$ (b) The initial total pressure (1.724 atm) is equal to the total pressure of the gases at equilibrium (1.724 atm) in this specific example because there are two moles of gas reacting and two moles of gas being produced, resulting in no overall change in the total number of moles. If the number of moles of reacting gases was different from the number of moles of produced gases, the relation would not hold true.

Step-by-step solution

Write the equilibrium expression.

The equilibrium expression is given by the equilibrium constant K, with pressure in this case, written as follows: $$ K_p = \frac{P_{SO2} \cdot P_{NO2}}{P_{NO} \cdot P_{SO3}} $$

Define the changes in pressure at equilibrium.

Let x represent the change in pressure for the gases involved in the reaction. The initial pressure of each gas is 0.862 atm. At equilibrium, the pressure of NO and SO3 will decrease by x, and the pressure of SO2 and NO2 will increase by x: $$ P_{NO} = 0.862 - x $$ $$ P_{SO3} = 0.862 - x $$ $$ P_{SO2} = x $$ $$ P_{NO2} = x $$

Substitute the expressions into the equilibrium constant expression.

Replace the pressures in the equilibrium expression with the expressions found in Step 2: $$ 0.0472 = \frac{x \cdot x}{(0.862 - x) \cdot (0.862 - x)} $$

Solve for x.

Now we need to solve for x, the change in pressure at equilibrium: $$ x^2 = 0.0472(0.862-x)^2 $$ $$ x^2 = 0.0472(0.743444 - 1.724x + x^2)$$ $$ x^2 = 0.035 \, 1801 - 0.081 \, 2968x + 0.0472x^2 $$ To solve for x, we can rearrange the equation to form a quadratic equation: $$ 0.9528x^2 + 0.081 \, 2968x - 0.035 \, 1801 = 0 $$ Using the quadratic formula, we find that x ≈ 0.283 atm.

Calculate the partial pressures of each gas at equilibrium.

Substitute the value of x back into the expressions for the pressures of each gas: $$ P_{NO} = 0.862 - 0.283 = 0.579 \, \mathrm{atm} $$ $$ P_{SO3} = 0.862 - 0.283 = 0.579 \, \mathrm{atm} $$ $$ P_{SO2} = 0.283 \, \mathrm{atm} $$ $$ P_{NO2} = 0.283 \, \mathrm{atm} $$

(a) Partial pressures of each gas at equilibrium

The partial pressures of the gases at equilibrium are: $$ P_{NO} = 0.579 \, \mathrm{atm} $$ $$ P_{SO3} = 0.579 \, \mathrm{atm} $$ $$ P_{SO2} = 0.283 \, \mathrm{atm} $$ $$ P_{NO2} = 0.283 \, \mathrm{atm} $$

Compare the initial total pressure with the total pressure of the gases at equilibrium.

The initial total pressure is 2 * 0.862 atm since there are two moles of gas initially: $$ P_{initial} = 2 \cdot 0.862 = 1.724 \, \mathrm{atm} $$ The total pressure at equilibrium is the sum of the partial pressures of the gases: $$ P_{equilibrium} = 0.579 + 0.579 + 0.283 + 0.283 = 1.724 \, \mathrm{atm} $$

(b) Comparison of initial and equilibrium total pressures

The initial total pressure is equal to the total pressure of the gases at equilibrium in this case (1.724 atm). This relation will not always hold true for all gaseous systems. It only happens to be true in this specific example because there are two moles of gas reacting and two moles of gas being produced, resulting in no overall change in the total number of moles. If the number of moles of reacting gases was different from the number of moles of produced gases, the relation would not hold true.

Key concepts

Partial Pressure

Partial pressure is a crucial concept when it comes to gaseous reactions and equilibria. It refers to the pressure that would be exerted by one of the gases in a mixture if it alone occupied the entire volume of the mixture at the same temperature. In essence, it's the 'part' of the total pressure that is due to each individual gas and can be calculated using Dalton's law of partial pressures. In the solution of our exercise, we learned to calculate the partial pressure at equilibrium by finding the change in pressure (represented by x) and then adjusting the initial pressures accordingly.

For example, if a reaction starts with all reactants and products at a pressure of 0.862 atm and the change in pressure at equilibrium is 0.283 atm, we can determine that the partial pressures of the reactants decrease by this change, while the pressures of the products increase. Understanding how to calculate these pressures is essential for predicting how the reaction proceeds under different conditions.

Gaseous Equilibrium

Gaseous equilibrium occurs when the rates of the forward and reverse reactions are equal, leading to constant concentrations of products and reactants over time. In our given reaction, the gaseous system reaches equilibrium when the process of NO and SO3 turning into SO2 and NO2 occurs at the same rate as the reverse reaction. At this point, the partial pressures of all species involved in the reaction are constant.

The significance of understanding gaseous equilibria lies in its ability to predict the behavior of gases during reactions. For students, grasping this concept of dynamic balance - where changes still occur but with no net effect on concentration or pressure - is often a stepping stone to mastering more complex chemical principles.

Equilibrium Expression

Right at the heart of chemical equilibria lies the equilibrium expression. The equilibrium constant, designated as K, quantifies the ratio of product concentrations to reactant concentrations at equilibrium. When dealing with gases, we often use partial pressures instead of concentrations, hence the equilibrium expression might involve K_p, as seen in the problem we worked on. For the reaction NO(g) + SO3(g) ⇌ SO2(g) + NO2(g), the equilibrium expression is given by \[ K_p = \frac{P_{SO2} \cdot P_{NO2}}{P_{NO} \cdot P_{SO3}} \].

This expression shows the relationship between the partial pressures of the gases at equilibrium. By manipulating the equilibrium expression - either by calculation as seen in the step-by-step solution or conceptually through Le Chatelier's principle - chemists can predict how changes in conditions might affect the equilibrium state.

Le Chatelier's Principle

Le Chatelier's principle is a guideline for predicting how a system at equilibrium reacts to changes in concentration, temperature, or pressure. Essentially, if a stress is applied to a system at equilibrium, the system will adjust to counteract that stress. For instance, if we increase the pressure by decreasing the volume of the container for our given reaction, the system will try to reduce that pressure. This often involves the reaction shifting to favor the side with fewer moles of gas.

In the exercise, when asked if the initial total pressure is always equal to the total pressure at equilibrium, the answer is 'no' due to Le Chatelier’s principle. Different reactions involve different numbers of moles of gases, so changes in conditions such as temperature or pressure can shift the reaction to favor the production of either more or fewer moles, thus changing the total pressure at equilibrium.