Problem 61 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

At \(1000^{\circ} \mathrm{C}\) and a pressure of \(16 \mathrm{~atm}\), the equilibrium constant of the reaction: \(\mathrm{CO}_{2}(\mathrm{~g})+\mathrm{C}(\mathrm{s}) \rightleftharpoons 2 \mathrm{CO}(\mathrm{g})\) is such that for every nine moles of \(\mathrm{CO}\), there is one mole of \(\mathrm{CO}_{2}\). For what pressure of the mixture, is the ratio \(\mathrm{CO}: \mathrm{CO}_{2}=4: 1 ?\) The temperature remains \(1000^{\circ} \mathrm{C}\). (a) \(40.5 \mathrm{~atm}\) (b) 81 atm (c) \(33.75 \mathrm{~atm}\) (b) \(6.7 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The total pressure of the mixture when the ratio \(\mathrm{CO}: \mathrm{CO}_{2}=4:1\) is \(40.5 \mathrm{~atm}\).

Step 1: Understand the Given Information

The problem gives the equilibrium for the reaction \( \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{C}(\mathrm{s}) \rightleftharpoons 2\mathrm{CO}(\mathrm{g}) \). It is stated that initially for every nine moles of \( \mathrm{CO} \) produced, there is one mole of \( \mathrm{CO}_{2} \) at a pressure of 16 atm. The goal is to find a new pressure where the ratio of \( \mathrm{CO} \) to \( \mathrm{CO}_{2} \) is 4:1.

Step 2: Define the Equilibrium Expression

The equilibrium constant expression for the reaction based on the mole ratio is \(K_p = \frac{\left[CO\right]^2}{\left[CO_2\right]} = \frac{P_{CO}^2}{P_{CO_2}}\) since \( \mathrm{C}(\mathrm{s}) \) has no effect on the pressure.

Step 3: Calculate the Initial Partial Pressures

Given the 9:1 mole ratio at 16 atm and the total moles of gases as 10 (nine moles of \( \mathrm{CO} \) and one mole of \( \mathrm{CO}_{2} \) ), we can calculate the partial pressures as \((P_{CO})_{initial} = \frac{9}{10} \times 16 \mathrm{~atm} = 14.4 \mathrm{~atm}\), \((P_{CO_2})_{initial} = \frac{1}{10} \times 16 \mathrm{~atm} = 1.6 \mathrm{~atm}\).

Step 4: Apply the Equilibrium Constant

Using the initial conditions, solve for the equilibrium constant: \(K_p = (P_{CO})_{initial}^2 / (P_{CO_2})_{initial} = \frac{(14.4 \mathrm{~atm})^2}{1.6 \mathrm{~atm}} \). Calculate \(K_p \) to use in the next steps.

Step 5: Calculate the New Partial Pressures with the Desired Ratio

With the desired 4:1 ratio, if \( P_{CO_2} \) is \( x \), then \( P_{CO} \) is \( 4x \).

Step 6: Apply the Equilibrium Constant for the New Conditions

Solve for \( x \) using the equilibrium equation \(K_p = \frac{(4x)^2}{x} \). Since \( K_p \) is constant, use the value calculated in step 4.

Step 7: Calculate the New Total Pressure

Add the partial pressures at the new ratio to find the new total pressure: \(P_{total} = P_{CO} + P_{CO_2} = 4x + x = 5x\). Solve for \( x \) using the value of \( K_p \) obtained previously and then calculate \( P_{total} \) given this \( x \).

Step 8: Determine the Correct Answer

Compare the calculated total pressure with the given answer options to find the correct one.

Key Concepts

Equilibrium Constant

The equilibrium constant, denoted as K or Kp when referring to partial pressures, is a measure of the relative quantities of reactants and products at equilibrium in a chemical reaction. In layman's terms, it tells us how far a reaction will proceed before reaching a state where the forward and reverse reactions occur at the same rate.

For the reaction in our exercise, where carbon dioxide gas (\text{CO}_2) and solid carbon (\text{C}) are in equilibrium with carbon monoxide gas (\text{CO}), the equilibrium constant expression is formulated using the partial pressures of the gases, as the solid carbon does not contribute to the pressure of the system. Mathematically, it is represented as: \[K_p = \frac{[CO]^2}{[CO_2]}\], which simplifies to: \[K_p = \frac{P_{CO}^2}{P_{CO_2}}\], where \(P_{CO}\) and \(P_{CO_2}\) are the partial pressures of carbon monoxide and carbon dioxide, respectively.

The value of the equilibrium constant is determined by the conditions of temperature and pressure and remains constant unless these conditions change. In the example, solving for the equilibrium constant with the initial pressure conditions allows us to predict the behavior of the reaction under new pressure conditions while keeping the temperature constant.

Partial Pressure Calculation

Partial pressure is the pressure that a gas in a mixture would exert if it occupied the entire volume by itself. Calculating partial pressures is essential for solving equilibrium problems involving gases. To find the partial pressure of a gas, we use the mole fraction of the gas times the total pressure.

In the provided problem, we calculated the initial partial pressures by applying the mole fraction of each gas to the total given pressure of the system. For instance, the initial partial pressure of \text{CO} was determined using the equation: \[(P_{CO})_{initial} = \frac{9}{10} \times 16 \text{ atm}\]. Subsequently, the partial pressure when the ratio of \text{CO} to \text{CO}_2 is 4:1 was calculated using a variable for the pressure of \text{CO}_2 (\(x\)) and then obtaining that of \text{CO} as 4 times \(x\) to satisfy the desired ratio. Both of these calculations hinge on the mole ratio, and solving for the new total pressure, \(P_{total}\), requires the summation of these new partial pressures, giving us the ability to find the total pressure under the new equilibrium conditions.

Le Chatelier's Principle

Le Chatelier's Principle is a qualitative tool used in chemistry to predict the effect of a change in conditions on a chemical equilibrium. This principle states that if an external stress (such as a change in temperature, pressure, or concentration of reactants or products) is applied to a system at equilibrium, the system will adjust itself in such a way as to counteract that change.

Common applications include predicting the direction a reaction will shift when there are changes in pressure or concentration. For gas-phase reactions, increasing the pressure generally causes the equilibrium to shift toward the side with fewer moles of gas, as this would decrease the pressure. In the context of our exercise, while the temperature remains constant at 1000°C, a change in the pressure of the system will lead to a shift in equilibrium to maintain the value of the equilibrium constant. By assessing these changes quantitatively as done in our step-by-step solution, Le Chatelier's Principle gives us a roadmap for understanding and calculating the response of the system to such changes.