Question

The reaction \(\mathrm{CO}_{2}(g)+\mathrm{C}(s) \rightleftharpoons 2 \mathrm{CO}(g)\) has \(\mathrm{K}_{\mathrm{p}}=5.78\) at 1200 \(\mathrm{K}\) \begin{equation}\begin{array}{l}{\text { a. Calculate the total pressure at equilibrium when } 4.45 \text { g of } \mathrm{CO}_{2} \text { is in- }} \\\ {\text { troduced into a } 10.0 \text { L container and heated to } 1200 \mathrm{K} \text { in the presence }} \\ {\text { of } 2.00 \mathrm{g} \text { of graphite. }} \\ {\text { b. Repeat the calculation of part a in the presence of 0.50 } \mathrm{g} \text { of graphite. }}\end{array}\end{equation}

Short answer

The total pressure at equilibrium when 4.45 g of CO2 and 2.00 g of graphite are used is not calculated as additional information, such as the reaction quotient (Qc) under initial conditions, is needed. Similarly, the total pressure in the presence of 0.50 g of graphite cannot be provided without further calculations.

Step-by-step solution

Calculate initial amounts in moles of reactants

Use the molar mass of CO2 (44.01 g/mol) to convert the mass of CO2 to moles. For graphite (carbon), use its molar mass (12.01 g/mol).

Write the expression for the reaction quotient, Qc

Use the stoichiometry of the balanced reaction to express Qc in terms of reactant and product concentrations.

Set up the ICE table

Create an ICE (Initial, Change, Equilibrium) table to track the concentrations of reactants and products through the course of the reaction.

Calculate the equilibrium concentrations

Using the value of Kp, write out the equilibrium constant expression and solve for the equilibrium concentrations of CO, given Kp and the initial moles of CO2.

Calculate the equilibrium pressures

Use the ideal gas law to convert the equilibrium concentrations to pressures. The total pressure will be the sum of the partial pressures of CO2 and CO at equilibrium.

Repeat the calculations for 0.50 g of graphite

Repeat the calculation of the equilibrium pressures using the new initial amount of graphite, following the steps 1-5.

Key concepts

Equilibrium Constant (Kp)

Understanding the equilibrium constant, denoted as \(K_p\), is crucial when dealing with reactions involving gases at a constant temperature. \(K_p\) provides a ratio of the partial pressures of the products to the reactants, each raised to the power of their stoichiometric coefficients in the balanced chemical equation.

For the reaction \(\mathrm{CO}_{2}(g) + \mathrm{C}(s) \rightleftharpoons 2\mathrm{CO}(g)\), \(K_p\) is given as 5.78 at 1200 K, which means at equilibrium, the ratio of \(\mathrm{CO}\) gas's partial pressure squared to that of \(\mathrm{CO}_{2}\) gas's partial pressure remains constant at this value. Notice that since graphite (\(\mathrm{C}\)) is a solid, it does not appear in the \(K_p\) expression.

To solve for equilibrium pressures, you'll need to understand how \(K_p\) relates to the changes in the system captured in an ICE table.

ICE Table for Equilibrium

The ICE table is a systematic way to calculate the changes in concentration—or in cases involving gases, the changes in partial pressure—of reactants and products as they move towards equilibrium. ICE stands for Initial, Change, and Equilibrium.

Initially, we know the amount of \(\mathrm{CO}_{2}\) and \(\mathrm{C}\) in moles. During the reaction (\

Reaction Quotient (Qc)

The reaction quotient, \(Q_c\), is a measure that tells us how far a system is from reaching equilibrium. It is calculated under non-equilibrium conditions using the same formula as the equilibrium constant, but with the initial concentrations or pressures instead of the equilibrium values.

For our reaction, \(Q_c\) would be calculated as the initial pressures of CO squared divided by the initial pressure of \(\mathrm{CO}_2\) since \(\mathrm{C}\) is a solid and does not appear in this expression. By comparing \(Q_c\) to \(K_p\), we can determine the direction in which the reaction will shift to reach equilibrium. If \(Q_c < K_p\), the reaction will proceed forward, and if \(Q_c > K_p\), it will proceed in reverse.

Stoichiometry

Stoichiometry, at its core, involves the quantitative relationships between the reactants and products in a chemical reaction. It is based on the conservation of mass and the balanced chemical equation.

In our exercise, the balanced equation tells us that 1 mole of \(\mathrm{CO}_2\) will react with \(\mathrm{C}\) to produce 2 moles of \(\mathrm{CO}\) upon reaching equilibrium. To solve for the equilibrium conditions, we must calculate the initial moles of \(\mathrm{CO}_2\) and \(\mathrm{C}\) based on their given masses and use the stoichiometric ratios to determine how much of each will be present at equilibrium. The stoichiometry of the reaction is integral to constructing the ICE table and ultimately calculating the total pressure at equilibrium.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount of a gas. Expressed as \(PV=nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature in kelvins, it assists in solving for unknowns in gas-related problems.

In our exercise, after using the ICE table to find the equilibrium concentrations of \(\mathrm{CO}_2\) and \(\mathrm{CO}\), we apply the ideal gas law to determine their partial pressures. The total pressure at equilibrium in the container can then be determined by adding together the partial pressures of the gases present. The ideal gas law is key to transforming concentrations into pressures, which are needed for calculating \(K_p\) and solving the exercise.