Question

A quantity of 0.20 mole of carbon dioxide was heated at a certain temperature with an excess of graphite in a closed container until the following equilibrium was reached: $$ \mathrm{C}(s)+\mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g) $$ Under this condition, the average molar mass of the gases was found to be \(35 \mathrm{~g} / \mathrm{mol}\). (a) Calculate the mole fractions of \(\mathrm{CO}\) and \(\mathrm{CO}_{2}\). (b) What is the \(K_{P}\) for the equilibrium if the total pressure was 11 atm? (Hint: The average molar mass is the sum of the products of the mole fraction of each gas and its molar mass.)

Short answer

After calculating the mole fraction for CO2 as 0.22 and for CO as 0.78, the partial pressure for CO comes out to be 8.58 atm and for CO2 is 2.42 atm. The equilibrium constant for the reaction at the given conditions is then found to be 2.9.

Step-by-step solution

Calculate the Mole Fractions

The Average molar mass of a mixture of gases can be determined using the expression \[\text{Average molar mass} = \Sigma \text{(mole fraction * molar mass of individual gas)}\] Solving this for each gas:\[x_{\text{CO2}} * M_{\text{CO2}} + x_{\text{CO}} * M_{\text{CO}} = \text{Average molar mass}\]where \(x\) represents the mole fraction and \(M\) denotes the molar mass. The molar masses for CO2 and CO are 44 g/mol and 28 g/mol respectively. Thus, we can record the equation as:\[x_{\text{CO2}} * 44 + x_{\text{CO}} * 28 = 35\] Since \(x_{\text{CO2}}\) and \(x_{\text{CO}}\) are mole fractions, their sum is 1. Therefore, \(x_{\text{CO}} = 1 - x_{\text{CO2}}\), and this value can be substituted into the initial equation which can be solved for the mole fractions.

Calculate the Partial Pressures

After obtaining the mole fractions, the partial pressures of CO and CO2 can be calculated. The partial pressure of a gas is obtained by multiplying the total pressure of the gas mixture by the mole fraction of the gas in the mixture. Therefore, the partial pressures for CO2 and CO are \(p_{\text{CO2}} = x_{\text{CO2}} * P_{\text{total}}\) and \(p_{\text{CO}} = x_{\text{CO}} * P_{\text{total}}\) respectively, where \(P_{\text{total}}\) is the total pressure (11 atm in this case).

Calculate the Equilibrium Constant

The equilibrium constant \(K_{P}\) for a gas reaction is the ratio of the partial pressure of the products to the reactants, raised to the power of their stoichiometric coefficients. The reaction given is: \( C(s) + CO_{2}(g) \leftrightharpoons 2CO(g) \). The equilibrium constant is therefore \(K_{P}=\left(\frac{p_{\text{CO}}^{2}}{p_{\text{CO2}}}\right)\), substituting the values of \(p_{\text{CO2}}\) and \(p_{\text{CO}}\) gives \(K_{P}\).

Key concepts

Mole Fraction Calculation

Understanding how to calculate mole fractions is essential when dealing with chemical mixtures, especially gases. The mole fraction is a way of expressing the composition of a mixture by indicating what fraction of the total moles is made up by each component. To calculate the mole fraction, you simply divide the number of moles of the substance of interest by the total number of moles present in the mixture.

For instance, in the provided exercise, to find the mole fractions of CO and CO2, we start from the equation for average molar mass:\[x_{\text{CO2}} \times 44 + x_{\text{CO}} \times 28 = 35\]
This equation is the sum of the products of each gas's mole fraction and its respective molar mass. Knowing that the sum of the mole fractions must equal to 1, we can solve for one mole fraction and find the other by subtraction from 1. This gives us the ratio of CO2 to CO in the gas phase.

Partial Pressure

In a mixture of gases, like the one we're examining in the equilibrium reaction, each gas exerts a pressure as if it were alone in the container. This is known as the gas's partial pressure, and it directly depends on the mole fraction of that gas in the mixture. Dalton's Law of Partial Pressures tells us that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas.

To calculate partial pressures, you simply multiply the mole fraction of the gas by the total pressure of the mixture:\[p_{\text{gas}} = x_{\text{gas}} \times P_{\text{total}}\]
In our exercise, the total pressure is given as 11 atm. After determining the mole fractions, as described previously, multiplying them by the total pressure yields the partial pressures for CO and CO2. These values are fundamental in determining the equilibrium constant for the reaction.

Equilibrium Constant (Kp)

The equilibrium constant (Kp) is a dimensionless number that provides a measure of the extent of a reaction; it offers insight into the position of equilibrium. For reactions involving gases, Kp is expressed in terms of partial pressures. It is calculated by taking the ratio of the partial pressures of the products to the reactants, with each raised to the power of its stoichiometric coefficient, as seen in the chemical equation of the reaction.

The reaction in the exercise is:\[\mathrm{C}(s) + \mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)\]Therefore, the equilibrium constant is expressed as:\[K_{P}=\left(\frac{p_{\text{CO}}^{2}}{p_{\text{CO2}}}\right)\]
After determining the partial pressures of CO and CO2 using the mole fractions and the total pressure, we substitute these values into the equation to calculate Kp. The value of Kp helps predict the direction of the reaction under given conditions and is a crucial parameter in chemical equilibrium.