Question

A mixture of 3 mol of \(\mathrm{CO}_{2}\) and 3 mol of \(\mathrm{O}_{2}\) is heated to \(3400 \mathrm{K}\) at a pressure of 2 atm. Determine the equilibrium composition of the mixture, assuming that only \(\mathrm{CO}_{2}, \mathrm{CO}, \mathrm{O}_{2},\) and \(\mathrm{O}\) are present.

Short answer

Answer: To find the equilibrium composition, calculate the moles of CO2, CO, O2, and O at equilibrium using the value of x obtained from the equilibrium equations. Then, use these moles to calculate the mole fraction of each species in the mixture. The equilibrium composition will be represented by the mole fractions of CO2, CO, O2, and O in the mixture.

Step-by-step solution

- Identify the chemical reactions and their stoichiometry

The chemical reactions taking place are: (i) CO2 <=> CO + O2 (ii) CO + O2 <=> CO2

- Determine the equilibrium constants for the reactions

Besides using the equilibrium constant directly, it can also be calculated using the expression: \(K_p = e^{-\frac{\Delta G^0}{RT}}\) For both reactions, we have to find the value of \(\Delta G^0\), which can be calculated using the expression: \(\Delta G^0 = \Delta H^0 - T\Delta S^0\) The values of \(\Delta H^0\) and \(\Delta S^0\) for the given temperature can be looked up in standard thermodynamic tables and used to calculate the equilibrium constants \(K_p\) for both reactions.

- Write the expressions for equilibrium concentrations

In order to find the equilibrium concentration, first, we express the equilibrium constants in terms of moles: (i) \(K_{p,1} = \frac{[CO] [O]}{[CO_2]}\) (ii) \(K_{p,2} = \frac{[CO_2]}{[CO] [O_2]}\)

- Calculate the change in moles for each compound and write the equilibrium conditions

Let \(x\) moles of CO2 dissociate, then: \([CO_2] = 3 - x\) \([CO] = x\) \([O] = x\) \([O_2] = 3 - x\) Substituting these in the expressions for equilibrium constants: (i) \(K_{p,1} = \frac{x^2}{3-x}\) (ii) \(K_{p,2} = \frac{3-x}{x(3-x)}\)

- Solve the equilibrium equations for x

Since both reactions are in equilibrium, their product of their equilibrium constants will be equal to 1: \(K_{p,1} \times K_{p,2} = 1\) Substituting the expressions for \(K_{p,1}\) and \(K_{p,2}\) obtained in Step 4: \(\frac{x^2}{3-x} \times \frac{3-x}{x(3-x)} = 1\) Solve for x to get the equilibrium moles of the species.

- Calculate the equilibrium composition

Using the value of x obtained in Step 5, calculate the final moles of each species and convert them to concentrations using the total volume. Then, divide each concentration by the sum of all concentrations to get the mole fraction of each species present in the mixture: Mole fractions: CO2: \(\frac{[CO_2]}{[CO]+[CO_2]+[O_2]+[O]}\) CO: \(\frac{[CO]}{[CO]+[CO_2]+[O_2]+[O]}\) O2: \(\frac{[O_2]}{[CO]+[CO_2]+[O_2]+[O]}\) O: \(\frac{[O]}{[CO]+[CO_2]+[O_2]+[O]}\) These mole fractions give us the equilibrium composition of the gaseous mixture.