Question

Phosgene \(\left(\mathrm{COCl}_{2}\right)\) is a toxic substance that forms readily from carbon monoxide and chlorine at elevated temperatures: $$ \mathrm{CO}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{COCl}_{2}(g) $$ If \(0.350 \mathrm{~mol}\) of each reactant is placed in a 0.500 - L flask at \(600 \mathrm{~K}\), what are the concentrations of all three substances at equilibrium \(\left(K_{c}=4.95\right.\) at this temperature)?

Short answer

Equilibrium concentrations are: \( [\mathrm{COCl}_{2}] = 0.419 \text{ M} \), \( [\mathrm{CO}] = 0.281 \text{ M} \), \( [\mathrm{Cl}_{2}] = 0.281 \text{ M} \).

Step-by-step solution

- Write the balanced chemical equation

Identify the balanced chemical equation for the reaction: \[ \mathrm{CO}(g) + \mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{COCl}_{2}(g) \]

- Determine initial concentrations

Given 0.350 moles of each reactant in a 0.500 L flask, calculate the initial concentrations: \[ \left[ \mathrm{CO} \right]_0 = \frac{0.350 \text{ mol}}{0.500 \text{ L}} = 0.700 \text{ M} \] \[ \left[ \mathrm{Cl}_2 \right]_0 = \frac{0.350 \text{ mol}}{0.500 \text{ L}} = 0.700 \text{ M} \] \[ \left[ \mathrm{COCl}_2 \right]_0 = 0 \text{ M} \]

- Define the change in concentration

Let the change in concentration of CO and \( \mathrm{Cl}_2 \) at equilibrium be \( -x \). The change in concentration of \( \mathrm{COCl}_2 \) at equilibrium will be \( +x \). Write the expressions: \[ \left[ \mathrm{CO} \right] = 0.700 - x \] \[ \left[ \mathrm{Cl}_2 \right] = 0.700 - x \] \[ \left[ \mathrm{COCl}_2 \right] = x \]

- Write the equilibrium constant expression

The equilibrium constant expression for the reaction is: \[ K_c = \frac{\left[ \mathrm{COCl}_{2} \right]}{\left[ \mathrm{CO} \right] \left[ \mathrm{Cl}_2 \right]} \] Substitute the equilibrium concentrations: \[ 4.95 = \frac{x}{(0.700 - x)(0.700 - x)} \]

- Solve for x

Solve the quadratic equation for x: \[ 4.95 = \frac{x}{(0.700 - x)^2} \] Simplify to get the quadratic form: \[ 4.95 (0.700 - x)^2 = x \] \[ 4.95 (0.49 - 1.4x + x^2) = x \] \[ 2.4255 - 6.93x + 4.95x^2 = x \] \[ 4.95x^2 - 7.93x + 2.4255 = 0 \] Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4.95 \), \( b = -7.93 \), and \( c = 2.4255 \): \[ x = \frac{7.93 \pm \sqrt{(-7.93)^2 - 4(4.95)(2.4255)}}{2(4.95)} \] \[ x = \frac{7.93 \pm \sqrt{62.8849 - 48.6039}}{9.9} \] \[ x = \frac{7.93 \pm \sqrt{14.281}}{9.9} \] \[ x = \frac{7.93 \pm 3.78}{9.9} \] Select the feasible positive root: \[ x = \frac{11.71}{9.9} = 1.183 \text{ M} \] (This value exceeds the initial concentration, so it is not possible. Use the smaller value) \[ x = \frac{4.15}{9.9} = 0.419 \text{ M} \]

- Determine equilibrium concentrations

Calculate the equilibrium concentrations using x = 0.419 M: \[ \left[ \mathrm{COCl}_{2} \right] = x = 0.419 \text{ M} \] \[ \left[ \mathrm{CO} \right] = 0.700 - x = 0.700 - 0.419 = 0.281 \text{ M} \] \[ \left[ \mathrm{Cl}_2 \right] = 0.700 - x = 0.700 - 0.419 = 0.281 \text{ M} \]

Key concepts

chemical equilibrium

Chemical equilibrium is the state where the concentrations of all reactants and products remain constant over time. In this state, the forward reaction rate equals the reverse reaction rate. For a given reaction, the equilibrium can be described using an equilibrium constant, often represented as \(K_c\). This constant is a ratio of the concentrations of products to the concentrations of reactants, each raised to the power of their respective coefficients in the balanced equation. Chemical equilibrium is dynamic, meaning reactions continue to occur, but there is no net change in concentrations.

balanced chemical equation

A balanced chemical equation ensures the principle of the conservation of mass is maintained, meaning the number of atoms of each element present in the reactants equals the number in the products. For the reaction CO(g) + Cl₂(g) ⇌ COCl₂(g), the equation is already balanced as written, meaning one molecule of carbon monoxide reacts with one molecule of chlorine to form one molecule of phosgene. A balanced equation is crucial for accurately describing the stoichiometry of the reaction and for any related calculations, including those needed to find equilibrium concentrations.

quadratic formula

To solve for the concentrations of reactants and products at equilibrium, we often end up with a quadratic equation. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). We can solve this using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In our exercise, substituting the appropriate values gives us:\[ 4.95(0.700 - x)^2 = x\]\[4.95(0.49 - 1.4x + x^2) = x\]which simplifies to the quadratic equation \(4.95x^2 - 7.93x + 2.4255 = 0\). Solving this provides the equilibrium concentration changes.

reaction stoichiometry

Reaction stoichiometry involves the quantitative relationships between reactants and products in a chemical reaction. It relies on the balanced chemical equation to provide the correct mole ratio for calculations. In our example, the stoichiometry of the reaction CO(g) + Cl₂(g) ⇌ COCl₂(g) is 1:1:1. This means that for every mole of CO and Cl₂ that react, one mole of COCl₂ is produced. By using stoichiometry, we can determine how the initial moles of reactants change and establish the equilibrium conditions for each species involved in the reaction.