Question

A mixture consisting of \(1.000 \mathrm{~mol} \mathrm{} \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) and \(1.000 \mathrm{~mol}\) \(\mathrm{CO}(\mathrm{g})\) is placed in a \(10.00-\mathrm{L}\) reaction vessel at 800 . K. At equilibrium, \(0.665 \mathrm{~mol} \mathrm{} \mathrm{CO}_{2}(\mathrm{~g})\) is present as a result of the reactio \(\mathrm{CO}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightleftharpoons \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g})\). What are (a) the equilibrium concentrations for all substances and (b) the value of \(K_{c}\) at \(800 . \mathrm{K}\) ?

Short answer

The equilibrium concentrations are: [CO] = [H2O] = 0.0335 M, [CO2] = [H2] = 0.0665 M. The equilibrium constant, Kc, at 800 K is 3.946.

Step-by-step solution

Write the balanced equation and determine moles at equilibrium

The balanced chemical equation is already provided: \[ \text{CO}(g) + \text{H}_2\text{O}(g) \rightleftharpoons \text{CO}_2(g) + \text{H}_2(g) \]. Initially, there are 1.000 mol of CO and H2O each, and 0 mol of CO2 and H2. At equilibrium, there are 0.665 mol of CO2, which means the same amount of CO and H2O reacted (because of the 1:1 stoichiometry), and the same amount of H2 was produced.

Calculate the equilibrium concentrations

The equilibrium concentrations (M) can be found by dividing the number of moles of each substance at equilibrium by the volume of the container. The container volume is 10.00 L. So the concentrations at equilibrium are: \[ [\text{CO}] = \frac{1.000 - 0.665}{10.00} \, M, [\text{H}_2\text{O}] = \frac{1.000 - 0.665}{10.00} \, M, [\text{CO}_2] = \frac{0.665}{10.00} \, M, [\text{H}_2] = \frac{0.665}{10.00} \, M \].

Calculate the Equilibrium Constant (Kc)

The equilibrium constant expression for the reaction is \[ K_c = \frac{[\text{CO}_2][\text{H}_2]}{[\text{CO}][\text{H}_2\text{O}]} \]. We will plug in the equilibrium concentrations from Step 2 to find the value of \(K_c\).

Solve for Kc

Using the concentrations from Step 2: \[ K_c = \frac{\left(\frac{0.665}{10}\right)\left(\frac{0.665}{10}\right)}{\left(\frac{1.000 - 0.665}{10}\right)\left(\frac{1.000 - 0.665}{10}\right)} = \frac{(0.0665)(0.0665)}{(0.0335)(0.0335)} = \frac{0.00442}{0.00112} = 3.946 \]. Therefore, \(K_c = 3.946\) at 800 K.

Key concepts

Equilibrium Constant (Kc)

The equilibrium constant, represented as \(K_c\), is a vital concept in chemical equilibrium, signifying the ratio of the concentrations of products to reactants at equilibrium. It is important to note that concentrations are raised to the power of their coefficients in the balanced chemical equation. The equilibrium constant is temperature dependent and gives us an idea of the extent of the reaction; a large \(K_c\) indicates that at equilibrium, the products predominate, while a small \(K_c\) suggests a reactant-favored mixture.

The calculation of \(K_c\) involves substituting the equilibrium concentrations of the reactants and products into the equilibrium expression. For instance, in the given exercise, we determined \(K_c\) by using the equilibrium concentrations of \(CO_2\), \(H_2\), \(CO\), and \(H_2O\). The higher value of \(K_c = 3.946\) suggests that at 800 K, the reaction favors the production of \(CO_2\) and \(H_2\).

Equilibrium Concentrations

Equilibrium concentrations refer to the amounts of reactants and products present when a chemical reaction has reached a state of balance. At this point, the rate of the forward reaction equals the rate of the reverse reaction, resulting in constant concentrations of all species involved in the reaction.

In order to calculate these concentrations, as shown in the exercise, we take the initial moles of substances and deduct the moles that reacted, then divide by the volume of the vessel. Being able to accurately determine these concentrations is crucial for calculating the equilibrium constant and understanding the reaction dynamics.

Reaction Quotient (Q)

The reaction quotient, \(Q\), is similar to the equilibrium constant in that it involves the same concentration terms and stoichiometric coefficients, but for a reaction mixture that has not necessarily reached equilibrium. It uses the current concentrations or partial pressures of the reactants and products at any point in time.

If \(Q\) is less than \(K_c\), the reaction will proceed in the forward direction to reach equilibrium. If \(Q\) is greater than \(K_c\), the reaction will go in the reverse direction. When \(Q\) is equal to \(K_c\), the system is at equilibrium. Calculating \(Q\) can help predict the shift in a reaction when conditions change.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction, governed by the conservation of mass and the coefficients in the balanced chemical equation. It's essentially 'chemical bookkeeping', ensuring that atoms are neither created nor destroyed in reactions.

In practice, stoichiometry allows chemists to predict the amounts of substances consumed or produced, such as the 1:1:1:1 ratio in our CO and H2O reaction, where 0.665 mol of CO2 production means that 0.665 mol of CO and H2O were consumed, and 0.665 mol of H2 were produced. Understanding stoichiometry is crucial for calculating equilibrium concentrations and constants accurately.