Question

The equilibrium constant \(K_{\mathrm{c}}\) for the reaction $$ \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g) $$ is 4.2 at \(1650^{\circ} \mathrm{C}\). Initially \(0.80 \mathrm{~mol} \mathrm{H}_{2}\) and \(0.80 \mathrm{~mol}\) \(\mathrm{CO}_{2}\) are injected into a \(5.0-\mathrm{L}\) flask. Calculate the concentration of each species at equilibrium.

Short answer

The final concentrations of the species at equilibrium can be calculated by using the ICE table approach, writing down the equation for the equilibrium constant \(K_{\mathrm{c}}\), solve for the unknown variable \(x\), and finally substitute back \(x\) to get the equilibrium concentrations.

Step-by-step solution

Write the equilibrium constant expression

The equilibrium constant expression for this reaction according to Law of Mass Action is: \(K_{\mathrm{c}} = \frac{[\mathrm{H}_{2}\mathrm{O}][\mathrm{CO}]}{[\mathrm{H}_{2}][\mathrm{CO}_{2}]}\)

Set up an ICE table

ICE stands for Initial, Change, and Equilibrium concentrations. The initial concentrations of \(\mathrm{H}_2\) and \(\mathrm{CO}_2\) are each 0.16 M, given by \(0.80 \, \mathrm{mol} \, / \, 5.0 \, \mathrm{L}\), and of \(\mathrm{H}_2\mathrm{O}\) and \(\mathrm{CO}\) are 0 since they are products. By denoting the change in concentrations as \(x\), together with their stoichiometric coefficients, the equilibrium concentrations are given by: \(0.16-x\) for \(\mathrm{H}_2\) and \(\mathrm{CO}_2\), and \(x\) for \(\mathrm{H}_2\mathrm{O}\) and \(\mathrm{CO}\).

Subsitute into \(K_{\mathrm{c}}\) expression and solve for x

Subsitute the equilibrium concentrations derived from the ICE table into the expression of \(K_{\mathrm{c}}\), set it equal to 4.2, gives the equation for \(x\): \(4.2 = \frac{x^2}{(0.16-x)^2}\). This is a quadratic equation. Solving it provides the value of \(x\), which then can be used to find the equilibrium concentrations.

Get the equilibrium concentrations

Use the calculated \(x\) value to substitute into the equilibrium expressions and get the equilibrium concentrations.

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is a foundational concept in chemistry, particularly in the study of reactions. It refers to the state at which the rate of the forward reaction equals the rate of the reverse reaction. This means that the concentrations of the reactants and products remain constant over time, though they are not necessarily equal.

At equilibrium, there is no net change in the concentration of reactants and products because the processes of formation and breakdown are balanced. It’s important to note that achieving equilibrium does not imply that the reactants and products are present in equal amounts, but rather that their quantities have stabilized in a fixed ratio.

Law of Mass Action

The Law of Mass Action describes the relationship between the concentrations of reactants and products at equilibrium. It states that for a reversible reaction at equilibrium and at a constant temperature, the ratio of the product of concentrations of the products raised to the power of their stoichiometric coefficients to the product of concentrations of the reactants raised to the power of their stoichiometric coefficients is constant.

This constant is known as the equilibrium constant, denoted as \( K_c \) for concentrations or \( K_p \) for partial pressures. The general form of the equilibrium constant expression for a reaction like \( aA + bB \rightleftharpoons cC + dD \) is \( K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} \). This expression is pivotal in calculating the equilibrium concentrations of reactants and products.

ICE Table Method

The ICE table method is an organized way to tackle equilibrium problems. ICE stands for Initial, Change, and Equilibrium, representing different stages of the reaction. By listing the concentrations of reactants and products at each stage, the ICE table visually aids in understanding the shifts that occur as a reaction reaches equilibrium. For initial concentrations, you list the starting amounts. For change, you represent how much reactant is converted to product, often using \( x \) or another variable to indicate the unknown changes. Finally, in the equilibrium row, you calculate the final concentrations, considering the initial amounts and the changes that occurred.

Equilibrium Concentration

Equilibrium concentration refers to the concentration of a species in a reaction mixture when the reaction has reached equilibrium. At this point, the concentration of all reactants and products stays constant. Determining the equilibrium concentrations typically involves using an ICE table, with the equilibrium constant expression derived from the Law of Mass Action. By substituting the equilibrium concentrations into the equilibrium constant expression and solving the resulting equation, you unravel the concentrations of each species at equilibrium, which is crucial in predicting the extent of the reaction and the composition of the final mixture.