Problem 68 - Chemistry: The Central Science

Question

The acid-dissociation constant for chlorous acid \(\left(\mathrm{HClO}_{2}\right)\) is \(1.1 \times 10^{-2}\). Calculate the concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}, \mathrm{ClO}_{2}^{-}\), and \(\mathrm{HClO}_{2}\) at equilibrium if the initial concentration of \(\mathrm{HClO}_{2}\) is \(0.0125 \mathrm{M} .\)

Step-by-Step Solution

Answer

The equilibrium concentrations for the given scenario are as follows: \([\mathrm{HClO}_2]_{eq} = 0.0088\, \mathrm{M} \) \([\mathrm{H}^+]_{eq} (or \,[\mathrm{H}_{3} O^{+}]_{eq}) = 0.0037\, \mathrm{M} \) \([\mathrm{ClO}_2^-]_{eq} = 0.0037\, \mathrm{M} \)

Step 1: Write the acid dissociation equation and Ka expression

For chlorous acid, we have the following acid dissociation reaction: \[ \mathrm{HClO}_2 \rightleftharpoons \mathrm{H}^+ + \mathrm{ClO}_2^- \] The Ka expression for this reaction is: \[ K_a = \frac{[\mathrm{H}^+][\mathrm{ClO}_2^-]}{[\mathrm{HClO}_2]} \]

Step 2: Set up the Initial-Change-Equilibrium (ICE) table

We will use the ICE table to determine the concentrations of the ions at equilibrium. The table is set up as follows for the reaction \( \mathrm{HClO}_2 \rightleftharpoons \mathrm{H}^+ + \mathrm{ClO}_2^- \) | | HClO2 | H+ | ClO2- | |-------------|------------|-----------|------------| | Initial | 0.0125 M | 0 M | 0 M | | Change | -x | +x | +x | | Equilibrium | 0.0125 - x | x | x |

Step 3: Substitute the equilibrium concentrations into the Ka expression

We will substitute the equilibrium concentrations from the ICE table into the Ka expression: \[K_a = 1.1 \times 10^{-2} = \frac{x^2}{0.0125 - x} \]

Step 4: Solve for x

In order to solve for x (the change in concentrations), we have to make an approximation. Since Ka is small (1.1 x 10^{-2}), it is reasonable to assume that x is small compared to 0.0125, therefore, we can write: \[1.1 \times 10^{-2} = \frac{x^2}{0.0125} \] Now, solve for x by multiplying both sides by 0.0125 and taking the square root of the result: \[ x = \sqrt{1.1 \times 10^{-2} \times 0.0125} \approx 0.0037\]

Step 5: Determine the equilibrium concentrations

Now that we have the value of x, we can determine the equilibrium concentrations of HClO2, H+, and ClO2-: \([\mathrm{HClO}_2]_{eq} = 0.0125 - x = 0.0125 - 0.0037 = 0.0088\, \mathrm{M} \) \([\mathrm{H}^+]_{eq} = x = 0.0037\, \mathrm{M} \) \([\mathrm{ClO}_2^-]_{eq} = x = 0.0037\, \mathrm{M} \) So, the equilibrium concentrations are: \([\mathrm{HClO}_2]_{eq} = 0.0088\, \mathrm{M} \) \([\mathrm{H}^+]_{eq} (or \,[\mathrm{H}_{3} O^{+}]_{eq}) = 0.0037\, \mathrm{M} \) \([\mathrm{ClO}_2^-]_{eq} = 0.0037\, \mathrm{M} \)

Key Concepts

Acid Dissociation Reaction

In chemistry, an acid dissociation reaction is a process where an acid, a substance that donates protons (hydrogen ions, H+), splits into ions when dissolved in water. For instance, chlorous acid ewline (ewline HClO2), a weak acid, partially dissociates in water to form H+ and ClO2- ions. This reversible reaction is represented by the chemical equation ewline HClO2 ewline rightleftharpoons H+ + ewline ClO2-. The extent of dissociation for weak acids is quantified by the acid-dissociation constant (Ka), which offers insight into the acid's strength. In our case, the Ka value for chlorous acid is 1.1 x 10^-2, indicating that it's a weak acid as it doesn't dissociate completely. Understanding this reaction is fundamental for calculating the concentration of the ions produced in solution at equilibrium.

Equilibrium Concentration

The term equilibrium concentration refers to the concentration of reactants and products in a chemical reaction that has reached a state of balance. At equilibrium, both the forward and reverse reactions occur at the same rate, resulting in a stable ratio of product and reactant concentrations. These concentrations remain constant over time unless the system is disturbed. In our exercise, the aim is to calculate the equilibrium concentrations of the reactants ewline (ewline HClO2) and the products ewline (ewline H3O+, ClO2-) of the acid-dissociation reaction. Accurate knowledge of these concentrations is crucial in various fields, including chemistry, biology, and environmental science, as it helps in predicting the direction of the reaction and the pH of the solution.

ICE Table

An ICE table, which stands for Initial-Change-Equilibrium, is a systematic tool used to keep track of changes in concentration throughout a chemical reaction. It is incredibly helpful for visualizing and solving equilibrium problems in chemistry. Constructing an ICE table involves three rows symbolizing the initial concentrations, the changes in concentration as the reaction proceeds, and the equilibrium concentrations of each species involved in the reaction.

For our chlorous acid dissociation, we start with the initial concentration of HClO2 and assume zero for the products H+ and ClO2- since they have not formed yet. Throughout the course of the reaction, the amount of HClO2 that dissociates is represented by 'x', which is also the amount of H+ and ClO2- ions produced. At equilibrium, the concentrations are adjusted accordingly. The ICE table simplifies the calculation process and, combined with the equilibrium constant expression, allows us to solve for 'x' and thus find the equilibrium concentrations.

Chemical Equilibrium

The state of chemical equilibrium is achieved in a reversible chemical reaction when the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of the reactants and products no longer change with time. It is significant to recognize that equilibrium does not imply equal concentrations of reactants and products but rather a constant ratio as defined by the equilibrium constant (K).

For the dissociation of chlorous acid, our goal is to find the point at which the chemical system reaches equilibrium. At this point, the ratio of the product concentrations to the reactant concentrations raised to their stoichiometric coefficients equates to the acid-dissociation constant, Ka. By understanding chemical equilibrium, we can predict how a reaction will respond to changes in conditions such as concentration, pressure, or temperature, using Le Chatelier's principle, and manipulate reactions to favor the production of desired substances.