Question

Calculate the percent ionization of propionic acid \((\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH})\) in solutions of each of the following concentrations (\(K_{a}\) is given in AppendixD): (a) \(0.250 M,(\mathbf{b}) 0.0800 M,\) \((\mathbf{c}) 0.0200 M .\)

Short answer

Percent ionization can be calculated using the formula: \(\text{Percent Ionization} = \frac{x}{c} \times 100\%\), where \(x\) is the concentration of dissociated ions and \(c\) is the initial concentration of propionic acid. Use the given \(K_{a}\) value and the equilibrium expression: \(K_{a} = \frac{x^2}{c - x}\), and assuming \(x << c\), approximate to: \(K_{a} = \frac{x^2}{c}\). (a) For the \(0.250 M\) concentration, calculate \(x\) and find the percent ionization. (b) For the \(0.0800 M\) concentration, calculate \(x\) and find the percent ionization. (c) For the \(0.0200 M\) concentration, calculate \(x\) and find the percent ionization.

Step-by-step solution

Write the dissociation reaction for propionic acid

When propionic acid \((\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH})\) dissociates in water, it forms hydrogen ions \((\mathrm{H}^+)\) and propionate ions \((\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COO}^-)\). The dissociation reaction can be written as: \[ \mathrm{C}_{2}\mathrm{H}_{5}\mathrm{COOH} \left( aq \right) \rightleftharpoons \mathrm{H}^+ \left( aq \right) + \mathrm{C}_{2}\mathrm{H}_{5}\mathrm{COO}^- \left( aq \right) \]

Write the equilibrium expression for \(K_{a}\)

The equilibrium expression for the dissociation of propionic acid is: \[ K_{a} = \frac{[\mathrm{H}^+] [\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{COO}^-]}{[\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{COOH}]} \] We will use this equilibrium expression to find the concentration of dissociated ions for each concentration of propionic acid and then calculate the percent ionization.

Calculate the percent ionization for each concentration of propionic acid

For each concentration of propionic acid, we will assume a change in its concentration due to ionization and set up an ICE (Initial, Change, Equilibrium) table. Since the concentrations of \(\mathrm{H}^+\) and \(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{COO}^-\) will have the same change, we will use a variable \(x\) to represent the change in concentration and find its value using the \(K_a\) expression. Let's denote the initial concentration of propionic acid by \(c\). We have the following ICE table: | | \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\) | \(\mathrm{H}^+\) | \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COO}^-\) | |------------------|------------------------|-------------|------------------------| | Initial (M) | c | 0 | 0 | | Change (M) | -x | +x | +x | | Equilibrium (M) | c - x | x | x | Now substitute these concentrations into the \(K_{a}\) expression and solve for \(x\): \[ K_{a} = \frac{x^2}{c - x} \] Since \(K_a\) is generally small for weak acids, we can approximate that \(x << c\), so \(c - x \approx c\). Thus, we simplify the expression to: \[ K_{a} = \frac{x^2}{c} \] Now, we will use the given values of \(K_{a}\) and \(c\) (concentrations) to find the value of \(x\) for each part of the exercise. Finally, we will calculate the percent ionization using the following formula: \[ \text{Percent Ionization} = \frac{x}{c} \times 100\% \] Note that \(K_{a}\) values can be found in the Appendix D table of your textbook: (a) For the \(0.250 M\) concentration of propionic acid: Calculate the value of \(x\) for this concentration, and then the percent ionization. (b) For the \(0.0800 M\) concentration of propionic acid: Calculate the value of \(x\) for this concentration, and then the percent ionization. (c) For the \(0.0200 M\) concentration of propionic acid: Calculate the value of \(x\) for this concentration, and then the percent ionization.

Key concepts

Propionic Acid

Propionic acid, a carboxylic acid with the chemical formula \( \mathrm{C}_2\mathrm{H}_5\mathrm{COOH} \), plays a significant role in chemical equilibrium calculations involving weak acid dissociation. This acid is classified as weak because it partially ionizes in aqueous solutions, forming hydrogen ions \( \mathrm{H}^+ \) and propionate ions \( \mathrm{C}_2\mathrm{H}_5\mathrm{COO}^- \). This partial dissociation is what makes propionic acid a good candidate for studying chemical equilibria and percent ionization.When dissolved in water, propionic acid establishes a dynamic equilibrium—a balanced state where the rate of the forward reaction (dissociation of the acid) equals the rate of the reverse reaction (recombination of ions to form the acid again). The ability of propionic acid to ionize weakly thus allows us to use equilibrium expressions and other calculations to determine the extent of its ionization and the related concentrations of its ions and molecules. This is why understanding propionic acid's behavior in solution is essential for grasping the foundational concepts of weak acid equilibria.

Equilibrium Expression

The equilibrium expression is a mathematical representation that relates the concentrations of reactants and products in a chemical equilibrium system. For the dissociation of propionic acid, the equilibrium expression helps us understand how much of the acid dissociates to form its ions.The equilibrium constant, \( K_a \), for propionic acid specific equilibrium is expressed as:\[ K_a = \frac{[\mathrm{H}^+][\mathrm{C}_2\mathrm{H}_5\mathrm{COO}^-]}{[\mathrm{C}_2\mathrm{H}_5\mathrm{COOH}]} \]This formula shows the ratio of the product of the concentrations of the ions \( \mathrm{H}^+ \) and \( \mathrm{C}_2\mathrm{H}_5\mathrm{COO}^- \) to the concentration of the undissociated acid \( \mathrm{C}_2\mathrm{H}_5\mathrm{COOH} \). The value of \( K_a \) is given and depends on the temperature but is constant for a given substance under standard conditions.Knowing \( K_a \), we can predict how much the acid ionizes at a certain concentration, which is particularly important in calculating the percent ionization of the acid. This is a key part of understanding weak acids and their behavior in solution.

ICE Table

ICE tables, an acronym for Initial, Change, and Equilibrium, are useful organizational tools in chemistry for solving equilibrium problems. They help visualize and calculate changes in the concentrations of reactants and products over the course of a reaction, providing clarity in understanding equilibrium dynamics.For propionic acid, the ICE table is set up based on the initial concentration \(c\) of the acid and the assumption that initially there are 0 concentrations of products \( \mathrm{H}^+ \) and \( \mathrm{C}_2\mathrm{H}_5\mathrm{COO}^- \). As the reaction proceeds, the changes in concentration are described using a variable \( x \), representing the amount of acid that has dissociated:

• Initial: The starting concentration \(c\) is for propionic acid, with zero for both ions.
• Change: The concentration of the acid decreases by \( x \), while the concentration of each ion increases by \( x \).
• Equilibrium: These are the result concentrations: \(c-x\) for the acid and \(x\) for each ion.

By substituting these values into the equilibrium expression \( K_a \) and solving for \( x \), we get the equilibrium concentrations. From here, the percent ionization is calculated as:\[ \text{Percent Ionization} = \frac{x}{c} \times 100\% \]This calculation provides meaningful insights into the extent of ionization of propionic acid under different concentrations, illustrating its weak acid nature and how it behaves under varying conditions.