Question

For propanoic acid \(\left(\mathrm{HC}_{3} \mathrm{H}_{5} \mathrm{O}_{2}, K_{\mathrm{a}}=1.3 \times 10^{-5}\right),\) determine the concentration of all species present, the \(\mathrm{pH},\) and the percent dissociation of a \(0.100-M\) solution.

Short answer

The concentration of all species in the $0.100$ M propanoic acid solution are as follows: $[HC_3H_5O_2] = 0.0996$ M, $[C_3H_5O_2^-] = 3.6 \times 10^{-4}$ M, and $[H^+] = 3.6 \times 10^{-4}$ M. The pH of the solution is $3.44$, and the percent dissociation of propanoic acid is $0.36 \%$.

Step-by-step solution

Write the dissociation equation for propanoic acid

Propanoic acid dissociates into propanoate ions and hydrogen ions as follows: \[HC_3H_5O_2 \rightleftharpoons C_3H_5O_2^- + H^+\]

Use the ICE table to determine the change in concentration

Let's set up the ICE table for propanoic acid dissociation: | | HC3H5O2 | C3H5O2- | H+ | |-------|---------|---------|-----| |Initial| 0.100 | 0 | 0 | |Change | -x | +x | +x | |Equil. | 0.100-x | x | x | Here, x represents the change in concentration of each species during the dissociation process.

Write the expression for Ka and solve for x

The expression for Ka is: \[K_a = \frac{[C_3H_5O_2^-][H^+]}{[HC_3H_5O_2]}\] Substitute the equilibrium concentration values from the ICE table and the given Ka value: \[1.3 \times 10^{-5} = \frac{x \times x}{0.100 - x}\] We can assume that x is small compared to 0.100, so 0.100 - x is approximately 0.100. Therefore, the equation becomes: \[1.3 \times 10^{-5} = \frac{x^2}{0.100}\] Solve for x: \[x = \sqrt{1.3 \times 10^{-5} \times 0.100} = 3.6 \times 10^{-4}\]

Calculate the concentration of all species present

Now we can find the equilibrium concentration of each species by plugging x back into the ICE table: | | HC3H5O2 | C3H5O2- | H+ | |-------|---------|---------|-----| |Initial| 0.100 | 0 | 0 | |Change | -x | +x | +x | |Equil. | 0.100-x | x | x | | | HC3H5O2 | C3H5O2- | H+ | |-------|---------|---------|-----| |Equil. | 0.100-3.6×10⁻⁴ | 3.6×10⁻⁴ | 3.6×10⁻⁴ | Equilibrium concentrations: HC3H5O2: 0.100 - 3.6 × 10⁻⁴ = 0.0996 M C3H5O2-: 3.6 × 10⁻⁴ M H+: 3.6 × 10⁻⁴ M

Determine the pH of the solution

The pH of the solution can be calculated using the following formula: \[pH = -\log[H^+]\] \[pH = -\log(3.6 \times 10^{-4}) = 3.44\]

Calculate the percent dissociation

Percent dissociation is the ratio of the dissociated acid concentration to the initial acid concentration, multiplied by 100: \[% \thinspace dissociation = \frac{[H^+]}{[HC_3H_5O_2]_{initial}} \times 100\] \[% \thinspace dissociation = \frac{3.6 \times 10^{-4}}{0.100} \times 100 = 0.36\%\] The concentration of all species in the solution are as follows: [HC3H5O2] = 0.0996 M, [C3H5O2-] = 3.6 × 10⁻⁴ M, and [H+] = 3.6 × 10⁻⁴ M. The pH of the solution is 3.44, and the percent dissociation of propanoic acid is 0.36%.

Key concepts

Propanoic Acid

Propanoic acid is a weak organic acid with the chemical formula \(HC_3H_5O_2\). When dissolved in water, it partially dissociates into propanoate ions \((C_3H_5O_2^-)\) and hydrogen ions \((H^+)\).

This dissociation process is characteristic of weak acids that do not completely ionize in solution. The strength of an acid is quantitatively described by its acid dissociation constant \(K_a\). For propanoic acid, the \(K_a\) value is \(1.3 \times 10^{-5}\), indicating a relatively low degree of ionization compared to strong acids.

Understanding the dissociation behavior of weak acids is essential in predicting the composition of solutions and determining the acidity level (pH). This understanding is crucial in fields like chemistry, biology, environmental science, and various industrial applications.

ICE Table

An ICE table is a valuable tool for organizing data and solving equilibrium problems related to weak acids like propanoic acid. "ICE" stands for Initial, Change, and Equilibrium, representing different stages of the reaction.

In this setup, the initial concentrations of the reactants and products are recorded first. For propanoic acid, we begin with 0.100 M of \(HC_3H_5O_2\), while the concentrations of \(C_3H_5O_2^-\) and \(H^+\) are initially zero, as the acid has not yet dissociated.

• Initial: [HC3H5O2] = 0.100 M, [C3H5O2-] = 0, [H+] = 0.

Next, the change in concentration due to dissociation is represented. In this case, we denote the change by \(x\), where \(x\) is the amount of acid that dissociates. As the reaction proceeds, the concentration of \(HC_3H_5O_2\) decreases by \(x\), and the concentrations of \(C_3H_5O_2^-\) and \(H^+\) increase by \(x\).

• Change: Δ[HC3H5O2] = -x, Δ[C3H5O2-] = +x, Δ[H+] = +x.

Finally, the equilibrium concentrations are calculated. The equilibrium step shows how much of each substance is present once the reaction reaches a steady state. This setup aids in solving the equilibrium expressions necessary to find unknown concentrations.

• Equilibrium: [HC3H5O2] = 0.100 - x, [C3H5O2-] = x, [H+] = x.

pH Calculation

The pH of a solution is a measure of its acidity or alkalinity and is calculated based on the concentration of hydrogen ions \((H^+)\) present. For weak acids like propanoic acid, the dissociation results in a relatively low concentration of \(H^+\), which impacts the overall pH.

Using the change determined from the ICE table, we substitute the value of \(x\) (which is equal to the \([H^+]\)) into the equation for pH:

\[pH = -\log[H^+]\]

For our specific calculation, \(x = 3.6 \times 10^{-4}\). Therefore, the pH of a 0.100 M solution of propanoic acid becomes:

\[pH = -\log(3.6 \times 10^{-4}) \approx 3.44\]

The pH provides insight into the extent of dissociation of the acid and helps to predict the behavior of the solution. A lower pH indicates a higher concentration of hydrogen ions and a more acidic solution.

Percent Dissociation

Percent dissociation quantifies the fraction of original acid molecules that dissociate into ions in solution, expressed as a percentage. It is calculated using the concentration of \([H^+]\) and the initial concentration of the acid.

The formula to calculate percent dissociation is:

\[% \, \text{dissociation} = \left(\frac{[H^+]}{[\text{HC}_3\text{H}_5\text{O}_2]_{\text{initial}}} \right) \times 100\%\]

For propanoic acid with \([H^+] = 3.6 \times 10^{-4} \, \text{M}\), and an initial concentration of \(0.100 \, \text{M}\), the percent dissociation is:

\[% \, \text{dissociation} = \left(\frac{3.6 \times 10^{-4}}{0.100} \right) \times 100 = 0.36\%\]

This indicates that only a small percentage of the propanoic acid molecules dissociate in solution, reflecting its weak acid nature. Understanding percent dissociation is important in predicting the behavior of acid solutions in various chemical environments.