Question

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

Short answer

The percent ionization of propionic acid in solutions with concentrations 0.250 M, 0.0800 M, and 0.0200 M are approximately 0.444%, 1.07%, and 2.61%, respectively.

Step-by-step solution

Using given concentration and \(K_{a}\)

First, we will set up the equations for the three concentrations using the \(K_{a}\) equation and the given concentrations: (a) For 0.250 M: \(K_{a} = \frac{x^2}{0.250 - x}\) (b) For 0.0800 M: \(K_{a} = \frac{x^2}{0.0800 - x}\) (c) For 0.0200 M: \(K_{a} = \frac{x^2}{0.0200 - x}\)

Calculate the concentration of H⁺

Next, we need to solve these equations for x (the concentration of \(\mathrm{H}^+\)): (a) For 0.250 M: \(1.3\times10^{-5} = \frac{x^2}{0.250-x}\) Solving for x, we get: \(x \approx 0.00111\) (b) For 0.0800 M: \(1.3\times10^{-5} = \frac{x^2}{0.0800-x}\) Solving for x, we get: \(x \approx 0.000854\) (c) For 0.0200 M: \(1.3\times10^{-5} = \frac{x^2}{0.0200-x}\) Solving for x, we get: \(x \approx 0.000521\)

Calculate the percent ionization

Finally, we will calculate the percent ionization using the equation: Percent Ionization = \(\frac{[\mathrm{H}^+]}{[\mathrm{HA}]} \times 100\) (a) For 0.250 M: Percent Ionization = \(\frac {0.00111} {0.250} \times 100 = 0.444 \%\) (b) For 0.0800 M: Percent Ionization = \(\frac {0.000854} {0.0800} \times 100 = 1.07 \%\) (c) For 0.0200 M: Percent Ionization = \(\frac {0.000521} {0.0200} \times 100 = 2.61 \%\) So we have calculated the percent ionization for each concentration as follows: (a) 0.444 % for 0.250 M (b) 1.07 % for 0.0800 M (c) 2.61 % for 0.0200 M

Key concepts

Acid Ionization Constant

The acid ionization constant, denoted as \( Ka \), is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for the chemical reaction where an acid donates a proton to water, forming its conjugate base and a hydronium ion. The higher the value of \( Ka \), the stronger the acid and the more it dissociates in solution, leading to higher concentrations of protons (\( H^+ \)). Understanding \( Ka \) is essential in determining the extent of acid ionization, which in turn plays a crucial role in calculating pH and percent ionization for a given acid concentration.
In the case of propionic acid \( (C_2H_5COOH) \), which is a weak acid, we consider the ionization reaction \( C_2H_5COOH \rightarrow C_2H_5COO^- + H^+ \). The value of \( Ka \) given in the appendix helps us predict how much of the acid will ionize when dissolved in water.

Equilibrium Concentration

Equilibrium concentration refers to the concentration of reactants and products in a chemical reaction that has reached a state of dynamic equilibrium. When dealing with weak acids like propionic acid, not all of the acid molecules ionize. Some remain intact, and an equilibrium is established between the un-ionized form and the ions produced.
To solve for percent ionization, we first express the equilibrium concentrations of the products and reactants using the stoichiometry of the ionization reaction. By setting up an ICE (Initial, Change, Equilibrium) table, we can see how the initial concentration changes as ionization progresses and use the equilibrium values to find the concentration of protons (\( H^+ \)). This process involves some approximations, particularly when the acid concentration is much larger than the \( Ka \), where the change in concentration due to ionization can be deemed negligible.

Propionic Acid

Propionic acid \( (C_2H_5COOH) \) is a carboxylic acid and a weak acid commonly employed in the food industry as a preservative. Being a weak acid means that it only partially ionizes in aqueous solutions. During the ionization of propionic acid, it releases protons (\( H^+ \)), leading to an increase in the hydrogen ion concentration of the solution. This process impacts both the pH of the solution and the observable percent ionization.
In weak acid calculations, it's important to factor in that these acids have a relatively low percent ionization, which means that when you dissolve propionic acid in water, most of the molecules remain non-ionized. The percent ionization will also depend on the concentration of the acid; dilute solutions usually result in a higher percent ionization compared to concentrated ones.

pH Calculation

The pH of a solution is a measure of its acidity or alkalinity, expressed on a logarithmic scale from 0 to 14, with 7 being neutral. Calculating pH involves finding the negative logarithm of the hydrogen ion concentration (\( -\text{log}[H^+] \)). For weak acids like propionic acid, this process requires first finding the \( H^+ \) concentration at equilibrium, which we obtain using the \( Ka \) value and initial concentration of the acid.
Once the \( H^+ \) concentration is known, calculating the pH is straightforward. However, remember that for dilute solutions of weak acids, the autoionization of water becomes significant and should be considered. This can slightly impact the final pH value of the solution. Additionally, temperature and ionic strength can influence the pH, although these variables are often held constant in introductory chemistry problems.