Question

Acid Solutions. Find the \(\mathrm{pH}\) and percent ionization of a 0.100 \(\mathrm{M}\) solution of a weak monoprotic acid having the given \(K_{\mathrm{a}}\) values. $$\begin{array}{ll}{\text { a. } K_{\mathrm{a}}} {=1.0 \times 10^{-5}} \\\ {\text { b. }} {K_{\mathrm{a}}=1.0 \times 10^{-3}} \\ {\text { c. }} {K_{\mathrm{a}}=1.0 \times 10^{-1}}\end{array}$$

Short answer

For each Ka value, the pH is calculated using -log(x), and percent ionization using the formula (x/0.100)×100%, where x is the ionization concentration ( a. pH = 3.00, Percent Ionization = 0.10%; b. pH = 2.00, Percent Ionization = 1.00%; c. pH = 1.00, Percent Ionization = 10.00%).

Step-by-step solution

Write the Equilibrium Equation

For a weak monoprotic acid, HA, the dissociation in water can be written as: ewline HA(aq) + H2O(l) ⇌ H3O+(aq) + A−(aq). From this equation, you can express the equilibrium constant expression, Ka, as: ewline Ka=[H3O+][A−]/[HA].

Define the Variables

Let 'x' be the concentration of H3O+ and A− at equilibrium. Because HA is a weak acid and only partially ionizes, the change in concentration of HA can also be considered 'x'. The equilibrium concentrations of the species involved would be [HA]=[HA]initial−x, [H3O+]=x, and [A−]=x.

Write the Ka Expression and Solve for x

Based on the defined variables, the Ka expression for the weak acid can be written as: ewline Ka=x^2/([HA]initial−x). Since x is much less than 0.100 M, we can assume that [HA]initial−x ≈ [HA]initial. Thus, Ka ≈ x^2/([HA]initial), and x can be calculated as x = √(Ka×[HA]initial).

Calculate the pH for Each Ka

The pH is the negative logarithm of the H3O+ concentration (x). So, pH = -log(x). Calculate this for each given Ka value to find the corresponding pH.

Calculate the Percent Ionization

Percent ionization is found using the formula: (concentration ionized/original concentration)×100%. For this, use the value of x for each case as the concentration ionized, and the given 0.100 M as the original concentration to find the percent ionization for each Ka.

Determine the pH and Percent Ionization for Each Case

Use the Ka values to determine x for each given case and follow the formulas provided in Steps 4 and 5 to obtain the pH and percent ionization for each of the following: ewline a. Ka = 1.0 × 10−5 b. Ka = 1.0 × 10−3 c. Ka = 1.0 × 10−1.

Key concepts

Weak Acid Dissociation

Understanding weak acid dissociation is fundamental when studying the behavior of acids in water. A weak acid partially ionizes in solution, meaning not all the acid molecules will donate a proton (H+) to water. This incomplete ionization is what distinguishes weak acids from strong acids, which fully dissociate.

The process of dissociation for a weak monoprotic acid, represented by the generic formula HA, can be illustrated by the reversible reaction:
\[\begin{equation}HA(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + A^-(aq)\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\textnormal{}\end{equation}\]In this equation, HA is the acid that dissociates, H3O+ is the hydronium ion, and A- is the conjugate base of the acid. The reversible nature of the reaction implies that an equilibrium is established between the reactants and the products.

Equilibrium Constant Expression (Ka)

The extent to which a weak acid dissociates into ions is measured by its acid dissociation constant, denoted as Ka. The Ka value is a reflection of the acid's strength—the smaller the Ka, the weaker the acid, and vice versa. The equilibrium constant expression for a weak acid is written like so:
\[\begin{equation}Ka = \frac{[H_3O^+][A^-]}{[HA]}\end{equation}\]
In this expression, [H3O+] and [A-] represent the concentrations of the hydronium ion and the conjugate base at equilibrium, and [HA] is the concentration of the undissociated acid. The square brackets indicate concentration in moles per liter (Molarity, M). In situations where the initial concentration of the acid and its degree of ionization are known, the Ka expression allows us to calculate the equilibrium concentrations of all species present in the system.

pH Calculation

The pH of a solution is a measure of its acidity and is defined by the negative logarithm of the H3O+ ion concentration. The pH scale typically runs from 0 to 14, with 7 being neutral. Lower pH values correspond to higher acidity.

To calculate the pH of a weak acid solution, one would follow these steps:

• Determine the concentration of H3O+ ions at equilibrium, which we often denote as 'x'.
• Use the equilibrium constant expression to solve for x.
• Once x is found, use the formula \(\text{pH} = -\log_{10}(x)\)to find the pH.

In the process of solving for x, one may make the simplification that the change in the weak acid's concentration due to ionization is negligible, thus making calculations more straightforward. This is because x is typically much smaller than the initial concentration of the weak acid.

Percent Ionization

Percent ionization expresses the extent of ionization of an acid in solution. It helps in understanding the degree to which an acid releases protons when dissolved in water.

To calculate the percent ionization of a weak acid, use the following formula: \[\text{Percent Ionization} = \left(\frac{\text{Concentration Ionized}}{\text{Original Concentration}}\right) \times 100\%\]
The 'Concentration Ionized' refers to the concentration of H3O+ ions at equilibrium, while the 'Original Concentration' denotes the initial concentration of the weak acid. For example, if a 0.100 M solution of a weak acid yields a 0.005 M concentration of H3O+ at equilibrium, then the percent ionization is (0.005 M / 0.100 M) x 100%, resulting in a 5% ionization. This calculation is valuable in estimating how a weak acid behaves in solution, and it provides insight into the relative strengths of weak acids.