Question

Saccharin, a sugar substitute, is a weak acid with \(\mathrm{p} K_{a}=2.32\) at \(25^{\circ} \mathrm{C}\). It ionizes in aqueous solution as follows: \(\mathrm{HNC}_{7} \mathrm{H}_{4} \mathrm{SO}_{3}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{NC}_{7} \mathrm{H}_{4} \mathrm{SO}_{3}^{-}(a q)\) What is the \(\mathrm{pH}\) of a \(0.10 \mathrm{M}\) solution of this substance?

Short answer

The pH of the 0.10 M saccharin solution can be calculated using the following steps: 1. Calculate the Ka from the given pKa value: \(Ka = 10^{-2.32}\) 2. Set up the ICE table and write the expression for Ka. 3. Solve the equation for x, assuming x is much smaller than 0.10: \(x^2 = Ka(0.10)\) 4. Calculate the equilibrium concentration of H⁺ ions (x): \(x =\sqrt{(10^{-2.32})(0.10)}\) 5. Calculate the pH of the solution using the calculated x value: \(pH = -\log{x}\)

Step-by-step solution

Find the Ka from the given pKa value

Using the relationship between Ka and pKa, where pKa = -log(Ka), we calculate the Ka for saccharin: \[Ka = 10^{-pKa} = 10^{-2.32}\]

Write the balanced ionization reaction and set up the ICE table

The ionization reaction is given as: \[HNC_7H_4SO_3(aq) \rightleftharpoons H^+(aq) + NC_7H_4SO_3^−(aq)\] The ICE table can be set up as: | | Initial (M) | Change (M) | Equilibrium (M) | |-------------------|-------------|------------|-----------------| | HNC₇H₄SO₃ | 0.10 | -x | 0.10 - x | | H⁺ | 0 | +x | x | | NC₇H₄SO₃⁻ | 0 | +x | x | We want to find the concentration of H⁺ ions at equilibrium, which is x.

Write the expression for Ka and substitute the equilibrium concentrations

Ka is the acid-dissociation constant: \[Ka = \frac{[H^+][NC_{7}H_{4}SO_{3}^{-}]}{[HNC_{7}H_{4}SO_{3}]}\] Substitute the equilibrium concentrations from the ICE table: \[Ka = \frac{x^2}{0.10 - x}\] Solve this equation for x: \[x^2 = Ka(0.10 - x)\] Because saccharin is a weak acid, we can assume that x is much smaller than 0.10, so we have: \[x^2 = Ka(0.10)\]

Calculate the equilibrium concentration of H⁺ ions (x)

Now we can solve for x using the Ka value calculated in Step 1: \[x =\sqrt{Ka(0.10)} = \sqrt{(10^{-2.32})(0.10)}\]

Calculate the pH of the solution

Use the concentration of H⁺ ions to calculate the pH: \[pH = -\log[H^+] = -\log{x}\] After plugging in the value of x from Step 4, we can find the pH of the 0.10 M saccharin solution.

Key concepts

Weak Acid Ionization

Understanding the behavior of weak acids in water is essential when working with pH calculations. Unlike strong acids which fully dissociate, weak acids only partially ionize in solution. This partial ionization is what makes the acid 'weak' and it’s represented by a reversible reaction. For example, saccharin (a weak acid) ionizes in water to form hydrogen ions (H⁺) and the saccharinate anion (NC₇H₄SO₃⁻).

The degree of ionization is not only determined by the acid's inherent strength but also influenced by the concentration of the acid and the properties of the solvent. In dilute solutions, weak acids are less dissociated, as evidenced by the equilibrium that sets up between the non-ionized acid and the ions produced.

Ka and pKa Relationship

The strength of a weak acid is quantified by its acid-dissociation constant (Ka). The larger the Ka, the stronger the acid, meaning it ionizes more in solution. However, working with these constant values when they are extremely small can be cumbersome. To simplify calculations, chemists often use the pKa value, which is the negative base-10 logarithm of the Ka value: \( pKa = -\log(Ka) \).

This relationship also works the other way around, allowing you to find Ka if you know the pKa, which is particularly useful in pH calculations. The pKa provides a more convenient scale for expressing the strength of an acid. In the case of saccharin with a pKa of 2.32, we can calculate its Ka using \( Ka = 10^{-pKa} \) to find the concentration of hydrogen ions at equilibrium.

ICE Table

An ICE table, standing for Initial, Change, and Equilibrium, is an organizational tool used to keep track of the changes in concentrations of reactants and products as a chemical reaction approaches equilibrium. For weak acid ionizations, we start by listing the initial concentrations of the acid and ions, the changes that will occur during reaction, and the concentrations at equilibrium.

When dealing with weak acid ionization, it is generally safe to assume that the change in concentration of the acid (\(x\)) is small compared to the initial concentration, leading to simplifications in calculations. The ICE table helps in clearly defining the relationships between the concentrations of different species involved in the reaction and is instrumental in setting up the equation to calculate Ka.

Acid-Dissociation Constant

The acid-dissociation constant (Ka) provides a measure of the extent to which an acid can donate protons in an aqueous solution. The formula for Ka is \( Ka = \frac{[H^+][A^-]}{[HA]} \), where [H⁺] is the concentration of hydrogen ions, [A⁻] is the concentration of the acid's anion, and [HA] is the concentration of the acid itself.

For weak acids like saccharin, Ka is relatively small, indicating limited ionization. By substituting the equilibrium concentrations derived from the ICE table into the Ka expression, we can solve for the concentration of hydrogen ions. This, in turn, enables us to calculate the pH of the solution. The understanding of Ka is pivotal in making these types of calculations and understanding acid-base equilibria.