Question

Calculate the ionization constant for the given acids. Each acid ionizes as follows: \(\mathrm{HA} \rightleftharpoons \mathrm{H}^{+}+\mathrm{A}^{-}\). $$ \begin{array}{|l|c|c|} \hline \text { Acid } & \begin{array}{c} \text { Acid } \\ \text { concentration } \end{array} & {\left[\mathrm{H}^{+}\right]} \\ \hline \text { Hypochlorous, } \mathrm{HOCl} & 0.10 M & 5.9 \times 10^{-5} \mathrm{~mol} / \mathrm{L} \\ \hline \text { Propanoic, } \mathrm{HC}_{3} \mathrm{H}_{5} \mathrm{O}_{2} & 0.15 M & 1.4 \times 10^{-3} \mathrm{~mol} / \mathrm{L} \\ \hline \text { Hydrocyanic, } \mathrm{HCN} & 0.20 M & 8.9 \times 10^{-6} \mathrm{~mol} / \mathrm{L} \\ \hline \end{array} $$

Short answer

\( K_a \) for hypochlorous, propanoic, and hydrocyanic acids are \( 3.481 \times 10^{-7} \), \( 1.31 \times 10^{-5} \), and \( 3.961 \times 10^{-9} \), respectively.

Step-by-step solution

Identify the Equilibrium Expression for Ionization

Write the equilibrium expression for the ionization of an acid HA. The ionization constant, Ka, is given by the expression: \( K_a = \frac{[\mathrm{H}^{+}][\mathrm{A}^{-}]}{[\mathrm{HA}]} \).

Calculate the Ionization Constant for Hypochlorous Acid

Use the given concentration of hypochlorous acid (0.10 M) and the concentration of hydrogen ions (\(5.9 \times 10^{-5} \) M) to find the ionization constant (\(K_a\)). Since the equation shows a 1:1 ratio of acid to hydrogen ions, \([A^{-}]\) will be equal to \([H^{+}]\). Therefore, \( K_a = \frac{(5.9 \times 10^{-5})(5.9 \times 10^{-5})}{0.10} \).

Calculate the Ionization Constant for Propanoic Acid

For propanoic acid, use the given concentration of propanoic acid (0.15 M) and the concentration of hydrogen ions (\(1.4 \times 10^{-3} \) M). Since the ratio is still 1:1, \( K_a = \frac{(1.4 \times 10^{-3})(1.4 \times 10^{-3})}{0.15} \).

Calculate the Ionization Constant for Hydrocyanic Acid

Finally, for hydrocyanic acid, use its concentration (0.20 M) and the concentration of hydrogen ions (\(8.9 \times 10^{-6} \) M) to find the ionization constant. With the 1:1 ratio, \( K_a = \frac{(8.9 \times 10^{-6})(8.9 \times 10^{-6})}{0.20} \).

Compute the Values

Perform the calculations to get the values of the ionization constants: For hypochlorous acid \( K_a = 3.481 \times 10^{-7} \), for propanoic acid \( K_a = 1.31 \times 10^{-5} \), and for hydrocyanic acid \( K_a = 3.961 \times 10^{-9} \).

Key concepts

Acid Ionization

When we talk about acid ionization, we're referring to the process by which an acid (HA) releases a proton (H+) to form its conjugate base (A-). This is a fundamental chemical reaction that occurs when an acid is dissolved in water. Acid ionization is a balance, or an equilibrium, between the un-ionized acid (HA) and the ions produced (H+ and A-).

Imagine a crowded dance floor representing a solution where dancers (molecules of HA) are freely moving. Occasionally, a dancer steps out and takes off a hat (the proton) and becomes a different kind of dancer (A-). Likewise, in an acidic solution, not all acid molecules will release a proton, but only a certain amount will ionize at any given moment. This equilibrium is crucial for understanding the acid's strength, which is directly related to how readily it donates protons.

Equilibrium Expression

Discovering the equilibrium expression for acid ionization is akin to finding the recipe for the perfect balance of flavors in a dish. It is a mathematical representation of the balance between the reactants and the products in a chemical reaction at equilibrium.

The equilibrium expression is the ratio of the concentration of the products to the concentration of the reactants, each raised to the power of their coefficients in the balanced chemical equation. For the ionization of an acid HA, the equilibrium expression is
\[\begin{equation}K_a = \frac{[H+][A-]}{[HA]}\end{equation}\]where \([H+]\) is the concentration of hydrogen ions, \([A-]\) is the concentration of the conjugate base, and \([HA]\) is the concentration of the un-ionized acid. The equilibrium constant for this reaction is symbolized by Ka, and it is unique for each acid, giving us essential information about the acid's propensity to ionize.

Ka Calculation

Calculating the ionization constant, or Ka, of an acid is like cracking a secret code that tells us how strong or weak an acid is in aqueous solution. The lower the Ka value, the weaker the acid, and vice versa.

In practice, to calculate Ka, we need to know the concentrations of the reactants and the products at equilibrium. The steps illustrated previously for hypochorous, propanoic, and hydrocyanic acid simplistically assume a 1:1 ratio in their ionization, which means that for every molecule of acid that ionizes, one molecule of H+ and one molecule of A- are formed. This direct proportionality allows us to use the given concentrations directly in the equilibrium expression without further adjustments for the stoichiometry of the reaction.

By calculating Ka, as shown in the steps, we quantify the acid's ability to donate H+ ions, which in turn enhances a student's ability to predict the behavior of acids in chemical reactions. The calculations involve squared concentrations because both the H+ and A- ions originate from the same HA molecule in equal amounts. Thus, understanding how to manipulate these expressions mathematically can reveal much about the intrinsic properties of acids.