Problem 49 - Chemistry The Central Science

Question

Write the chemical equation and the \(K_{a}\) expression for the ionization of each of the following acids in aqueous solution. First show the reaction with \(\mathrm{H}^{+}(a q)\) as a product and then with the hydronium ion: (a) \(\mathrm{HBrO}_{2}\), (b) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\).

Step-by-Step Solution

Answer

(a) For \(\mathrm{HBrO}_{2}\), the chemical equations are: \[ \mathrm{HBrO}_2 (aq) \rightleftharpoons \mathrm{H}^+(aq) + \mathrm{BrO}_2^-(aq) \] and \[ \mathrm{HBrO}_2(aq) + \mathrm{H}_2\mathrm{O}(l) \rightleftharpoons \mathrm{H}_3\mathrm{O}^+(aq) + \mathrm{BrO}_2^-(aq) \] The \(K_{a}\) expression is: \[K_{a} = \frac{[\mathrm{H}^+][\mathrm{BrO}_2^-]}{[\mathrm{HBrO}_2]}\] (b) For \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\), the chemical equations are: \[ \mathrm{C}_2\mathrm{H}_5\mathrm{COOH}(aq) \rightleftharpoons \mathrm{H}^+(aq) + \mathrm{C}_2\mathrm{H}_5\mathrm{COO}^-(aq) \] and \[ \mathrm{C}_2\mathrm{H}_5\mathrm{COOH}(aq) + \mathrm{H}_2\mathrm{O}(l) \rightleftharpoons \mathrm{H}_3\mathrm{O}^+(aq) + \mathrm{C}_2\mathrm{H}_5\mathrm{COO}^-(aq) \] The \(K_{a}\) expression is: \[K_{a} = \frac{[\mathrm{H}^+][\mathrm{C}_2\mathrm{H}_5\mathrm{COO}^-]}{[\mathrm{C}_2\mathrm{H}_5\mathrm{COOH}]}\]

Step 1: Chemical equation with \(\mathrm{H}^{+}(a q)\) as product

When \(\mathrm{HBrO}_{2}\) dissociates in water, it loses one proton (\(\mathrm{H}^{+}\)) and forms the anion \(\mathrm{BrO}_{2}^{-}\). The chemical equation for the ionization is: \[ \mathrm{HBrO}_2 (aq) \rightleftharpoons \mathrm{H}^+(aq) + \mathrm{BrO}_2^-(aq) \]

Step 2: Chemical equation with hydronium ion

When considering the hydronium ion in the ionization, a \(\mathrm{H}_{2}\mathrm{O}\) molecule accepts the proton from \(\mathrm{HBrO}_{2}\), forming a hydronium ion \(\mathrm{H}_{3}\mathrm{O}^{+}\). The chemical equation is: \[ \mathrm{HBrO}_2(aq) + \mathrm{H}_2\mathrm{O}(l) \rightleftharpoons \mathrm{H}_3\mathrm{O}^+(aq) + \mathrm{BrO}_2^-(aq) \]

Step 3: \(K_{a}\) expression for \(\mathrm{HBrO}_{2}\)

The acid dissociation constant expression, \(K_{a}\), for the ionization of \(\mathrm{HBrO}_{2}\) is given by: \[ K_{a} = \frac{[\mathrm{H}^+][\mathrm{BrO}_2^-]}{[\mathrm{HBrO}_2]} \] (b) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\)

Step 4: Chemical equation with \(\mathrm{H}^{+}(a q)\) as product

When \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\) dissociates in water, it loses one proton (\(\mathrm{H}^{+}\)) and forms the anion \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COO}^{-}\). The chemical equation for the ionization is: \[ \mathrm{C}_2\mathrm{H}_5\mathrm{COOH}(aq) \rightleftharpoons \mathrm{H}^+(aq) + \mathrm{C}_2\mathrm{H}_5\mathrm{COO}^-(aq) \]

Step 5: Chemical equation with hydronium ion

When considering the hydronium ion in the ionization, a \(\mathrm{H}_{2}\mathrm{O}\) molecule accepts the proton from \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\), forming a hydronium ion \(\mathrm{H}_{3}\mathrm{O}^{+}\). The chemical equation is: \[ \mathrm{C}_2\mathrm{H}_5\mathrm{COOH}(aq) + \mathrm{H}_2\mathrm{O}(l) \rightleftharpoons \mathrm{H}_3\mathrm{O}^+(aq) + \mathrm{C}_2\mathrm{H}_5\mathrm{COO}^-(aq) \]

Step 6: \(K_{a}\) expression for \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\)

The acid dissociation constant expression, \(K_{a}\), for the ionization of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\) is given by: \[ K_{a} = \frac{[\mathrm{H}^+][\mathrm{C}_2\mathrm{H}_5\mathrm{COO}^-]}{[\mathrm{C}_2\mathrm{H}_5\mathrm{COOH}]} \]

Key Concepts

Chemical Equations

Chemical equations are a symbolic representation of chemical reactions between substances. They showcase the transformation of reactants into products. In the context of acid reactions, a chemical equation is important because it helps us visualize how an acid donates a proton, or \(\text{H}^{+}\), to other molecules. In the dissociation of \(\text{HBrO}_2\) and \(\text{C}_2\text{H}_5\text{COOH}\) in water:

• \(\mathrm{HBrO}_2(aq) \rightleftharpoons \mathrm{H}^+(aq) + \mathrm{BrO}_2^-(aq)\)
• \(\mathrm{C}_2\mathrm{H}_5\mathrm{COOH}(aq) \rightleftharpoons \mathrm{H}^+(aq) + \mathrm{C}_2\mathrm{H}_5\mathrm{COO}^-(aq)\)

These equations represent the ionization process where the acids break apart, releasing \(\mathrm{H}^+\) ions into the solution. Understanding these equations is crucial because it reflects how different acids ionize differently, affecting their reactivity and behavior in chemical environments.

Ionization

Ionization is the process where an atom or molecule gains a negative or positive charge by gaining or losing electrons, often in an aqueous solution. For acids, it specifically involves the loss of a proton, transforming the acid into its conjugate base. In the example of \(\text{HBrO}_2\) and \(\text{C}_2\text{H}_5\text{COOH}\):

• \(\text{HBrO}_2\) loses a proton to become \(\text{BrO}_2^\ -\)
• \(\text{C}_2\text{H}_5\text{COOH}\) loses a proton to become \(\text{C}_2\text{H}_5\text{COO}^\ -\)

This exchange is fundamental to acid-base chemistry. Ionization is vital because it determines how an acid will react in solution, influencing the acidity and basicity of the solution. Ionization also helps in defining the strength of an acid, as stronger acids ionize more completely whereas weaker acids ionize less.

Acid Dissociation Constant

The acid dissociation constant, represented as \(K_{a}\), is an essential concept in understanding the strength of acids in a solution. It provides a numerical description of an acid's ability to donate protons, which is a core characteristic of an acid. The higher the \(K_{a}\) value, the stronger the acid, as it implies a greater degree of ionization in solution. For the acids \(\text{HBrO}_2\) and \(\text{C}_2\text{H}_5\text{COOH}\), their dissociation constants are expressed as:

• For \(\mathrm{HBrO}_2\), \[K_a = \frac{[\text{H}^+][\text{BrO}_2^-]}{[\text{HBrO}_2]}\]
• For \(\mathrm{C}_2\mathrm{H}_5\mathrm{COOH}\), \[K_a = \frac{[\text{H}^+][\text{C}_2\text{H}_5\text{COO}^-]}{[\text{C}_2\text{H}_5\text{COOH}]}\]

Calculating \(K_{a}\) helps in predicting the behavior of an acid in various chemical systems, assisting chemists in selecting appropriate acids for desired reactions.