Problem 45 - General Chemistry

Question

What is the hydronium-ion concentration of a \(2.00 M\) solution of 2,6 -dinitrobenzoic acid, \(\left(\mathrm{NO}_{2}\right)_{2} \mathrm{C}_{6} \mathrm{H}_{3} \mathrm{COOH},\) for which \(K_{a}=7.94 \times 10^{-2} ?\)

Step-by-Step Solution

Answer

The hydronium-ion concentration is approximately \(0.3985\, M\).

Step 1: Understanding the Dissociation Equation

2,6-dinitrobenzoic acid, \(\text{(NO}_2\text{)}_2\text{C}_6\text{H}_3\text{COOH}\), is a weak acid. When it dissociates in water, it forms hydronium ions (\(\text{H}_3\text{O}^+\)) and the corresponding anion ([\(\text{(NO}_2\text{)}_2\text{C}_6\text{H}_3\text{COO}^-\)]). The equation of dissociation is given as: \[\text{(NO}_2\text{)}_2\text{C}_6\text{H}_3\text{COOH(aq)} + \text{H}_2\text{O(l)} \rightleftharpoons \text{H}_3\text{O}^+(aq) + \text{(NO}_2\text{)}_2\text{C}_6\text{H}_3\text{COO}^-(aq)\]

Step 2: Apply the Expression for the Acid Dissociation Constant

The dissociation constant \(K_a\) for the acid can be expressed as: \[K_a = \frac{[\text{H}_3\text{O}^+][\text{(NO}_2\text{)}_2\text{C}_6\text{H}_3\text{COO}^-]}{[\text{(NO}_2\text{)}_2\text{C}_6\text{H}_3\text{COOH}]}\]Here, the initial concentration of the acid before dissociation is given as \(C = 2.00\, M\).

Step 3: Define Changes in Concentration

At the start, before dissociation, the concentration of 2,6-dinitrobenzoic acid is \(2.00\, M\), while concentrations of hydronium ions and anions are \(0\, M\). If \(x\) is the concentration of \(\text{H}_3\text{O}^+\) ions formed at equilibrium, then:- \([\text{(NO}_2\text{)}_2\text{C}_6\text{H}_3\text{COO}^-] = x\)- \([\text{(NO}_2\text{)}_2\text{C}_6\text{H}_3\text{COOH}] = 2.00 - x\) Since \(K_a\) is quite high, let's assume \(x\) is not too small.

Step 4: Solve for Hydronium Ion Concentration

Now substitute these relationships into the \(K_a\) expression: \[7.94 \times 10^{-2} = \frac{x^2}{2.00 - x}\]Assume \(x\) is small compared to \(2.00\), thus \(2.00 - x \approx 2.00\). Then,\[7.94 \times 10^{-2} \approx \frac{x^2}{2.00}\]Solving for \(x\):\[x^2 = 2.00 \times 7.94 \times 10^{-2}\]\[x^2 = 0.1588\]\[x = \sqrt{0.1588} \approx 0.3985\, M\]Thus, the hydronium ion concentration \([\text{H}_3\text{O}^+]\) is approximately \(0.3985\, M\).

Key Concepts

Acid Dissociation Constant

The acid dissociation constant, often symbolized as \( K_a \), is fundamental in understanding the behavior of weak acids in solution. It is a quantitative measure of the strength of an acid in solution. More specifically, \( K_a \) is the equilibrium constant for the dissociation (ionization) of the acid into its ions. This constant equation, \[ K_a = \frac{[\text{H}_3\text{O}^+][\text{A}^-]}{[\text{HA}]} \], tells us about the balance between the non-ionized form of the acid \( \text{HA} \) and its ionized form \( \text{H}_3\text{O}^+ \) and \( \text{A}^- \).

• Higher \( K_a \) value indicates a stronger acid which dissociates more in solution.
• Lower \( K_a \) implies that the acid is weak and dissociates less.

Understanding \( K_a \) is crucial for calculating the concentrations of components in a weak acid solution at equilibrium and therefore determining the hydronium ion concentration.

Weak Acid

A weak acid is one that only partially dissociates into ions when dissolved in water. This means that not all of the acid molecules donate protons to water molecules to form hydronium ions \( \text{H}_3\text{O}^+ \). In the dissociation of weak acids, the equilibrium lies to the left, favoring the non-dissociated form.

• Examples of weak acids include acetic acid and, as mentioned in this exercise, 2,6-dinitrobenzoic acid.
• Due to partial dissociation, weak acids have specific \( K_a \) values that help to predict the degree of ionization in solution.

The behavior of weak acids is significant in various fields, including chemistry and biology, where buffering and pH control are vital.

Chemical Equilibrium

Chemical equilibrium refers to a state in a reversible reaction where the rates of the forward and reverse reactions are equal. At this point, the concentrations of reactants and products remain unchanged over time. For the dissociation of a weak acid, this concept is essential as it determines the specific concentrations of all species involved at equilibrium.

• In the acid dissociation, both the non-ionized acid and the ionized products are present in a balance at equilibrium.
• This can be represented by a balanced chemical reaction: \( \text{HA} + \text{H}_2\text{O} \rightleftharpoons \text{H}_3\text{O}^+ + \text{A}^- \).

Understanding chemical equilibrium helps in solving equilibrium problems like finding the hydronium ion concentration using the known \( K_a \) value.

Concentration Changes

In a reaction involving a weak acid, such as the dissociation equation for 2,6-dinitrobenzoic acid, concentration changes occur from the initial state to equilibrium. Initially, the concentration of the acid is known, but the concentrations of ions form as the reaction progresses to equilibrium.

• The initial concentration of the weak acid before dissociation is called \( C \).
• As the dissociation reaches equilibrium, changes in concentration are small relative to the initial concentration.
• These changes can be expressed through variables: let \( x \) represent the change in concentration of ions formed.

This concept is the basis for establishing the concentrations in the equilibrium expression and is crucial for calculating the actual concentration of the hydronium ions in the solution.