Problem 53 - Chemistry: Principles and Reactions

Question

Calculate \(\left[\mathrm{H}^{+}\right]\) and the \(\mathrm{pH}\) of a \(0.10 \mathrm{M}\) solution of chloroacetic acid \(\left(K_{\mathrm{a}}=1.5 \times 10^{-3}\right)\)

Step-by-Step Solution

Answer

Answer: The concentration of hydrogen ions (\(\mathrm{H}^+\)) is approximately \(0.012 \mathrm{M}\), and the pH of the 0.10 M chloroacetic acid solution is approximately 1.92.

Step 1: Write the dissociation reaction and equilibrium expression

Chloroacetic acid (HA) dissociates in water to form the hydrogen ion (H+) and the chloride ion (A-): $$HA \rightleftharpoons H^+ + A^-$$ The equilibrium expression for the dissociation reaction is given by: $$K_a = \frac{[H^+][A^-]}{[HA]}$$

Step 2: Create an equilibrium table

Now, we will create an equilibrium table to find the concentrations of all species (HA, H+, and A-) present in the solution at equilibrium: | | Initial | Change | Equilibrium | |-------------|---------|--------|-------------| | HA | 0.10 M | -x | 0.10-x M | | H+ | 0 | +x | x M | | A- | 0 | +x | x M | Where x is the change in concentration of hydrogen ions at equilibrium.

Step 3: Solve for the concentration of H+ ions using equilibrium expression

Substitute the concentrations of the species into the equilibrium expression and solve for x: $$K_a = \frac{[H^+][A^-]}{[HA]}$$ $$1.5 \times 10^{-3} = \frac{x \cdot x}{0.10-x}$$ This expression can be solved either by the quadratic equation or by making an assumption that the change in concentration of chloroacetic acid is negligible compared to its initial concentration. Here we will make that assumption because the given Ka value is relatively small: $$1.5 \times 10^{-3} \approx \frac{x^2}{0.10}$$ $$x = \sqrt{1.5 \times 10^{-3} \cdot 0.10}$$ $$x \approx 0.012$$ Thus, the equilibrium concentration of hydrogen ions [H+] is approximately 0.012 M.

Step 4: Calculate the pH value

Now that we have the concentration of hydrogen ions, we can easily calculate the pH of the solution using the following formula: $$\mathrm{pH} = -\log_{10}([H^+])$$ Plug in the value of the H+ concentration, and we find: $$\mathrm{pH} = -\log_{10}(0.012)$$ $$\mathrm{pH} \approx 1.92$$ So, the hydrogen ion concentration (\(\mathrm{H}^+\)) is approximately \(0.012 \mathrm{M}\), and the pH of the 0.10 M chloroacetic acid solution is approximately 1.92.

Key Concepts

Acid Dissociation Constant

The acid dissociation constant, represented as \(K_a\), is an important value in chemistry. It essentially tells us how strongly an acid dissociates in a solution. The higher the value of \(K_a\), the stronger the acid, and it dissociates more in the solution. For chloroacetic acid, the given \(K_a\) is \(1.5 \times 10^{-3}\). This value indicates it is a weak acid, as it does not dissociate completely in water.
The dissociation can be represented by the reaction:

• Chloroacetic acid (HA): \(HA \rightleftharpoons H^+ + A^-\)

The equilibrium expression associated with this reaction is defined by:

• \(K_a = \frac{[H^+][A^-]}{[HA]}\)

In this context, \([H^+]\) represents the concentration of hydrogen ions, \([A^-]\) denotes the concentration of the conjugate base, and \([HA]\) is the concentration of the undissociated chloroacetic acid. This expression allows us to determine how the concentrations of these species relate at equilibrium.

Equilibrium Concentration

Understanding equilibrium concentration involves analyzing how the concentrations of reactants and products settle into a balanced state. In the dissociation of chloroacetic acid, we set up an equilibrium table to predict these concentrations. The initial concentration of chloroacetic acid is given as 0.10 M, with the assumption that there is no initial concentration of \(H^+\) or \(A^-\) ions.
As the acid dissociates:

• The concentration of \(HA\) decreases by \(x\).
• The concentrations of \(H^+\) and \(A^-\) increase by \(x\).

At equilibrium, this becomes:

• \([HA] = 0.10 - x\)
• \([H^+] = x\)
• \([A^-] = x\)

By forming this table, we can substitute values back into the equilibrium expression to solve for \(x\). This helps us find the exact concentration values for each species in the solution at equilibrium.

pH Calculation

Calculating the pH of a solution involves using the concentration of hydrogen ions \([H^+]\). The pH is a measure of the acidity or alkalinity of a solution. It's calculated using the formula:

• \(\mathrm{pH} = -\log_{10}([H^+])\)

Given the hydrogen ion concentration for our chloroacetic acid solution is approximately 0.012 M, the calculation becomes:

• \(\mathrm{pH} = -\log_{10}(0.012)\)

By solving this, we derive that:

• \(\mathrm{pH} \approx 1.92\)

This reflects that the solution is quite acidic. The pH scale ranges from 0 to 14, where values below 7 indicate acidity, 7 is neutral, and values above 7 suggest alkalinity. Thus, a pH of 1.92 confirms the high acidic nature of our solution.

Chloroacetic Acid Dissociation

Chloroacetic acid is an example of a weak acid. When it dissolves in water, it partially dissociates into hydrogen ions and its conjugate base, which is a chloride ion in this case. This dissociation is an equilibrium process, meaning that the forward and reverse reactions occur at the same rate once equilibrium is reached. The reaction is:

• \(\text{Chloroacetic acid (HA)} \rightleftharpoons H^+ + A^-\)

The degree of dissociation and the resulting equilibrium concentrations depend on the initial concentration of the acid and the \(K_a\) value. Because \(K_a\) is relatively low for chloroacetic acid, we assume that the amount of dissociation doesn't greatly reduce the initial concentration of the acid. Thus, for practical calculations, you can neglect the term \(-x\) in \(0.10-x\), simplifying computations.Understanding this balance helps in predicting the behavior of chloroacetic acid in solutions and its effects on the solution's overall pH.