Question

The beaker on the right contains 0.1 Macetic acid solution with methyl orange as an indicator. The beaker on the left contains a mixture of 0.1$M$ acetic acid and 0.1$M$ sodium acetate with methyl orange. (a) Using Figures 16.8 and 16.9, which solution has a higher pH? (b) Which solution is better able to maintain its pH when small amounts of NaOH are added? Explain. [Sections 17.1 and 17.2]

Short answer

The pH of the 0.1M acetic acid solution is lower than the pH of the buffer solution (0.1M acetic acid and 0.1M sodium acetate) based on the calculations using the equilibrium constant and Henderson-Hasselbalch equation. The buffer solution is better able to maintain its pH when small amounts of NaOH are added, as it can resist changes in pH due to the presence of both the weak acid and its conjugate base.

Step-by-step solution

Identify the solutions

In the beaker on the right, there is a 0.1M acetic acid solution with methyl orange as an indicator. Acetic acid is a weak acid, and its dissociation in water can be represented as: $$C{H}_{3}COO{H}_{(aq)}\rightleftharpoons C{H}_{3}CO{O}_{(aq)}^{-}+H_{(aq)}^{+}$$ In the beaker on the left, there is a mixture of 0.1M acetic acid and 0.1M sodium acetate with methyl orange as an indicator. Sodium acetate is the sodium salt of the weak acid, acetic acid. The acetate ion can react with water as a base: $$C{H}_{3}CO{O}_{(aq)}^{-}+H_2{O}_{(l)}\rightleftharpoons C{H}_{3}COO{H}_{(aq)}+O{H}_{(aq)}^{-}$$ This solution is an example of a buffer solution, which is a mixture of a weak acid and its conjugate base.

Determine the pH of the two solutions

In order to find the pH of the two solutions, we need to identify the dissociation constants ($K_a$) of acetic acid and the acetate ion. The $K_a$ for acetic acid is $1.8\times {10}^{-5}$. We can use the pH equation to find the pH of the acetic acid solution in the beaker on the right: $$pH=-\log [H^{+}]$$ Since we are given the molarity of the acetic acid solution, we can calculate the [H+] concentration using the equilibrium expression: $$K_a=\frac{[C{H}_{3}CO{O}^{-}][H^{+}]}{[C{H}_{3}COOH]}$$ For the buffer solution in the beaker on the left, we can use the Henderson-Hasselbalch equation to find the pH: $$pH=p{K}_a+\log \frac{[A^{-}]}{[HA]}$$ where $p{K}_a$ is the negative logarithm of $K_a$, $[A^{-}]$ is the concentration of the acetate ion, and $[HA]$ is the concentration of acetic acid. Since both acetic acid and sodium acetate have the same concentration (0.1 M), we can calculate the pH of the buffer solution.

Compare the pH of the two solutions

Let’s first find the pH of the acetic acid solution. Using the equilibrium expression, we can estimate [H+] then calculate the pH. Now find the pH of the buffer solution, by plugging in the values into the Henderson-Hasselbalch equation. Finally, compare both pH values and identify which solution has a higher pH.

Determine which solution maintains its pH better when NaOH is added

When NaOH (a strong base) is added to the solutions, it reacts with any available hydrogen ions present. In the case of the acetic acid solution, the addition of NaOH will neutralize free hydrogen ions creating water and acetate ions which will push the reaction further to the right, decreasing the pH value. In a buffer solution, the addition of NaOH will react with the weak acid (acetic acid) initially, causing the production of its conjugate base (acetate ions). The acetate ions will then react with water as shown in the second reaction, producing hydroxide ions and acetic acid. The presence of both the weak acid and its conjugate base will resist changes in the pH when small amounts of NaOH are added. The solution that will maintain its pH better when small amounts of NaOH are added is the buffer solution (the mixture of acetic acid and sodium acetate) because both the weak acid and its conjugate base are present.

Key concepts

Acetic Acid Dissociation

Understanding the dissolution of acetic acid in water is crucial for grasping the concept of buffer solutions. When acetic acid \textbf{(CH${}_3$COOH)} enters water, it does not fully dissociate into ions like strong acids do. Instead, it establishes an equilibrium between the acid and its dissociated form, leading to far fewer hydrogen ions \textbf{(H${}^{+}$)} in solution. This partial dissociation is represented by the following chemical equation:
$$C{H}_{3}COO{H}_{(aq)}\rightleftharpoons C{H}_{3}CO{O}_{(aq)}^{-}+H_{(aq)}^{+}$$
The degree to which acetic acid dissociates is given by its acid dissociation constant \textbf{(K${}_a$)}, which is relatively small, indicating that acetic acid is indeed a weak acid. This concept is the starting point for understanding why buffer solutions, such as those combining acetic acid with sodium acetate, are effective at maintaining a stable pH.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a cornerstone formula in buffer chemistry, relating the pH of a solution to the pKa (the negative logarithm of the acid dissociation constant) and the ratio of the concentrations of the conjugate base \textbf{(A${}^{-}$)} and the acid \textbf{(HA)}. For a weak acid and its conjugate base, the equation is expressed as:
$$pH=p{K}_a+\text{log}\frac{[A^{-}]}{[HA]}$$
This equation provides a way to approximate the pH of a buffer solution when you know the concentrations of the acid and its conjugate base. It's particularly useful because it demonstrates how a buffer solution works to maintain pH when small amounts of acid or base are added. Buffers maintain pH due to the presence of both a weak acid and a conjugate base in equilibrium, ready to neutralize added acids or bases.

pH Calculation

pH calculation is essential in understanding the acidity or basicity of a solution. The pH scale ranges from 0 to 14, with 7 being neutral. Acids have a pH less than 7, while bases have a pH greater than 7. To determine the pH, we take the negative logarithm of the hydrogen ion concentration \textbf{([H${}^{+}$])}.
$$pH=-\text{log}([H^{+}])$$
In the context of our problem, knowing how to use both the dissociation expression for a weak acid and the Henderson-Hasselbalch equation allows for accurate calculation of pH in different scenarios. For a weak acid like acetic acid, you would estimate the \textbf{[H${}^{+}$]} using its \textbf{Ka}, while for a buffer solution, you would apply the Henderson-Hasselbalch equation.

Weak Acid and Conjugate Base

When discussing weak acids, it's important to note that they have a unique partner called the conjugate base. The conjugate base forms when a weak acid loses a proton. This duality is the foundation of a buffer system. For example, acetic acid (\textbf{CH${}_3$COOH}) has a conjugate base known as acetate (\textbf{CH${}_3$COO${}^{-}$}).
In a buffer solution, such as our mixture of acetic acid and sodium acetate, the balance between the weak acid and its conjugate base allows the solution to mitigate fluctuations in pH. This resistance to changes in the pH when strong acids or bases are added is what makes buffer solutions so valuable in chemical and biological applications where maintaining a stable pH is crucial.