Question

Calculate the \(\mathrm{pH}\) of a \(0.20 \mathrm{M}\) ammonium acetate \(\left(\mathrm{CH}_{3} \mathrm{COONH}_{4}\right)\) solution.

Short answer

The pH of the 0.20 M ammonium acetate solution is approximately 4.74.

Step-by-step solution

Understanding the Components

Ammonium acetate ( \( \text{CH}_3 \text{COONH}_4 \)) is a salt formed by acetic acid ( \( \text{CH}_3 \text{COOH} \)) and ammonia ( \( \text{NH}_3 \)). It's important to recognize it acts as a buffer solution, capable of maintaining \( \text{pH} \) stability.

Determine the Acid and Base Dissociation Constants

For ammonium acetate, use the acid dissociation constant \( K_a \) of acetic acid and the base dissociation constant \( K_b \) of ammonia. Typically, \( K_a = 1.8 \times 10^{-5} \) for acetic acid and \( K_b = 1.8 \times 10^{-5} \) for ammonium.

Calculate the Concentrations of Buffer Components

Since ammonium acetate fully dissociates in water, the concentration of acetic acid \( \text{CH}_3 \text{COOH} \) and ammonium \( \text{NH}_4^+ \) in solution will both be \( 0.20 ext{ M} \).

Apply the Henderson-Hasselbalch Equation

The \( \text{pH} \) of a buffer solution can be calculated using the Henderson-Hasselbalch equation: \[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \] where \([\text{A}^-] = \text{0.20 M} \, (\text{acetate})\) and \([\text{HA}] = \text{0.20 M} \, (\text{acetic acid})\). Since the concentrations of \([\text{A}^-]\) and \([\text{HA}]\) are equal, the log term equals zero.

Calculated the pKa

The \( \text{pKa} \) of acetic acid is calculated by taking the negative logarithm of \( K_a \): \( \text{pKa} = -\log(1.8 \times 10^{-5}) \approx 4.74 \).

Final Calculation of pH

Using the equation from the Henderson-Hasselbalch equation and knowing \( \log(1) = 0 \), we find \( \text{pH} = \text{pKa} \approx 4.74 \).

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a valuable tool in chemistry, particularly when dealing with buffer solutions. This equation helps in calculating the pH of a solution where the concentrations of an acid and its conjugate base (or a base and its conjugate acid) are known. The equation is given by: \[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \] where \([\text{A}^-]\) represents the concentration of the conjugate base and \([\text{HA}]\) represents the concentration of the acid.
To use this equation effectively, it's essential to know both the pKa of the acid and the respective concentrations of the components in the buffer solution.
In cases where the concentrations of \([\text{A}^-]\) and \([\text{HA}]\) are equal, the logarithmic part of the equation becomes zero, and thus \(\text{pH} = \text{pKa}\). This simplification makes it particularly convenient when the system is equimolar.

Buffer solution

A buffer solution is a special type of solution that resists changes in pH when small amounts of acid or base are added. This stability is achieved by having a balance between a weak acid and its conjugate base, or a weak base and its conjugate acid.
For instance, in the case of ammonium acetate, the solution acts as a buffer because it contains both acetate ions \(\text{CH}_3\text{COO}^-\) and ammonium ions \(\text{NH}_4^+\).
Such compositions help to maintain the pH of a solution within a particular range, which is essential in many biological and chemical processes. Buffers are widely used in laboratory settings to maintain a stable environment for reactions, and in our bodies to keep the blood at a consistent pH.

Acid dissociation constant

The acid dissociation constant, denoted as \(K_a\), measures the strength of an acid in a solution. It reflects the ability of an acid to donate protons to the solution, thus dissociating into hydrogen and its conjugate base.
Mathematically, it is expressed as: \[ K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \]A larger \(K_a\) value indicates a stronger acid, which means it dissociates in water more completely.
In the context of the example given, acetic acid has a \(K_a\) of \(1.8 \times 10^{-5}\), which suggests that it is a weak acid, only partially dissociating in solution.
This concept is crucial in pH calculations, as it helps determine the pKa, which is simply the negative logarithm of \(K_a\), providing insight into the acid's behavior in a buffer solution.

Base dissociation constant

The base dissociation constant, \(K_b\), measures the extent to which a base dissociates in an aqueous solution. It quantifies the strength of a base by indicating its ability to accept protons.
The equation for \(K_b\) is given by:\[ K_b = \frac{[\text{BH}^+][\text{OH}^-]}{[\text{B}]} \] Here, \([\text{BH}^+]\) and \([\text{OH}^-]\) are the concentrations of the conjugate acid and hydroxide ions formed, respectively.
For ammonia in our example, \(K_b\) is also \(1.8 \times 10^{-5}\), indicating that ammonia is a weak base, partially accepting protons to form ammonium.
Understanding \(K_b\) is essential in predicting how bases will behave in chemical reactions and adjusting conditions to maintain a target pH, especially in buffer solutions.