Question

The pH of a glycine solution, in which the $\alpha\text{-NH}_3^+$ group (having $\text{pK}_a$ 9.6) is one third dissociated would be equal to [given $\log 2 = 0.3010$ and $\log 3 = 0.4771$]}

• A. 9.3
• B. 9.9
• C. 10.1
• D. 9.1

Hint:

When an acidic group is less than half dissociated ($< 50\%$), the pH of the solution must be lower than its $\text{pK}_a$.
This relationship helps quickly eliminate options higher than the $\text{pK}_a$ of $9.6$ (like $9.9$ and $10.1$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The pH of a weak acid or weak base solution undergoing ionization is calculated using the Henderson-Hasselbalch equation.
This equation quantitatively relates pH, the acid dissociation constant ($\text{pK}_a$), and the relative ratio of the conjugate base and its conjugate acid.
Key Formula or Approach:
The Henderson-Hasselbalch equation is:
\[ \text{pH} = \text{pK}_a + \log \frac{[\text{Conjugate Base}]}{[\text{Acid}]} \]

Step 2: Detailed Explanation:
In this system, the chemical equilibrium represents the deprotonation of the cationic amino group of glycine:
\[ \text{Glycine-NH}_3^+ \rightleftharpoons \text{Glycine-NH}_2 + \text{H}^+ \]
Here, the conjugate acid is $\text{Glycine-NH}_3^+$, and the conjugate base is $\text{Glycine-NH}_2$.
The problem states that the amino group is "one-third dissociated."
This indicates that:
\[ [\text{Conjugate Base}] = \frac{1}{3} \text{ of the total concentration} \]
\[ [\text{Acid}] = 1 - \frac{1}{3} = \frac{2}{3} \text{ of the total concentration} \]
Substituting these values into the ratio of base to acid:
\[ \frac{[\text{Conjugate Base}]}{[\text{Acid}]} = \frac{1/3}{2/3} = \frac{1}{2} \]
Now, substitute the ratio and the given $\text{pK}_a$ value ($9.6$) into the Henderson-Hasselbalch equation:
\[ \text{pH} = 9.6 + \log \left(\frac{1}{2}\right) \]
\[ \text{pH} = 9.6 - \log 2 \]
Using the given value of $\log 2 = 0.3010$:
\[ \text{pH} = 9.6 - 0.3010 = 9.299 \approx 9.3 \]

Step 3: Final Answer:
The pH of the glycine solution is approximately 9.3, which corresponds to Option (A).