The pH of 0.001 M solution of a weak acid `(HA)` is 4.50. It is neutralised with certain ammount of `NaOH` solution to decrease the acid content to half of initial value. Calculate the pH of the resulting solution.

Answer

To solve the problem step-by-step, we will follow the reasoning laid out in the video transcript.

### Step 1: Determine the hydrogen ion concentration from pH
Given that the pH of the weak acid solution (HA) is 4.5, we can calculate the hydrogen ion concentration \([H^+]\) using the formula:
\[ [H^+] = 10^{-\text{pH}} = 10^{-4.5} \]
Calculating this gives:
\[ [H^+] = 3.16 \times 10^{-5} \, \text{M} \]

**Hint:** Remember that the hydrogen ion concentration can be derived directly from the pH using the formula \( [H^+] = 10^{-\text{pH}} \).

### Step 2: Set up the dissociation of the weak acid
The weak acid dissociates according to the equation:
\[ HA \rightleftharpoons H^+ + A^- \]
Let \( C \) be the initial concentration of the weak acid, which is 0.001 M. Let \( \alpha \) be the degree of dissociation. At equilibrium, the concentrations will be:
- \([HA] = C(1 - \alpha)\)
- \([H^+] = C\alpha\)
- \([A^-] = C\alpha\)

From the information given, we know that at half neutralization, the concentration of the acid is halved, which means:
\[ C(1 - \alpha) = \frac{C}{2} \]
This implies:
\[ 1 - \alpha = \frac{1}{2} \quad \Rightarrow \quad \alpha = \frac{1}{2} \]

**Hint:** Understand that half neutralization means the concentration of the weak acid is reduced to half its initial value.

### Step 3: Calculate the dissociation constant \( K_a \)
The dissociation constant \( K_a \) for the weak acid can be calculated using:
\[ K_a = \frac{[H^+][A^-]}{[HA]} = \frac{(C\alpha)(C\alpha)}{C(1 - \alpha)} \]
Substituting the values we have:
\[ K_a = \frac{(C\cdot\frac{1}{2})(C\cdot\frac{1}{2})}{C\cdot\frac{1}{2}} = \frac{C^2 \cdot \frac{1}{4}}{C \cdot \frac{1}{2}} = \frac{C}{2} = \frac{0.001}{2} = 5 \times 10^{-4} \]

**Hint:** The dissociation constant can be calculated using the equilibrium concentrations of the acid and its ions.

### Step 4: Determine the pKa
To find \( pK_a \):
\[ pK_a = -\log(K_a) = -\log(5 \times 10^{-4}) \approx 3.3 \]

**Hint:** The \( pK_a \) is simply the negative logarithm of the dissociation constant \( K_a \).

### Step 5: Calculate the pH after half neutralization
After half neutralization, we have a buffer solution consisting of the weak acid \( HA \) and its salt \( A^- \). The pH of a buffer solution can be calculated using:
\[ pH = pK_a + \log\left(\frac{[A^-]}{[HA]}\right) \]
Since the concentrations of \( [A^-] \) and \( [HA] \) are equal after half neutralization:
\[ pH = pK_a + \log(1) = pK_a \]
Thus:
\[ pH = 3.3 \]

**Hint:** In a buffer solution at half neutralization, the concentrations of the weak acid and its salt are equal, simplifying the pH calculation.

### Final Answer
The pH of the resulting solution after neutralization is approximately **3.3**.