The pH of 0.1 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 calculate the pH of the resulting solution after neutralizing a weak acid (HA) with NaOH to reduce its concentration by half, we can follow these steps:

### Step 1: Determine the initial concentration of H⁺ ions.
Given that the pH of the 0.1 M solution of the weak acid HA is 4.50, we can calculate the concentration of H⁺ ions using the formula:

\[ \text{pH} = -\log[H^+] \]

From this, we can find [H⁺]:

\[ [H^+] = 10^{-\text{pH}} = 10^{-4.50} \approx 3.16 \times 10^{-5} \, \text{M} \]

**Hint:** Remember that the concentration of H⁺ ions can be found using the inverse logarithmic relationship with pH.

### Step 2: Calculate the dissociation constant (Kₐ) of the weak acid.
For a weak acid, we can relate the concentration of H⁺ ions to the acid dissociation constant (Kₐ) using the formula:

\[ K_a = \frac{[H^+]^2}{[HA]} \]

Where [HA] is the concentration of the acid. Initially, [HA] = 0.1 M. Thus,

\[ K_a = \frac{(3.16 \times 10^{-5})^2}{0.1} = \frac{10^{-10}}{0.1} = 10^{-9} \]

**Hint:** Use the relationship between the concentrations of H⁺, the weak acid, and Kₐ to find the dissociation constant.

### Step 3: Calculate the new concentration of the weak acid after neutralization.
Since we are neutralizing the acid to reduce its concentration by half, the new concentration of HA will be:

\[ [HA]_{\text{new}} = \frac{0.1}{2} = 0.05 \, \text{M} \]

**Hint:** To find the new concentration after neutralization, simply divide the initial concentration by 2.

### Step 4: Calculate the new concentration of the conjugate base (A⁻).
When HA is neutralized with NaOH, it forms its conjugate base (A⁻). Since we have reduced the acid concentration by half, the concentration of A⁻ will be equal to the amount of HA that was neutralized:

\[ [A^-] = 0.1 - 0.05 = 0.05 \, \text{M} \]

**Hint:** The concentration of the conjugate base formed is equal to the amount of weak acid that has been neutralized.

### Step 5: Use the Henderson-Hasselbalch equation to find the new pH.
The Henderson-Hasselbalch equation is given by:

\[ \text{pH} = \text{pK}_a + \log\left(\frac{[A^-]}{[HA]}\right) \]

First, we need to find pKₐ:

\[ \text{pK}_a = -\log(10^{-9}) = 9 \]

Now substituting the values into the equation:

\[ \text{pH} = 9 + \log\left(\frac{0.05}{0.05}\right) = 9 + \log(1) = 9 + 0 = 9 \]

### Final Answer:
The pH of the resulting solution after neutralizing the weak acid to half its initial concentration is **9.00**.

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