If `pK_a = 4 and k_a = Ca^2`, then find van't Hoff factor for a weak HA type acid, when C = 0.01M.

Options

A. 1.01
B. 1.02
C. `1.10`
D. `1.20`

Answer

To find the van't Hoff factor (i) for the weak acid HA, we can follow these steps:

### Step 1: Determine the dissociation constant (K_a)
Given that \( pK_a = 4 \), we can calculate \( K_a \) using the formula:
\[ K_a = 10^{-pK_a} = 10^{-4} = 1 \times 10^{-4} \]

### Step 2: Set up the equilibrium expression
For the dissociation of the weak acid HA:
\[ HA \rightleftharpoons H^+ + A^- \]
Let \( C \) be the initial concentration of HA, which is given as \( C = 0.01 \, M \). If \( \alpha \) is the degree of dissociation, then at equilibrium:
- Concentration of HA = \( C - C\alpha = C(1 - \alpha) \)
- Concentration of \( H^+ \) = \( C\alpha \)
- Concentration of \( A^- \) = \( C\alpha \)

### Step 3: Write the expression for \( K_a \)
The expression for \( K_a \) is given by:
\[ K_a = \frac{[H^+][A^-]}{[HA]} = \frac{(C\alpha)(C\alpha)}{C(1 - \alpha)} = \frac{C^2\alpha^2}{C(1 - \alpha)} = \frac{C\alpha^2}{1 - \alpha} \]
Substituting \( K_a = 1 \times 10^{-4} \) and \( C = 0.01 \):
\[ 1 \times 10^{-4} = \frac{0.01\alpha^2}{1 - \alpha} \]

### Step 4: Solve for \( \alpha \)
Rearranging the equation gives:
\[ 1 \times 10^{-4}(1 - \alpha) = 0.01\alpha^2 \]
\[ 1 \times 10^{-4} - 1 \times 10^{-4}\alpha = 0.01\alpha^2 \]
\[ 0.01\alpha^2 + 1 \times 10^{-4}\alpha - 1 \times 10^{-4} = 0 \]
This is a quadratic equation in \( \alpha \):
\[ 0.01\alpha^2 + 1 \times 10^{-4}\alpha - 1 \times 10^{-4} = 0 \]

Using the quadratic formula \( \alpha = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 0.01, b = 1 \times 10^{-4}, c = -1 \times 10^{-4} \):
\[ \alpha = \frac{-1 \times 10^{-4} \pm \sqrt{(1 \times 10^{-4})^2 - 4(0.01)(-1 \times 10^{-4})}}{2(0.01)} \]
Calculating the discriminant:
\[ = \frac{-1 \times 10^{-4} \pm \sqrt{1 \times 10^{-8} + 4 \times 10^{-6}}}{0.02} \]
\[ = \frac{-1 \times 10^{-4} \pm \sqrt{4.01 \times 10^{-6}}}{0.02} \]
\[ = \frac{-1 \times 10^{-4} \pm 0.0020025}{0.02} \]
Calculating the positive root:
\[ \alpha = \frac{-1 \times 10^{-4} + 0.0020025}{0.02} \approx 0.1 \]

### Step 5: Calculate the van't Hoff factor (i)
The van't Hoff factor \( i \) is given by:
\[ i = 1 + \alpha \]
Substituting \( \alpha = 0.1 \):
\[ i = 1 + 0.1 = 1.1 \]

### Final Answer
The van't Hoff factor for the weak acid HA is \( i = 1.1 \).