If `alpha` is the degree of ionization, C the concentration of a weak electrolyte and `K_(a)` the acid ionization constant , then the correct relationship between `alpha` and C is

Options

A. `alpha^(2) = sqrt(K_(a)/C)`
B. `alpha^(2) = sqrt(C/K_(a))`
C. `alpha= sqrt(K_(a)/C)`
D. `alpha = sqrt(C/K_(a))`

Answer

To find the correct relationship between the degree of ionization (α) and the concentration (C) of a weak electrolyte, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( \alpha \) be the degree of ionization.
- Let \( C \) be the initial concentration of the weak electrolyte.
- Let \( K_a \) be the acid ionization constant.

2. **Set Up the Ionization Equation**:
- For a weak electrolyte, we can represent the ionization as:
\[ HA \rightleftharpoons H^+ + A^- \]
- Initially, we have:
- Concentration of \( HA \) = \( C \)
- Concentration of \( H^+ \) = 0
- Concentration of \( A^- \) = 0

3. **Change in Concentration at Equilibrium**:
- At equilibrium, the concentration of \( HA \) will be \( C - \alpha C \) (since \( \alpha C \) is the amount that ionizes).
- The concentrations of \( H^+ \) and \( A^- \) will both be \( \alpha C \).

4. **Write the Expression for \( K_a \)**:
- The expression for the acid ionization constant \( K_a \) is given by:
\[ K_a = \frac{[H^+][A^-]}{[HA]} = \frac{(\alpha C)(\alpha C)}{C(1 - \alpha)} \]
- Simplifying this gives:
\[ K_a = \frac{\alpha^2 C}{1 - \alpha} \]

5. **Assumption for Weak Electrolytes**:
- For weak electrolytes, the degree of ionization \( \alpha \) is usually small (less than 0.01), which allows us to approximate \( 1 - \alpha \approx 1 \).
- Thus, we can simplify the equation to:
\[ K_a \approx \alpha^2 C \]

6. **Rearranging for \( \alpha \)**:
- From the equation \( K_a \approx \alpha^2 C \), we can express \( \alpha \) as:
\[ \alpha \approx \sqrt{\frac{K_a}{C}} \]

### Final Relationship:
The correct relationship between the degree of ionization \( \alpha \) and the concentration \( C \) of a weak electrolyte is:
\[ \alpha \approx \sqrt{\frac{K_a}{C}} \]