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

### Step 1: Define the Weak Electrolyte
Consider a weak electrolyte represented as HA, which dissociates in water as follows:
\[ HA \rightleftharpoons H^+ + A^- \]

### Step 2: Establish Initial Conditions
Let the initial concentration of the weak electrolyte (HA) be C. Initially, the concentrations of \( H^+ \) and \( A^- \) ions are both 0.

### Step 3: Set Up the Equilibrium Expression
At equilibrium, if α is the degree of ionization, the concentration of the dissociated ions will be:
- Concentration of \( H^+ \) ions = \( C\alpha \)
- Concentration of \( A^- \) ions = \( C\alpha \)
- Concentration of undissociated HA = \( C - C\alpha = C(1 - \alpha) \)

### Step 4: Write the Expression for the Acid Ionization Constant (Ka)
The acid ionization constant \( K_a \) is defined as:
\[ K_a = \frac{[H^+][A^-]}{[HA]} = \frac{(C\alpha)(C\alpha)}{C(1 - \alpha)} \]

### Step 5: Simplify the Expression
Substituting the concentrations into the equation gives:
\[ K_a = \frac{C^2\alpha^2}{C(1 - \alpha)} = \frac{C\alpha^2}{1 - \alpha} \]

### Step 6: Make an Assumption for Weak Electrolytes
For weak electrolytes, the degree of ionization (α) is typically very small (α << 1). Therefore, we can approximate:
\[ 1 - \alpha \approx 1 \]

### Step 7: Substitute the Approximation
Using this approximation, the equation simplifies to:
\[ K_a \approx C\alpha^2 \]

### Step 8: Rearranging the Equation
From the above equation, we can express α in terms of \( K_a \) and C:
\[ \alpha^2 \approx \frac{K_a}{C} \]
Taking the square root of both sides gives:
\[ \alpha \approx \sqrt{\frac{K_a}{C}} \]

### Step 9: Final Relationship
Thus, the relationship between the degree of ionization (α) and the concentration (C) of the weak electrolyte is:
\[ \alpha \approx \frac{K_a}{C} \]

### Conclusion
The correct relationship between the degree of ionization (α) and the concentration (C) of a weak electrolyte is:
\[ \alpha = \frac{K_a}{C} \]