For a weak acid HA of concentration `C(mol l^-1)` and degree of dissociation `(alpha)`, Ostwald's dilution law is represented by the equation

Options

A. `K_a=(C^2alpha)/(1-alpha)`
B. `K_a=(alpha^2C)/(1-alpha)`
C. `K_a= Calpha`
D. `K_a= (Calpha^2)/(1-alpha^2)`

Answer

To solve the problem regarding Ostwald's dilution law for a weak acid HA, we will follow these steps:

### Step 1: Understand the dissociation of the weak acid
The weak acid HA dissociates in water according to the following equilibrium reaction:
\[ \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \]

### Step 2: Define initial concentration and degree of dissociation
Let the initial concentration of the weak acid HA be \( C \) (in mol/L). The degree of dissociation is denoted by \( \alpha \), which represents the fraction of the acid that has dissociated.

### Step 3: Set up equilibrium concentrations
At equilibrium:
- The concentration of undissociated acid (HA) will be \( C(1 - \alpha) \).
- The concentration of \( \text{H}^+ \) ions will be \( C\alpha \).
- The concentration of \( \text{A}^- \) ions will also be \( C\alpha \).

### Step 4: Write the expression for the equilibrium constant \( K_a \)
The equilibrium constant \( K_a \) for the dissociation of the weak acid can be expressed as:
\[ K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \]
Substituting the equilibrium concentrations into this expression gives:
\[ K_a = \frac{(C\alpha)(C\alpha)}{C(1 - \alpha)} \]

### Step 5: Simplify the equation
This simplifies to:
\[ K_a = \frac{C^2 \alpha^2}{C(1 - \alpha)} \]
Cancelling one \( C \) from the numerator and denominator results in:
\[ K_a = \frac{C \alpha^2}{1 - \alpha} \]

### Step 6: Identify the correct option
From the derived equation, we can see that the correct representation of Ostwald's dilution law for the weak acid HA is:
\[ K_a = \frac{C \alpha^2}{1 - \alpha} \]
This corresponds to option 2 in the provided choices.

### Final Answer
The correct equation representing Ostwald's dilution law for the weak acid HA is:
\[ K_a = \frac{C \alpha^2}{1 - \alpha} \]

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