The degree of dissociation of weak electrolyde is inversely proportional to the square root fo concentration. It is called Ostwald's dilution law.
`alpha = sqrt((K_(a))/(c))` As the tempertaure increases, degree of dissociation will increase.
`(alpha_(1))/(alpha_(2)) = sqrt((K_(a_(1)))/(K_(a_(2))))` if concentration is same.
`(alpha_(1))/(alpha_(2)) = sqrt((c_(2))/(c_(1)))` if acid is same.
`a_(1)` and `a_(2)` are in ratio of `1:2, K_(a_(1)) = 2xx10^(-4)`. What will be `K_(a_(2))`?

Options

A. `8 xx 10^(-4)`
B. `2xx10^(-4)`
C. `4 xx 10^(-4)`
D. `1xx10^(-4)`

Explanation

To solve the problem, we will use the provided relationships and the given information step by step.

### Step 1: Understand the given information
We know that:
- The degree of dissociation (α) of a weak electrolyte is inversely proportional to the square root of its concentration (c).
- The relationship is given by Ostwald's dilution law:
\[ \alpha = \sqrt{\frac{K_a}{c}} \]
- We have two acids with degrees of dissociation α₁ and α₂, where α₁:α₂ = 1:2.
- The dissociation constant for the first acid is given as \( K_{a1} = 2 \times 10^{-4} \).

### Step 2: Set up the ratio of degrees of dissociation
From the problem, we have:
\[ \frac{\alpha_1}{\alpha_2} = \frac{1}{2} \]

### Step 3: Use the relationship for the same concentration
Since the concentration is the same for both acids, we can use the formula:
\[ \frac{\alpha_1}{\alpha_2} = \sqrt{\frac{K_{a1}}{K_{a2}}} \]

### Step 4: Substitute the known values into the equation
Substituting the values we have:
\[ \frac{1}{2} = \sqrt{\frac{2 \times 10^{-4}}{K_{a2}}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{1}{2}\right)^2 = \frac{2 \times 10^{-4}}{K_{a2}} \]
\[ \frac{1}{4} = \frac{2 \times 10^{-4}}{K_{a2}} \]

### Step 6: Rearrange to solve for \( K_{a2} \)
Rearranging the equation to solve for \( K_{a2} \):
\[ K_{a2} = 2 \times 10^{-4} \times 4 \]
\[ K_{a2} = 8 \times 10^{-4} \]

### Conclusion
Thus, the value of \( K_{a2} \) is:
\[ K_{a2} = 8 \times 10^{-4} \]