Given the two concentration of HCN `(K_(a)=10^(-9))` are `0.1` M and `0.001` M respectively. What will be the ratio of degree of dissociation?

Options

A. 1
B. `0.1`
C. `0.003`
D. `0.01`

Answer

To solve the problem of finding the ratio of the degree of dissociation of HCN at two different concentrations, we can follow these steps:

### Step 1: Understand the relationship between degree of dissociation and concentration
According to Ostwald's dilution law, the dissociation constant \( K_a \) for a weak acid can be expressed as:
\[ K_a = \frac{C \alpha^2}{1 - \alpha} \]
where:
- \( C \) is the concentration of the acid,
- \( \alpha \) is the degree of dissociation.

For weak acids, when \( \alpha \) is small, we can approximate \( 1 - \alpha \) as 1. Thus, the equation simplifies to:
\[ K_a \approx C \alpha^2 \]

### Step 2: Express the degree of dissociation in terms of \( K_a \) and concentration
From the simplified equation, we can rearrange it to find \( \alpha \):
\[ \alpha = \sqrt{\frac{K_a}{C}} \]

### Step 3: Calculate the degree of dissociation for both concentrations
Let’s denote:
- \( C_1 = 0.1 \, \text{M} \)
- \( C_2 = 0.001 \, \text{M} \)

Now we can write the degree of dissociation for each concentration:
\[ \alpha_1 = \sqrt{\frac{K_a}{C_1}} \quad \text{and} \quad \alpha_2 = \sqrt{\frac{K_a}{C_2}} \]

### Step 4: Find the ratio of the degrees of dissociation
To find the ratio \( \frac{\alpha_1}{\alpha_2} \):
\[ \frac{\alpha_1}{\alpha_2} = \frac{\sqrt{\frac{K_a}{C_1}}}{\sqrt{\frac{K_a}{C_2}}} \]
This simplifies to:
\[ \frac{\alpha_1}{\alpha_2} = \sqrt{\frac{C_2}{C_1}} \]

### Step 5: Substitute the values of concentrations
Substituting the values of \( C_1 \) and \( C_2 \):
\[ \frac{\alpha_1}{\alpha_2} = \sqrt{\frac{0.001}{0.1}} = \sqrt{\frac{1}{100}} = \sqrt{0.01} = 0.1 \]

### Conclusion
Thus, the ratio of the degree of dissociation \( \frac{\alpha_1}{\alpha_2} \) is \( 0.1 \).