The ratio of acid strength of `HOCN` and `HCN` is about
Given `K_(a)` of `HOCN=1.2xx10^(-4) and K_(a)" of "HCN "=4.2 xx10^(-10)`

Options

A. `535:1`
B. `1:535`
C. `2.86xx10^(5):1`
D. `2.86xx10^(4):1`

Answer

To find the ratio of the acid strength of HOCN and HCN, we will use the given acid dissociation constants \( K_a \) for both acids.

### Step-by-Step Solution:

1. **Identify the given values:**
- \( K_a \) for HOCN = \( 1.2 \times 10^{-4} \)
- \( K_a \) for HCN = \( 4.2 \times 10^{-10} \)

2. **Understand the relationship between \( K_a \) and acid strength:**
- The strength of an acid is directly related to its \( K_a \) value. A higher \( K_a \) indicates a stronger acid because it means that the acid dissociates more in solution, producing more \( H^+ \) ions.

3. **Set up the ratio of acid strengths:**
- The ratio of the acid strengths can be expressed as:
\[ \text{Ratio} = \sqrt{\frac{K_{a \text{ (HOCN)}}}{K_{a \text{ (HCN)}}}} \]

4. **Substitute the values into the equation:**
\[ \text{Ratio} = \sqrt{\frac{1.2 \times 10^{-4}}{4.2 \times 10^{-10}}} \]

5. **Calculate the fraction inside the square root:**
- First, calculate the fraction:
\[ \frac{1.2 \times 10^{-4}}{4.2 \times 10^{-10}} = \frac{1.2}{4.2} \times 10^{6} \approx 0.2857 \times 10^{6} \]
- This simplifies to:
\[ 2.857 \times 10^{5} \]

6. **Take the square root:**
\[ \text{Ratio} = \sqrt{2.857 \times 10^{5}} \approx 534 \]

7. **Final Result:**
- The ratio of the acid strength of HOCN to HCN is approximately \( 534:1 \).