A graph of `sqrt(v)` ( where `v` is the frequency of `K_(alpha)` line of the characteristic X-ray spectrum) is plotted against the atomic number `Z` of the elements emitting the characteristic X-ray .The intercept of the graph on the `z` -axis is 1 and the slope of the graph is `0.5 xx 10^(8)` in SI units. The frequency of the `K_(alpha)` line for an element of atomic number 41 is given as `alpha times 10^(16) Hz`. Find the value of `alpha`.

Answer

To solve the problem, we need to analyze the relationship between the frequency of the K-alpha line of characteristic X-ray spectrum and the atomic number \( Z \) of the elements. The relationship is given by the linear equation derived from the graph of \( \sqrt{v} \) versus \( Z \).

### Step-by-Step Solution:

1. **Understanding the Graph**:
The graph of \( \sqrt{v} \) (where \( v \) is the frequency of the K-alpha line) against the atomic number \( Z \) can be expressed in the linear form:
\[ \sqrt{v} = mZ + c \]
where \( m \) is the slope and \( c \) is the intercept.

2. **Given Values**:
- The intercept \( c = 1 \)
- The slope \( m = 0.5 \times 10^8 \)

3. **Equation of the Line**:
Substituting the values into the linear equation:
\[ \sqrt{v} = (0.5 \times 10^8)Z + 1 \]

4. **Finding \( v \) for \( Z = 41 \)**:
We need to find the frequency \( v \) when \( Z = 41 \):
\[ \sqrt{v} = (0.5 \times 10^8)(41) + 1 \]

5. **Calculating \( \sqrt{v} \)**:
Calculate \( (0.5 \times 10^8)(41) \):
\[ (0.5 \times 10^8)(41) = 20.5 \times 10^8 \]
Now add 1:
\[ \sqrt{v} = 20.5 \times 10^8 + 1 \approx 20.5 \times 10^8 \quad (\text{since } 1 \text{ is negligible in comparison}) \]

6. **Squaring to Find \( v \)**:
Now square both sides to find \( v \):
\[ v = (20.5 \times 10^8)^2 \]
\[ v = 420.25 \times 10^{16} \quad (\text{since } (10^8)^2 = 10^{16}) \]

7. **Expressing \( v \) in terms of \( \alpha \)**:
We know that \( v = \alpha \times 10^{16} \). Therefore:
\[ \alpha \times 10^{16} = 420.25 \times 10^{16} \]

8. **Finding \( \alpha \)**:
From the above equation, we can equate:
\[ \alpha = 420.25 \]

### Final Answer:
The value of \( \alpha \) is \( 420.25 \).