X-ray from a tube with a target `A` of atomic number `Z` shows strong `K` lines for target `A` and weak`K` lines for impurities. The wavelength og `K_(alpha)` lines is `lambda_(z)` for target `A` and `lambda_(1) and lambda_(2)` for two impurities.
`(lambda_(z))/(lambda_(1)) = 4 and (lambda_(z))/(lambda_(2)) = (1)/(4)`
Assuming the screeining contant of `K_(alpha)` lines to be unity select the correct statement(s).

Options

A. The atomic number of first impurity is `2 z - 1`.
B. The atomic number of first impurity is `2 z + 1`.
C. The atomic number of second impurity is ` ((z + 1))/(2)`.
D. The atomic number of second impurity is ` (z)/(2) + 1`.

Explanation

To solve the problem, we will use Moseley's law, which relates the frequency of X-rays emitted by an atom to its atomic number. The law states that the frequency (ν) of the emitted X-rays is proportional to the square of the atomic number (Z) minus a screening constant (b):

\[ \nu \propto (Z - b)^2 \]

For K-alpha lines, we can assume the screening constant \( b \) is unity, so:

\[ \nu \propto (Z - 1)^2 \]

We also know that the wavelength (λ) is inversely proportional to the frequency:

\[ \nu = \frac{c}{\lambda} \]

This means:

\[ \frac{1}{\lambda} \propto (Z - 1)^2 \]

Thus, we can write:

\[ \lambda \propto \frac{1}{(Z - 1)^2} \]

### Step 1: Analyze the given ratios

We are given two ratios:

1. \(\frac{\lambda_z}{\lambda_1} = 4\)
2. \(\frac{\lambda_z}{\lambda_2} = \frac{1}{4}\)

From these ratios, we can express the wavelengths in terms of \( \lambda_z \):

- \( \lambda_1 = \frac{\lambda_z}{4} \)
- \( \lambda_2 = 4\lambda_z \)

### Step 2: Set up equations using Moseley's law

Using the relationship derived from Moseley's law, we can write:

For the target \( A \):
\[ \lambda_z \propto \frac{1}{(Z - 1)^2} \]

For impurity 1 (with atomic number \( Z_1 \)):
\[ \lambda_1 \propto \frac{1}{(Z_1 - 1)^2} \]

For impurity 2 (with atomic number \( Z_2 \)):
\[ \lambda_2 \propto \frac{1}{(Z_2 - 1)^2} \]

### Step 3: Substitute the wavelengths into the equations

Substituting the expressions for \( \lambda_1 \) and \( \lambda_2 \):

1. From \( \lambda_z = 4\lambda_1 \):
\[ \frac{1}{(Z - 1)^2} = 4 \cdot \frac{1}{(Z_1 - 1)^2} \]
\[ (Z_1 - 1)^2 = 4(Z - 1)^2 \]
Taking the square root:
\[ Z_1 - 1 = 2(Z - 1) \]
\[ Z_1 = 2Z - 1 \]

2. From \( \lambda_z = \frac{1}{4}\lambda_2 \):
\[ \frac{1}{(Z - 1)^2} = \frac{1}{4} \cdot \frac{1}{(Z_2 - 1)^2} \]
\[ (Z_2 - 1)^2 = 4(Z - 1)^2 \]
Taking the square root:
\[ Z_2 - 1 = \frac{1}{2}(Z - 1) \]
\[ Z_2 = \frac{Z + 1}{2} \]

### Step 4: Final results and options

From the calculations, we find:

- The atomic number of the first impurity \( Z_1 = 2Z - 1 \)
- The atomic number of the second impurity \( Z_2 = \frac{Z + 1}{2} \)

Now, we can check the options:

- Option A: Atomic number of first impurity is \( 2Z - 1 \) (Correct)
- Option B: Atomic number of first impurity is \( 2Z + 1 \) (Incorrect)
- Option C: Atomic number of second impurity is \( \frac{Z + 1}{2} \) (Correct)
- Option D: Atomic number of second impurity is \( \frac{Z}{2} + 1 \) (Incorrect)

### Conclusion

The correct statements are:
- Option A: Atomic number of first impurity is \( 2Z - 1 \)
- Option C: Atomic number of second impurity is \( \frac{Z + 1}{2} \)