The wavelengths of `K_alpha` X-rays from lead isotopes `Pb^(204) , Pb^(206) and Pb^(208)` are `lambda_1, lambda_2 and lambda_3` respectively. Choose the correct alternative.

Options

A. `lambda_1ltlambda_2ltlambda_3`
B. `lambda_1gtlambda_2gtlambda_3`
C. `lambda_1=lambda_2=lambda_3`
D. None of these

Explanation

To solve the problem regarding the wavelengths of K-alpha X-rays from lead isotopes \( Pb^{204} \), \( Pb^{206} \), and \( Pb^{208} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Isotopes**:
- The isotopes \( Pb^{204} \), \( Pb^{206} \), and \( Pb^{208} \) all belong to the same element, lead (Pb), which means they have the same atomic number \( Z \). The atomic number for lead is 82.

2. **K-alpha X-ray Emission**:
- K-alpha X-rays are emitted when an electron transitions from the L-shell (n=2) to the K-shell (n=1). The wavelength of the emitted X-ray can be described by the formula:
\[ \frac{1}{\lambda} = R \cdot (Z - B)^2 \cdot \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
- Here, \( R \) is a constant, \( Z \) is the atomic number, \( B \) is a screening constant, and \( n_1 \) and \( n_2 \) are the principal quantum numbers of the electron shells involved in the transition.

3. **Applying the Formula**:
- For K-alpha transitions, \( n_1 = 1 \) and \( n_2 = 2 \). Therefore, the formula simplifies to:
\[ \frac{1}{\lambda} \propto (Z - B)^2 \]
- Since \( B \) is typically taken as 1 for K-alpha lines, we can rewrite it as:
\[ \frac{1}{\lambda} \propto (Z - 1)^2 \]

4. **Comparing the Isotopes**:
- Since all three isotopes have the same atomic number \( Z = 82 \), the value of \( (Z - 1)^2 \) will be the same for all isotopes:
\[ (Z - 1)^2 = (82 - 1)^2 = 81^2 \]
- Therefore, the wavelengths \( \lambda_1 \), \( \lambda_2 \), and \( \lambda_3 \) for \( Pb^{204} \), \( Pb^{206} \), and \( Pb^{208} \) respectively will be the same.

5. **Conclusion**:
- Since the wavelengths are the same, we conclude that:
\[ \lambda_1 = \lambda_2 = \lambda_3 \]
- Hence, the correct alternative is that all three wavelengths are equal.

### Final Answer:
The correct option is **Option 3**: \( \lambda_1 = \lambda_2 = \lambda_3 \).