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_(1)=lambda_(2)=lambda_(3)`
B. `lambda_(1)gtlambda_(2)gtlambda_(3)`
C. `lambda_(1)ltlambda_(2)ltlambda_(3)`
D. `lambda_(2)=sqrt(lambda_(1)lambda_(3))`

Answer

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. **Identify the Isotopes**: We have three isotopes of lead: \( Pb^{204}, Pb^{206}, \) and \( Pb^{208} \). The atomic number \( Z \) for lead is 82, which is the same for all isotopes.

2. **Understand K-alpha X-rays**: K-alpha X-rays are emitted when an electron transitions from the L-shell (n=2) to the K-shell (n=1) of an atom. The energy and wavelength of these X-rays depend on the atomic number \( Z \).

3. **Use the Wavelength Formula**: The relationship between the wavelength \( \lambda \) of K-alpha X-rays and the atomic number \( Z \) can be expressed as:
\[ \frac{1}{\lambda} = \frac{a^2}{c} (Z - 1)^2 \]
where \( a \) is a constant and \( c \) is another constant.

4. **Analyze the Formula**: In this formula, since \( Z \) is the same for all isotopes of lead (82), the term \( (Z - 1)^2 \) will also be the same for all isotopes.

5. **Conclude the Wavelengths**: Since the formula shows that the wavelength \( \lambda \) is inversely proportional to \( (Z - 1)^2 \), and since \( Z \) is the same for all isotopes, it follows that:
\[ \lambda_1 = \lambda_2 = \lambda_3 \]
Thus, the wavelengths of K-alpha X-rays from all three isotopes are equal.

6. **Final Answer**: Therefore, the correct conclusion is that \( \lambda_1 = \lambda_2 = \lambda_3 \).