The shortest wavelength of Lyman series of the hydrogen atom is equal to the shortest wavelength of Balmer series of a hydrogen -like atom of atomic number Z. The value of Z is equal to

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem, we need to find the value of \( Z \) such that the shortest wavelength of the Lyman series of the hydrogen atom is equal to the shortest wavelength of the Balmer series of a hydrogen-like atom with atomic number \( Z \).

### Step-by-Step Solution:

1. **Understanding the Lyman Series**:
The Lyman series corresponds to transitions from higher energy levels to the first energy level (n=1) in hydrogen. The formula for the wavelength in the Lyman series is given by:
\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
For the shortest wavelength (which occurs when \( n_2 \to \infty \)):
\[ \frac{1}{\lambda_{Lyman}} = RZ^2 \left( \frac{1}{1^2} - 0 \right) = RZ^2 \]
Thus, the shortest wavelength for the Lyman series of hydrogen (where \( Z = 1 \)) is:
\[ \lambda_{Lyman} = \frac{1}{R} \]

2. **Understanding the Balmer Series**:
The Balmer series corresponds to transitions from higher energy levels to the second energy level (n=2) in hydrogen-like atoms. The formula for the wavelength in the Balmer series is:
\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{2^2} - \frac{1}{n_2^2} \right) \]
For the shortest wavelength (which occurs when \( n_2 \to \infty \)):
\[ \frac{1}{\lambda_{Balmer}} = RZ^2 \left( \frac{1}{2^2} - 0 \right) = RZ^2 \cdot \frac{1}{4} \]
Thus, the shortest wavelength for the Balmer series is:
\[ \lambda_{Balmer} = \frac{4}{RZ^2} \]

3. **Setting the Wavelengths Equal**:
According to the problem, the shortest wavelength of the Lyman series is equal to the shortest wavelength of the Balmer series:
\[ \frac{1}{R} = \frac{4}{RZ^2} \]

4. **Solving for \( Z^2 \)**:
We can simplify this equation by multiplying both sides by \( RZ^2 \):
\[ Z^2 = 4 \]

5. **Finding \( Z \)**:
Taking the square root of both sides gives:
\[ Z = 2 \]

### Conclusion:
The value of \( Z \) is \( 2 \).