The wave number of the limiting line in Lyman series of hydrogen is `109678 cm^(-1)` . The wave number of the limiting line in Balmer series of `He^+` would be :

Options

A. `54839cm^(-1)`
B. `109678 cm^(-1)`
C. `219356 cm^(-1)`
D. `438712 cm^(-1)`

Answer

To solve the problem of finding the wave number of the limiting line in the Balmer series of \( He^+ \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Wave Number**:
The wave number \( \bar{\nu} \) is given by the formula:
\[ \bar{\nu} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant, \( Z \) is the atomic number, \( n_1 \) is the principal quantum number of the lower energy level, and \( n_2 \) is the principal quantum number of the higher energy level.

2. **Identify Given Values for Hydrogen**:
For the Lyman series in hydrogen (\( H \)):
- The wave number of the limiting line is given as \( 109678 \, \text{cm}^{-1} \).
- For hydrogen, \( Z = 1 \).
- The limiting line corresponds to \( n_1 = 1 \) and \( n_2 = \infty \).

3. **Calculate the Rydberg Constant \( R \)**:
Using the formula for the Lyman series:
\[ 109678 = R \cdot 1^2 \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) \]
Since \( \frac{1}{\infty^2} = 0 \), we have:
\[ 109678 = R \cdot 1 \cdot (1 - 0) \implies R = 109678 \, \text{cm}^{-1} \]

4. **Identify Values for Helium Ion \( He^+ \)**:
For the Balmer series in \( He^+ \):
- The atomic number \( Z = 2 \).
- For the Balmer series, the lower energy level is \( n_1 = 2 \) and the limiting line corresponds to \( n_2 = \infty \).

5. **Substitute Values into the Wave Number Formula**:
Using the formula for the Balmer series:
\[ \bar{\nu} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Substituting the known values:
\[ \bar{\nu} = 109678 \cdot 2^2 \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) \]
Since \( \frac{1}{\infty^2} = 0 \):
\[ \bar{\nu} = 109678 \cdot 4 \left( \frac{1}{4} - 0 \right) = 109678 \cdot 4 \cdot \frac{1}{4} = 109678 \, \text{cm}^{-1} \]

6. **Conclusion**:
The wave number of the limiting line in the Balmer series of \( He^+ \) is:
\[ \bar{\nu} = 109678 \, \text{cm}^{-1} \]