The wavelength of the first Lyman lines of hydrogen `He^(+)` and `Li^(2+)` ions and `lamda_(1),lamda_(2)& lamda_(3)` the ratio of these wavelengths is

Options

A. `1:4:9`
B. `9:4:1`
C. `36:9:4`
D. `6:3:2`

Explanation

To solve the problem of finding the ratio of the wavelengths of the first Lyman lines for hydrogen (H), helium ion (He⁺), and lithium ion (Li²⁺), we will use the Rydberg formula for the wavelengths of spectral lines in hydrogen-like atoms.

### Step-by-Step Solution:

1. **Identify the Transition**:
The first Lyman line corresponds to the transition from n=2 to n=1. Thus, we will consider the transitions for each ion:
- For hydrogen (H): n1 = 1, n2 = 2
- For helium ion (He⁺): n1 = 1, n2 = 2
- For lithium ion (Li²⁺): n1 = 1, n2 = 2

2. **Rydberg Formula**:
The Rydberg formula for the wavelength (λ) of a spectral line is given by:
\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where:
- \( R \) is the Rydberg constant,
- \( Z \) is the atomic number,
- \( n_1 \) and \( n_2 \) are the principal quantum numbers.

3. **Calculate for Each Ion**:
- **For Hydrogen (H)**:
\[ \frac{1}{\lambda_1} = R \cdot 1^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \cdot \frac{3}{4} \]
Thus, \( \lambda_1 = \frac{4}{3R} \).

- **For Helium Ion (He⁺)**:
\[ \frac{1}{\lambda_2} = R \cdot 2^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \cdot 4 \left( 1 - \frac{1}{4} \right) = R \cdot 4 \cdot \frac{3}{4} = 3R \]
Thus, \( \lambda_2 = \frac{1}{3R} \).

- **For Lithium Ion (Li²⁺)**:
\[ \frac{1}{\lambda_3} = R \cdot 3^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \cdot 9 \left( 1 - \frac{1}{4} \right) = R \cdot 9 \cdot \frac{3}{4} = \frac{27R}{4} \]
Thus, \( \lambda_3 = \frac{4}{27R} \).

4. **Finding the Ratios**:
Now we can find the ratios of the wavelengths:
\[ \text{Ratio} = \frac{\lambda_1 : \lambda_2 : \lambda_3} = \frac{\frac{4}{3R} : \frac{1}{3R} : \frac{4}{27R}}{1} = \frac{4 : 1 : \frac{4}{27}}{1} \]
To simplify, multiply through by 27:
\[ = 4 \cdot 27 : 1 \cdot 27 : 4 = 108 : 27 : 4 \]
Thus, the final ratio is:
\[ = 108 : 27 : 4 \]

### Final Answer:
The ratio of the wavelengths \( \lambda_1 : \lambda_2 : \lambda_3 \) is \( 108 : 27 : 4 \).