The ratio of the wavelengths for `2 rarr 1` transition in `Li^(++), He^(+)` and `H` is

Options

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

Explanation

To find the ratio of the wavelengths for the \(2 \rightarrow 1\) transition in \(Li^{++}\), \(He^{+}\), and \(H\), we can use the Rydberg formula for the wavelengths of transitions in hydrogen-like atoms:

\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

Where:
- \(R\) is the Rydberg constant,
- \(Z\) is the atomic number,
- \(n_1\) and \(n_2\) are the principal quantum numbers of the lower and upper energy levels, respectively.

### Step 1: Identify the transitions
For the \(2 \rightarrow 1\) transition:
- \(n_1 = 1\)
- \(n_2 = 2\)

### Step 2: Write the formula for each ion
1. For \(H\) (Hydrogen, \(Z = 1\)):
\[ \frac{1}{\lambda_H} = R \cdot 1^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \cdot \frac{3}{4} \]

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

3. For \(Li^{++}\) (Lithium, \(Z = 3\)):
\[ \frac{1}{\lambda_{Li}} = R \cdot 3^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \cdot 9 \left( 1 - \frac{1}{4} \right) = R \cdot 9 \cdot \frac{3}{4} = \frac{27R}{4} \]

### Step 3: Find the wavelengths
Now, we can express the wavelengths:
- \(\lambda_H = \frac{4}{3R}\)
- \(\lambda_{He} = \frac{1}{3R}\)
- \(\lambda_{Li} = \frac{4}{27R}\)

### Step 4: Find the ratios of the wavelengths
Now, we can find the ratio of the wavelengths:
\[ \lambda_H : \lambda_{He} : \lambda_{Li} = \frac{4}{3R} : \frac{4}{9R} : \frac{4}{27R} \]

To simplify, we can multiply through by \(27R\) to eliminate the denominators:
\[ \lambda_H : \lambda_{He} : \lambda_{Li} = 36 : 12 : 4 \]

### Step 5: Simplify the ratio
Now, we can simplify this ratio:
\[ 36 : 12 : 4 = 9 : 3 : 1 \]

### Final Ratio
Thus, the final ratio of the wavelengths for the \(2 \rightarrow 1\) transition in \(Li^{++}\), \(He^{+}\), and \(H\) is:
\[ 9 : 3 : 1 \]