What is the ratio of minimum to maximum wavelength of radiation emitted by electron when it jump from as higher state to ground
state in `Li^(2+)` ion ?

Options

A. `(1)/(4)`
B. `(8)/(9)`
C. `(3)/(4)`
D. `(1)/(9)`

Explanation

To solve the problem of finding the ratio of the minimum to maximum wavelength of radiation emitted by an electron when it jumps from a higher state to the ground state in the \( \text{Li}^{2+} \) ion, we can follow these steps:

### Step 1: Understand the Energy Levels
The energy levels of the hydrogen-like ions can be described by the formula:
\[ E_n = -\frac{Z^2 \cdot R}{n^2} \]
where \( Z \) is the atomic number, \( R \) is the Rydberg constant, and \( n \) is the principal quantum number.

For \( \text{Li}^{2+} \), \( Z = 3 \).

### Step 2: Calculate the Minimum Wavelength
The minimum wavelength corresponds to the transition from the highest energy level (infinity) to the ground state (n=1). The formula for the wavelength is given by:
\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
For the minimum wavelength:
- \( n_f = 1 \) (ground state)
- \( n_i = \infty \)

Thus, we have:
\[ \frac{1}{\lambda_{\text{min}}} = R \cdot 3^2 \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = 9R \]
Therefore,
\[ \lambda_{\text{min}} = \frac{1}{9R} \]

### Step 3: Calculate the Maximum Wavelength
The maximum wavelength corresponds to the transition from the second energy level (n=2) to the ground state (n=1). Thus:
- \( n_f = 1 \)
- \( n_i = 2 \)

Using the same formula:
\[ \frac{1}{\lambda_{\text{max}}} = R \cdot 3^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = 9R \left( 1 - \frac{1}{4} \right) = 9R \cdot \frac{3}{4} = \frac{27R}{4} \]
Therefore,
\[ \lambda_{\text{max}} = \frac{4}{27R} \]

### Step 4: Find the Ratio of Minimum to Maximum Wavelength
Now we can find the ratio of minimum to maximum wavelength:
\[ \text{Ratio} = \frac{\lambda_{\text{min}}}{\lambda_{\text{max}}} = \frac{\frac{1}{9R}}{\frac{4}{27R}} = \frac{1}{9R} \cdot \frac{27R}{4} = \frac{27}{36} = \frac{3}{4} \]

### Final Answer
The ratio of the minimum to maximum wavelength of radiation emitted by the electron when it jumps from a higher state to the ground state in \( \text{Li}^{2+} \) ion is:
\[ \frac{3}{4} \]