What should be the ratio of minimum to maximum wavelength of radiation emitted by transition of an electron to ground state of Bohr's hydrogen atom ?

Options

A. `3/4`
B. `1/4`
C. `1/8`
D. `3/8`

Answer

To find the ratio of the minimum to maximum wavelength of radiation emitted by the transition of an electron to the ground state of Bohr's hydrogen atom, we can follow these steps:

### Step 1: Identify the Series and States
In the case of the hydrogen atom, the transition to the ground state corresponds to the Lyman series, where the electron transitions from a higher energy level (n2) to the ground state (n1 = 1).

### Step 2: Determine Maximum Wavelength
The maximum wavelength occurs when the electron transitions from n2 = 2 to n1 = 1. We can use the Rydberg formula for this transition:
\[ \frac{1}{\lambda_{\text{max}}} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Substituting \( n_1 = 1 \) and \( n_2 = 2 \):
\[ \frac{1}{\lambda_{\text{max}}} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]
Thus,
\[ \lambda_{\text{max}} = \frac{4}{3R} \]

### Step 3: Determine Minimum Wavelength
The minimum wavelength occurs when the electron transitions from n2 = ∞ to n1 = 1. Again, using the Rydberg formula:
\[ \frac{1}{\lambda_{\text{min}}} = R \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R \left( 1 - 0 \right) = R \]
Thus,
\[ \lambda_{\text{min}} = \frac{1}{R} \]

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

### Conclusion
The ratio of the minimum to maximum wavelength of radiation emitted by the transition of an electron to the ground state of Bohr's hydrogen atom is:
\[ \frac{\lambda_{\text{min}}}{\lambda_{\text{max}}} = \frac{3}{4} \]