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 a Bohr hydrogen atom, we can follow these steps:

### Step 1: Understand the transitions
In a hydrogen atom, when an electron transitions from a higher energy level to a lower energy level, it emits radiation. The maximum wavelength corresponds to the transition from the second energy level (n=2) to the ground state (n=1), while the minimum wavelength corresponds to the transition from an infinitely high energy level (n=∞) to the ground state (n=1).

### Step 2: Use the formula for wavelength
The formula for the wavelength of emitted radiation in a hydrogen atom is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
where \( R \) is the Rydberg constant, \( n_f \) is the final energy level, and \( n_i \) is the initial energy level.

### Step 3: Calculate the maximum wavelength
For the maximum wavelength, the transition is from \( n_i = 2 \) to \( n_f = 1 \):
\[ \frac{1}{\lambda_{\text{max}}} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]
Thus,
\[ \lambda_{\text{max}} = \frac{4}{3R} \]

### Step 4: Calculate the minimum wavelength
For the minimum wavelength, the transition is from \( n_i = \infty \) to \( n_f = 1 \):
\[ \frac{1}{\lambda_{\text{min}}} = R \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R \left( 1 - 0 \right) = R \]
Thus,
\[ \lambda_{\text{min}} = \frac{1}{R} \]

### Step 5: Find the ratio of minimum to maximum wavelength
Now, we can find the ratio of the minimum wavelength to the 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 a Bohr hydrogen atom is:
\[ \frac{\lambda_{\text{min}}}{\lambda_{\text{max}}} = \frac{3}{4} \]