What will be the wavelength of radiation released in nano-metre when an electron drops from the 5th Bohr orbit to 2nd Bohr orbit in hydrogen atom?

Options

A. 217nm
B. 434nm
C. 651nm
D. 868nm

Explanation

To find the wavelength of radiation released when an electron drops from the 5th Bohr orbit to the 2nd Bohr orbit in a hydrogen atom, we can use the Rydberg formula. Here’s the step-by-step solution:

### Step 1: Write the Rydberg formula
The Rydberg formula for the wavelength of emitted radiation is given by:
\[ \frac{1}{\lambda} = R_H \cdot Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where:
- \(\lambda\) is the wavelength,
- \(R_H\) is the Rydberg constant (\(1.09678 \times 10^7 \, \text{m}^{-1}\)),
- \(Z\) is the atomic number (for hydrogen, \(Z = 1\)),
- \(n_1\) is the lower energy level (2nd orbit, \(n_1 = 2\)),
- \(n_2\) is the higher energy level (5th orbit, \(n_2 = 5\)).

### Step 2: Substitute the values into the formula
Substituting the values into the Rydberg formula:
\[ \frac{1}{\lambda} = 1.09678 \times 10^7 \cdot 1^2 \left( \frac{1}{2^2} - \frac{1}{5^2} \right) \]
\[ = 1.09678 \times 10^7 \left( \frac{1}{4} - \frac{1}{25} \right) \]

### Step 3: Calculate the fractions
Calculating the fractions:
\[ \frac{1}{4} = 0.25 \]
\[ \frac{1}{25} = 0.04 \]
Now, subtract these two values:
\[ 0.25 - 0.04 = 0.21 \]

### Step 4: Substitute back into the equation
Now substitute back into the equation:
\[ \frac{1}{\lambda} = 1.09678 \times 10^7 \cdot 0.21 \]
\[ = 2.303218 \times 10^6 \, \text{m}^{-1} \]

### Step 5: Calculate \(\lambda\)
Now, take the reciprocal to find \(\lambda\):
\[ \lambda = \frac{1}{2.303218 \times 10^6} \approx 4.343 \times 10^{-7} \, \text{m} \]

### Step 6: Convert to nanometers
To convert meters to nanometers:
\[ \lambda \approx 4.343 \times 10^{-7} \, \text{m} = 434.3 \, \text{nm} \]

### Final Answer
Thus, the wavelength of radiation released when the electron drops from the 5th to the 2nd Bohr orbit in a hydrogen atom is approximately:
\[ \lambda \approx 434 \, \text{nm} \]