If, an electron in hydrogen atom jumps from an orbit of lelvel n=3 to an orbit of level n=2, emitted radiation has a freqwuency (R= Rydbertg's contant ,c = velocity of light)

Options

A. `(3Rc)/(27)`
B. `(3Rc)/(25)`
C. `(8Rc)/(9)`
D. `(5Rc)/(36)`

Explanation

To solve the problem of an electron in a hydrogen atom jumping from an orbit of level \( n = 3 \) to an orbit of level \( n = 2 \) and determining the frequency of the emitted radiation, we can follow these steps:

### Step 1: Identify the initial and final energy levels
The initial energy level \( n_i \) is 3 and the final energy level \( n_f \) is 2.

### Step 2: Use the Rydberg formula
The Rydberg formula for the wavelength \( \lambda \) of the emitted radiation 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.

### Step 3: Substitute the values into the formula
Substituting \( n_f = 2 \) and \( n_i = 3 \) into the formula:
\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \]
Calculating \( \frac{1}{2^2} \) and \( \frac{1}{3^2} \):
\[ \frac{1}{2^2} = \frac{1}{4}, \quad \frac{1}{3^2} = \frac{1}{9} \]
Thus,
\[ \frac{1}{\lambda} = R \left( \frac{1}{4} - \frac{1}{9} \right) \]

### Step 4: Find a common denominator and simplify
The common denominator of 4 and 9 is 36. Therefore:
\[ \frac{1}{4} = \frac{9}{36}, \quad \frac{1}{9} = \frac{4}{36} \]
Now substituting these values:
\[ \frac{1}{\lambda} = R \left( \frac{9}{36} - \frac{4}{36} \right) = R \left( \frac{5}{36} \right) \]
So,
\[ \frac{1}{\lambda} = \frac{5R}{36} \]

### Step 5: Calculate the frequency
The frequency \( \nu \) of the emitted radiation is related to the wavelength \( \lambda \) by the equation:
\[ \nu = \frac{c}{\lambda} \]
Substituting \( \lambda \) from the previous step:
\[ \nu = c \cdot \frac{1}{\frac{36}{5R}} = \frac{5Rc}{36} \]

### Final Answer
Thus, the frequency of the emitted radiation when the electron jumps from \( n = 3 \) to \( n = 2 \) is:
\[ \nu = \frac{5Rc}{36} \]
This corresponds to option 4.