When an electron makes a transition from `(n+1)` state of nth state, the frequency of emitted radiations is related to n according to `(n gtgt 1)`:

Options

A. (a) `v=(2cRZ^(2))/(n^(3))`
B. (b) `v=(cRZ^(2))/(n^(4))`
C. (c) `v=(cRZ^(2))/(n^(2))`
D. (d) `v=(2cRZ^(2))/(n^(2))`

Answer

To solve the problem of finding the relationship between the frequency of emitted radiation when an electron transitions from the (n+1) state to the nth state, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Transition**:
The electron is transitioning from the (n+1) energy level to the nth energy level. This transition results in the emission of radiation.

2. **Use the Rydberg Formula**:
The Rydberg formula for the wavelength (λ) of emitted radiation during such transitions is given by:
\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant, \( Z \) is the atomic number, \( n_1 \) is the lower energy level (n), and \( n_2 \) is the higher energy level (n+1).

3. **Substituting Values**:
Substitute \( n_1 = n \) and \( n_2 = n + 1 \):
\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n^2} - \frac{1}{(n+1)^2} \right) \]

4. **Simplify the Expression**:
To simplify \( \frac{1}{n^2} - \frac{1}{(n+1)^2} \):
\[ \frac{1}{n^2} - \frac{1}{(n+1)^2} = \frac{(n+1)^2 - n^2}{n^2(n+1)^2} = \frac{2n + 1}{n^2(n+1)^2} \]
Thus, we have:
\[ \frac{1}{\lambda} = RZ^2 \cdot \frac{2n + 1}{n^2(n+1)^2} \]

5. **Approximate for Large n**:
Since \( n \) is much greater than 1, we can approximate \( (n+1)^2 \approx n^2 \):
\[ \frac{1}{\lambda} \approx RZ^2 \cdot \frac{2n + 1}{n^4} \]
Ignoring the 1 for large \( n \):
\[ \frac{1}{\lambda} \approx \frac{2RZ^2}{n^3} \]

6. **Relate Wavelength to Frequency**:
The relationship between frequency (ν) and wavelength (λ) is given by:
\[ c = \nu \lambda \]
Rearranging gives:
\[ \nu = \frac{c}{\lambda} \]

7. **Substituting for λ**:
Substitute \( \lambda \) from our previous step:
\[ \nu = c \cdot \frac{1}{\frac{2RZ^2}{n^3}} = \frac{c n^3}{2RZ^2} \]

8. **Final Expression**:
The frequency of the emitted radiation when an electron transitions from the (n+1) state to the nth state is:
\[ \nu \propto \frac{1}{n^3} \]

### Conclusion:
The frequency of emitted radiation is inversely proportional to the cube of the principal quantum number \( n \) when \( n \) is much greater than 1.