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

Options

A. `v prop n^(-3)`
B. `v prop n^(2)`
C. ` v prop n^(3)`
D. `v prop n^((2)/(3))`

Explanation

To solve the problem of how the frequency of emitted radiation is related to the principal quantum number \( n \) when an electron transitions from the \( (n+1) \) state to the \( n \) state, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Transition**: When an electron transitions from a higher energy level (n+1) to a lower energy level (n), it emits radiation. The frequency of this radiation can be derived from the energy difference between these two states.

2. **Use the Rydberg Formula**: The Rydberg formula for the wavelength (\( \lambda \)) of the emitted radiation is given by:
\[ \frac{1}{\lambda} \propto \frac{1}{n_1^2} - \frac{1}{n_2^2} \]
where \( n_1 \) is the lower energy level and \( n_2 \) is the higher energy level.

3. **Assign Values**: In our case, \( n_1 = n \) and \( n_2 = n + 1 \). Therefore, we can rewrite the equation as:
\[ \frac{1}{\lambda} \propto \frac{1}{n^2} - \frac{1}{(n+1)^2} \]

4. **Simplify the Expression**: We can simplify the right-hand side:
\[ \frac{1}{\lambda} \propto \frac{(n+1)^2 - n^2}{n^2(n+1)^2} \]
Expanding the numerator:
\[ (n+1)^2 - n^2 = n^2 + 2n + 1 - n^2 = 2n + 1 \]
Thus, we have:
\[ \frac{1}{\lambda} \propto \frac{2n + 1}{n^2(n + 1)^2} \]

5. **Approximate for Large n**: For large values of \( n \) (i.e., \( n \gg 1 \)), we can approximate \( 2n + 1 \approx 2n \) and \( (n + 1)^2 \approx n^2 \). Therefore:
\[ \frac{1}{\lambda} \propto \frac{2n}{n^2 \cdot n^2} = \frac{2}{n^4} \]

6. **Relate Wavelength to Frequency**: The frequency \( \mu \) is related to the wavelength by the equation:
\[ \mu = \frac{C}{\lambda} \]
where \( C \) is the speed of light. Since \( \frac{1}{\lambda} \propto \frac{2}{n^4} \), we can say:
\[ \mu \propto \frac{1}{\lambda} \propto n^4 \]

7. **Final Relationship**: Therefore, the frequency of emitted radiation \( \mu \) is inversely proportional to \( n^3 \):
\[ \mu \propto \frac{1}{n^3} \]

### Conclusion:
The frequency of emitted radiation when an electron transitions from the \( (n + 1) \) state to the \( n \) state is related to \( n \) by:
\[ \mu \propto \frac{1}{n^3} \]