A Rydberg atom is a single-electron atom with a large quantum number n. Rydberg states are close together in energy, so transitions between from a state `n + 1` to a state n in hydrogen. The wavelength of the emitted photon varies with quantum number n as

Options

A. `(1)/(n^(2))`
B. `n`
C. `(1)/(n^(3))`
D. `n^(3)`

Explanation

To solve the problem regarding the wavelength of the emitted photon during the transition from a Rydberg state \( n+1 \) to \( n \) in hydrogen, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Levels**:
The energy of an electron in a hydrogen atom at a principal quantum number \( n \) is given by:
\[ E_n = -\frac{E_0}{n^2} \]
where \( E_0 \) is a constant (approximately \( 13.6 \, \text{eV} \)).

2. **Calculate the Energy Difference**:
The energy difference between the states \( n+1 \) and \( n \) is:
\[ E_{n+1} - E_n = \left(-\frac{E_0}{(n+1)^2}\right) - \left(-\frac{E_0}{n^2}\right) \]
This simplifies to:
\[ E_{n+1} - E_n = E_0 \left(\frac{1}{n^2} - \frac{1}{(n+1)^2}\right) \]

3. **Use the Formula for Energy Difference**:
The expression for the energy difference can be rewritten using the formula for the difference of squares:
\[ \frac{1}{n^2} - \frac{1}{(n+1)^2} = \frac{(n+1)^2 - n^2}{n^2(n+1)^2} = \frac{2n + 1}{n^2(n+1)^2} \]
Thus, we have:
\[ E_{n+1} - E_n = E_0 \cdot \frac{2n + 1}{n^2(n+1)^2} \]

4. **Relate Energy to Wavelength**:
The energy of the emitted photon is also given by:
\[ E = \frac{hc}{\lambda} \]
Setting the two expressions for energy equal gives:
\[ \frac{hc}{\lambda} = E_0 \cdot \frac{2n + 1}{n^2(n+1)^2} \]

5. **Solve for Wavelength**:
Rearranging for \( \lambda \):
\[ \lambda = \frac{hc \cdot n^2(n+1)^2}{E_0(2n + 1)} \]

6. **Approximate for Large n**:
For large values of \( n \), \( n+1 \approx n \) and \( 2n + 1 \approx 2n \). Thus, we can simplify:
\[ \lambda \approx \frac{hc \cdot n^2 n^2}{E_0 \cdot 2n} = \frac{hc \cdot n^4}{2E_0} \]
Therefore, we can conclude that:
\[ \lambda \propto n^3 \]

### Final Result:
The wavelength of the emitted photon varies with the quantum number \( n \) as:
\[ \lambda \propto n^3 \]