An excited hydrogen atom emits a photon of wavelength `lambda` in returning to the ground state. If 'R' is the Rydberg's constant, then the quantum number 'n' of the excited state is:

Options

A. `sqrt(lambdaR(lambdaR-1))`
B. `sqrt((lambdaR)/((lambdaR-1)))`
C. `sqrt(((lambdaR-1))/(lambdaR))`
D. `sqrt((1)/(lambdaR(lambdaR-1)))`

Explanation

To find the quantum number 'n' of the excited state of a hydrogen atom that emits a photon of wavelength \( \lambda \) while returning to the ground state, we can use the Rydberg formula for hydrogen. Here’s the step-by-step solution:

### Step 1: Understand the Rydberg Formula
The Rydberg formula relates the wavelength of emitted or absorbed light to the quantum states of the hydrogen atom:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
where:
- \( \lambda \) is the wavelength of the emitted photon,
- \( R \) is the Rydberg constant,
- \( n_f \) is the final quantum number (ground state, which is 1 for hydrogen),
- \( n_i \) is the initial quantum number (the excited state we want to find).

### Step 2: Set the Final State
Since the atom is returning to the ground state, we set:
\[ n_f = 1 \]

### Step 3: Substitute into the Rydberg Formula
Substituting \( n_f \) into the Rydberg formula gives:
\[ \frac{1}{\lambda} = R \left( 1 - \frac{1}{n_i^2} \right) \]

### Step 4: Rearrange the Equation
Rearranging the equation to isolate \( n_i^2 \):
\[ \frac{1}{\lambda} = R \left( 1 - \frac{1}{n_i^2} \right) \]
\[ \frac{1}{\lambda} = R - \frac{R}{n_i^2} \]
\[ \frac{R}{n_i^2} = R - \frac{1}{\lambda} \]
\[ \frac{1}{n_i^2} = \frac{R - \frac{1}{\lambda}}{R} \]

### Step 5: Simplify the Expression
Simplifying further, we have:
\[ \frac{1}{n_i^2} = 1 - \frac{1}{\lambda R} \]
\[ \frac{1}{n_i^2} = \frac{\lambda R - 1}{\lambda R} \]

### Step 6: Solve for \( n_i^2 \)
Taking the reciprocal gives:
\[ n_i^2 = \frac{\lambda R}{\lambda R - 1} \]

### Step 7: Take the Square Root
Finally, taking the square root to find \( n_i \):
\[ n_i = \sqrt{\frac{\lambda R}{\lambda R - 1}} \]

### Conclusion
Thus, the quantum number \( n \) of the excited state is:
\[ n = \sqrt{\frac{\lambda R}{\lambda R - 1}} \]