What is the wavelength of light for the least energetic photon emitted in the Lyman series of the hydrogen spectrum. (Take , hc =1240 eV -nm)

Options

A. 122nm
B. 82nm
C. 150nm
D. 120nm

Explanation

To find the wavelength of light for the least energetic photon emitted in the Lyman series of the hydrogen spectrum, we can follow these steps:

### Step 1: Understand the Lyman Series
The Lyman series corresponds to transitions of electrons in a hydrogen atom from higher energy levels (n_i) to the lowest energy level (n_f = 1). The least energetic photon will be emitted when the electron transitions from the first excited state (n_i = 2) to the ground state (n_f = 1).

### Step 2: Use the Rydberg Formula
The wavelength (λ) of the emitted photon can be calculated using the Rydberg formula for hydrogen:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
where \( R_H = 13.6 \, \text{eV} \) (the Rydberg constant for hydrogen).

### Step 3: Substitute Values
For the Lyman series:
- \( n_f = 1 \)
- \( n_i = 2 \)

Substituting these values into the formula:
\[ \frac{1}{\lambda} = 13.6 \, \text{eV} \cdot \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]
Calculating the terms:
\[ \frac{1}{\lambda} = 13.6 \, \text{eV} \cdot \left( 1 - \frac{1}{4} \right) = 13.6 \, \text{eV} \cdot \left( \frac{3}{4} \right) = 10.2 \, \text{eV} \]

### Step 4: Convert Energy to Wavelength
Using the relationship \( E = \frac{hc}{\lambda} \) and rearranging it gives:
\[ \lambda = \frac{hc}{E} \]
Given \( hc = 1240 \, \text{eV} \cdot \text{nm} \), we can substitute \( E = 10.2 \, \text{eV} \):
\[ \lambda = \frac{1240 \, \text{eV} \cdot \text{nm}}{10.2 \, \text{eV}} = \frac{1240}{10.2} \, \text{nm} \]

### Step 5: Calculate the Wavelength
Calculating the above expression:
\[ \lambda \approx 121.57 \, \text{nm} \]

### Step 6: Round the Result
Rounding to three significant figures, we get:
\[ \lambda \approx 122 \, \text{nm} \]

### Final Answer
The wavelength of light for the least energetic photon emitted in the Lyman series of the hydrogen spectrum is approximately **122 nm**.

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