The wavelength of the spectral line when the electron is the hydrogen atom undergoes a transition from the energy level `4` to energy level `2` is.

Options

A. 185.2 nm
B. 285.2 nm
C. 385.2 nm
D. 486.4nm

Answer

To find the wavelength of the spectral line when the electron in a hydrogen atom undergoes a transition from energy level 4 to energy level 2, we can follow these steps:

### Step 1: Identify the Energy Levels
We need to identify the initial and final energy levels of the electron transition.
- Initial level (n2) = 4
- Final level (n1) = 2

### Step 2: Use the Rydberg Formula
The Rydberg formula for calculating the wave number (ν̅) of the emitted or absorbed light during a transition is given by:
\[ \bar{\nu} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R \) is the Rydberg constant, approximately \( 109677 \, \text{m}^{-1} \).

### Step 3: Substitute the Values
Now, we substitute the values into the Rydberg formula:
\[ \bar{\nu} = 109677 \left( \frac{1}{2^2} - \frac{1}{4^2} \right) \]
Calculating the squares:
- \( 2^2 = 4 \)
- \( 4^2 = 16 \)

Thus, we have:
\[ \bar{\nu} = 109677 \left( \frac{1}{4} - \frac{1}{16} \right) \]

### Step 4: Simplify the Expression
Now, we need to find a common denominator to simplify the expression:
\[ \frac{1}{4} = \frac{4}{16} \]
So,
\[ \bar{\nu} = 109677 \left( \frac{4}{16} - \frac{1}{16} \right) = 109677 \left( \frac{3}{16} \right) \]

### Step 5: Calculate the Wave Number
Now we can calculate:
\[ \bar{\nu} = 109677 \times \frac{3}{16} = 20564.25 \, \text{m}^{-1} \]

### Step 6: Calculate the Wavelength
The wavelength \( \lambda \) is related to the wave number by the equation:
\[ \lambda = \frac{1}{\bar{\nu}} \]
Substituting the wave number:
\[ \lambda = \frac{1}{20564.25} \approx 4.86 \times 10^{-5} \, \text{m} \]

### Step 7: Convert to Nanometers
To convert meters to nanometers, we multiply by \( 10^9 \):
\[ \lambda \approx 4.86 \times 10^{-5} \times 10^9 \, \text{nm} = 486.4 \, \text{nm} \]

### Final Answer
The wavelength of the spectral line when the electron transitions from energy level 4 to energy level 2 is approximately **486.4 nm**.

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