What is the colour corresponding to the wavelength of light emitted when the electron in a hydrogen atom undergoes transition from n = 4 to n = 2?

Options

A. Blue
B. Red
C. Yellow
D. Green

Explanation

To determine the color corresponding to the wavelength of light emitted when an electron in a hydrogen atom transitions from n = 4 to n = 2, we can follow these steps:

### Step 1: Identify the Transition
The electron is transitioning from the n = 4 energy level to the n = 2 energy level in a hydrogen atom.

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength of light emitted during an electron transition is given by:

\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]

Where:
- \( \lambda \) is the wavelength,
- \( R_H \) is the Rydberg constant (approximately \( 109677 \, \text{cm}^{-1} \)),
- \( n_f \) is the final energy level (2 in this case),
- \( n_i \) is the initial energy level (4 in this case).

### Step 3: Substitute the Values
Substituting the values into the formula:

\[ \frac{1}{\lambda} = 109677 \left( \frac{1}{2^2} - \frac{1}{4^2} \right) \]

Calculating the squares:

\[ \frac{1}{\lambda} = 109677 \left( \frac{1}{4} - \frac{1}{16} \right) \]

### Step 4: Simplify the Expression
Calculating \( \frac{1}{4} - \frac{1}{16} \):

\[ \frac{1}{4} = \frac{4}{16} \]
\[ \frac{1}{4} - \frac{1}{16} = \frac{4}{16} - \frac{1}{16} = \frac{3}{16} \]

Now substituting back:

\[ \frac{1}{\lambda} = 109677 \times \frac{3}{16} \]

### Step 5: Calculate \( \frac{1}{\lambda} \)
Calculating \( 109677 \times \frac{3}{16} \):

\[ \frac{1}{\lambda} = 20564.4 \, \text{cm}^{-1} \]

### Step 6: Calculate the Wavelength \( \lambda \)
Now, take the reciprocal to find \( \lambda \):

\[ \lambda = \frac{1}{20564.4} \, \text{cm} \]

Calculating this gives:

\[ \lambda \approx 4.86 \times 10^{-7} \, \text{cm} \]

### Step 7: Convert to Meters
Convert centimeters to meters:

\[ \lambda \approx 4.86 \times 10^{-7} \, \text{cm} = 4.86 \times 10^{-9} \, \text{m} \]

### Step 8: Convert to Nanometers
Since \( 1 \, \text{m} = 10^9 \, \text{nm} \):

\[ \lambda \approx 486 \, \text{nm} \]

### Step 9: Determine the Color
Now, we need to determine the color corresponding to a wavelength of 486 nm. The visible spectrum ranges are approximately:
- Violet: 400 - 450 nm
- Blue: 450 - 500 nm
- Green: 500 - 550 nm

Since 486 nm falls between 450 nm and 500 nm, the color corresponding to this wavelength is **blue**.

### Final Answer
The color corresponding to the wavelength of light emitted when the electron in a hydrogen atom undergoes a transition from n = 4 to n = 2 is **blue**.

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