Hydrogen atom emits blue light when it changes from `n = 4` energy level to the `n = 2` level. Which colour of light would te atom emit when it changes from the `n = 5` level to the `n = 2` level ?

Options

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

Answer

To determine the color of light emitted when a hydrogen atom transitions from the \( n = 5 \) energy level to the \( n = 2 \) level, we can follow these steps:

### Step 1: Understand the energy levels
The energy levels of a hydrogen atom are given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

### Step 2: Calculate the energy for \( n = 4 \) to \( n = 2 \)
For the transition from \( n = 4 \) to \( n = 2 \):
- Energy at \( n = 4 \):
\[ E_4 = -\frac{13.6}{4^2} = -\frac{13.6}{16} = -0.85 \, \text{eV} \]
- Energy at \( n = 2 \):
\[ E_2 = -\frac{13.6}{2^2} = -\frac{13.6}{4} = -3.4 \, \text{eV} \]
- Change in energy (\( \Delta E \)):
\[ \Delta E = E_2 - E_4 = -3.4 - (-0.85) = -3.4 + 0.85 = -2.55 \, \text{eV} \]

### Step 3: Calculate the energy for \( n = 5 \) to \( n = 2 \)
For the transition from \( n = 5 \) to \( n = 2 \):
- Energy at \( n = 5 \):
\[ E_5 = -\frac{13.6}{5^2} = -\frac{13.6}{25} = -0.544 \, \text{eV} \]
- Change in energy (\( \Delta E \)):
\[ \Delta E = E_2 - E_5 = -3.4 - (-0.544) = -3.4 + 0.544 = -2.856 \, \text{eV} \]

### Step 4: Determine the color of the emitted light
The energy of the emitted photon corresponds to the wavelength of light emitted. The relationship between energy and wavelength is given by:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light. Higher energy corresponds to shorter wavelengths.

Since the transition from \( n = 5 \) to \( n = 2 \) has a greater energy change (\( -2.856 \, \text{eV} \)) compared to the transition from \( n = 4 \) to \( n = 2 \) (\( -2.55 \, \text{eV} \)), the emitted light will have a shorter wavelength and thus a higher energy.

### Step 5: Identify the color
The blue light emitted from the \( n = 4 \) to \( n = 2 \) transition has a certain energy. The transition from \( n = 5 \) to \( n = 2 \) has more energy than that, which corresponds to violet light.

### Final Answer
The hydrogen atom would emit violet light when it changes from the \( n = 5 \) level to the \( n = 2 \) level.

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