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

Options

A. red
B. yellow
C. green
D. voilet.

Explanation

To determine the color of light emitted by a hydrogen atom when it transitions from the n=5 energy level to the n=2 energy 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 at n=5
Using the formula for \( n=5 \):
\[ E_5 = -\frac{13.6 \, \text{eV}}{5^2} = -\frac{13.6 \, \text{eV}}{25} = -0.544 \, \text{eV} \]

### Step 3: Calculate the Energy at n=2
Now, calculate the energy for \( n=2 \):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV} \]

### Step 4: Calculate the Change in Energy (ΔE)
The change in energy when the electron transitions from \( n=5 \) to \( n=2 \) is given by:
\[ \Delta E = E_2 - E_5 \]
Substituting the values:
\[ \Delta E = -3.4 \, \text{eV} - (-0.544 \, \text{eV}) = -3.4 \, \text{eV} + 0.544 \, \text{eV} = -2.856 \, \text{eV} \]
The magnitude of the energy change is:
\[ |\Delta E| = 2.856 \, \text{eV} \]

### Step 5: Determine the Wavelength of the Emitted Light
To find the wavelength of the emitted light, we can use the energy-wavelength relationship:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant (\( 4.135667696 \times 10^{-15} \, \text{eV s} \)) and \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)).

Rearranging for \( \lambda \):
\[ \lambda = \frac{hc}{E} \]
Substituting the values:
\[ \lambda = \frac{(4.135667696 \times 10^{-15} \, \text{eV s})(3 \times 10^8 \, \text{m/s})}{2.856 \, \text{eV}} \approx 4.35 \times 10^{-7} \, \text{m} = 435 \, \text{nm} \]

### Step 6: Identify the Color
The wavelength of approximately 435 nm falls within the visible spectrum and corresponds to blue light.

### Final Answer
The hydrogen atom emits blue light when it transitions from the n=5 energy level to the n=2 energy level.

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