A proton traps a free electron to form the hydrogen atom of lowest energy level. The wavelength of photon emitted in this process will be (assume initial kinetic energy of the electron to be zero)

Options

A. 91.4 nm
B. 71.4 nm
C. 103.6 nm
D. 141.2 nm

Explanation

To find the wavelength of the photon emitted when a proton traps a free electron to form a hydrogen atom in its lowest energy level, we can follow these steps:

### Step 1: Understand the Energy Levels of Hydrogen
The energy of an electron in a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number. For the lowest energy level (ground state), \( n = 1 \).

### Step 2: Calculate the Energy of the Ground State
Substituting \( n = 1 \) into the energy formula:
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]

### Step 3: Determine the Change in Energy
Initially, the free electron has zero kinetic energy, so its initial energy is:
\[ E_{\text{initial}} = 0 \, \text{eV} \]
When the electron is captured by the proton to form a hydrogen atom, the final energy is:
\[ E_{\text{final}} = -13.6 \, \text{eV} \]
The change in energy (\( \Delta E \)) is:
\[ \Delta E = E_{\text{final}} - E_{\text{initial}} = -13.6 \, \text{eV} - 0 \, \text{eV} = -13.6 \, \text{eV} \]
The absolute value of the change in energy is:
\[ |\Delta E| = 13.6 \, \text{eV} \]

### Step 4: Convert Energy to Joules
To convert electron volts to joules, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ \Delta E = 13.6 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 2.176 \times 10^{-18} \, \text{J} \]

### Step 5: Use the Energy-Wavelength Relationship
The energy of a photon is related to its wavelength by the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of the photon,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength.

### Step 6: Rearrange to Solve for Wavelength
Rearranging the equation to solve for wavelength (\( \lambda \)):
\[ \lambda = \frac{hc}{E} \]

### Step 7: Substitute Values
Substituting the values into the equation:
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J s})(3 \times 10^8 \, \text{m/s})}{2.176 \times 10^{-18} \, \text{J}} \]

### Step 8: Calculate Wavelength
Calculating the wavelength:
\[ \lambda = \frac{1.9878 \times 10^{-25} \, \text{J m}}{2.176 \times 10^{-18} \, \text{J}} \approx 9.12 \times 10^{-8} \, \text{m} = 91.2 \, \text{nm} \]

### Conclusion
The wavelength of the photon emitted during the formation of the hydrogen atom in its lowest energy level is approximately:
\[ \lambda \approx 91.2 \, \text{nm} \]