If `13.6eV` energy is required to separate a hydrogen atom into a proton and an electron, then the orbital radius of electron in a hydrogen atom is

Options

A. `5.3xx10^(-11)m`
B. `4.3xx10^(-11)m`
C. `6.3xx10^(-11)m`
D. `7.3xx10^(-11)m`

Explanation

To find the orbital radius of the electron in a hydrogen atom given that the energy required to separate a hydrogen atom into a proton and an electron is 13.6 eV, we can follow these steps:

### Step 1: Understand the relationship between energy and radius
The total energy \( E \) of the electron in a hydrogen atom can be expressed as:
\[ E = -\frac{e^2}{8 \pi \epsilon_0 r} \]
where:
- \( e \) is the charge of the electron,
- \( \epsilon_0 \) is the permittivity of free space,
- \( r \) is the orbital radius.

Given that the energy required to separate the atom is \( 13.6 \, \text{eV} \), we can equate this to the absolute value of the total energy:
\[ |E| = 13.6 \, \text{eV} \]

### Step 2: Convert energy from eV to Joules
To use the formula, we need to convert the energy from electron volts to joules. The conversion factor is:
\[ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \]
Thus,
\[ E = 13.6 \, \text{eV} = 13.6 \times 1.6 \times 10^{-19} \, \text{J} = 2.176 \times 10^{-18} \, \text{J} \]

### Step 3: Rearrange the energy formula to solve for radius \( r \)
From the energy equation, we can rearrange it to solve for \( r \):
\[ r = -\frac{e^2}{8 \pi \epsilon_0 E} \]

### Step 4: Substitute known values
We know:
- \( e = 1.6 \times 10^{-19} \, \text{C} \)
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)

Now substituting these values into the equation:
\[ r = -\frac{(1.6 \times 10^{-19})^2}{8 \pi (8.85 \times 10^{-12}) (2.176 \times 10^{-18})} \]

### Step 5: Calculate the numerator and denominator
Calculating the numerator:
\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, \text{C}^2 \]

Calculating the denominator:
\[ 8 \pi (8.85 \times 10^{-12}) (2.176 \times 10^{-18}) \approx 4.83 \times 10^{-29} \, \text{N m}^2/\text{C}^2 \]

### Step 6: Final calculation for \( r \)
Now substituting back into the equation for \( r \):
\[ r = -\frac{2.56 \times 10^{-38}}{4.83 \times 10^{-29}} \approx 5.3 \times 10^{-11} \, \text{m} \]

Thus, the orbital radius of the electron in a hydrogen atom is approximately:
\[ r \approx 5.3 \times 10^{-11} \, \text{m} \]

### Conclusion
The orbital radius of the electron in a hydrogen atom is \( 5.3 \times 10^{-11} \, \text{m} \).

---