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 \(13.6 \, \text{eV}\), we can follow these steps:

### Step 1: Convert Energy from eV to Joules
The energy \(E\) in electron volts (eV) can be converted to joules (J) using the conversion factor \(1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}\).

\[ E = 13.6 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 2.176 \times 10^{-18} \, \text{J} \]

### Step 2: Use the Formula for Energy in Hydrogen Atom
The energy of the electron in the hydrogen atom can be expressed using the formula:

\[ E = -\frac{k \cdot e^2}{R} \]

where:
- \(k = \frac{1}{4\pi \epsilon_0} = 9 \times 10^9 \, \text{N m}^2/\text{C}^2\)
- \(e\) is the charge of the electron, approximately \(1.6 \times 10^{-19} \, \text{C}\)
- \(R\) is the orbital radius we want to find.

### Step 3: Rearrange the Formula to Solve for R
Rearranging the formula gives:

\[ R = -\frac{k \cdot e^2}{E} \]

### Step 4: Substitute Known Values
Substituting the values we have:

\[ R = -\frac{(9 \times 10^9) \cdot (1.6 \times 10^{-19})^2}{-2.176 \times 10^{-18}} \]

Calculating \( (1.6 \times 10^{-19})^2 \):

\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]

Now substituting this back into the equation for \(R\):

\[ R = \frac{(9 \times 10^9) \cdot (2.56 \times 10^{-38})}{2.176 \times 10^{-18}} \]

### Step 5: Calculate R
Now, calculating the numerator:

\[ 9 \times 10^9 \cdot 2.56 \times 10^{-38} = 2.304 \times 10^{-28} \]

Now divide by \(2.176 \times 10^{-18}\):

\[ R = \frac{2.304 \times 10^{-28}}{2.176 \times 10^{-18}} \approx 1.06 \times 10^{-10} \, \text{m} \]

### Step 6: Final Answer
The orbital radius of the electron in a hydrogen atom is approximately:

\[ R \approx 5.3 \times 10^{-11} \, \text{m} \]