The energy of electron in the first orbit of i `He^+ is -871.6*10^-20` J. The energy of the electron in the first orbit of hydrogen atom would be

Options

A. `-871.6× 10^-20` J
B. `-435.8 × 10^-20` J
C. `-108.9× 10^-20`J
D. `-217.9 × 10^-20` J

Answer

To find the energy of the electron in the first orbit of a hydrogen atom (H), given the energy of the electron in the first orbit of the helium ion (He⁺), we can use the formula for the energy of an electron in an orbit:

\[ E = -\frac{13.6 \, Z^2}{n^2} \, \text{eV} \]

Where:
- \( E \) is the energy of the electron,
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number (for the first orbit, \( n = 1 \)).

### Step 1: Identify the given values
- The energy of the electron in the first orbit of He⁺ is given as \( E_{He^+} = -871.6 \times 10^{-20} \, \text{J} \).
- The atomic number of hydrogen (H) is \( Z_H = 1 \).
- The atomic number of helium (He) is \( Z_{He} = 2 \).

### Step 2: Use the relationship between the energies of H and He⁺
The energy of the electron in hydrogen can be related to the energy of the electron in He⁺ using the formula:

\[ \frac{E_H}{E_{He^+}} = \frac{Z_H^2}{Z_{He}^2} \]

### Step 3: Substitute the values into the equation
Substituting the known values into the equation:

\[ \frac{E_H}{-871.6 \times 10^{-20}} = \frac{1^2}{2^2} \]

This simplifies to:

\[ \frac{E_H}{-871.6 \times 10^{-20}} = \frac{1}{4} \]

### Step 4: Solve for \( E_H \)
Now, we can solve for \( E_H \):

\[ E_H = -871.6 \times 10^{-20} \times \frac{1}{4} \]

Calculating this gives:

\[ E_H = -217.9 \times 10^{-20} \, \text{J} \]

### Final Answer
Thus, the energy of the electron in the first orbit of the hydrogen atom is:

\[ E_H = -217.9 \times 10^{-20} \, \text{J} \]