According to Bohr's theory, the electronic energy of H-atom in Bohr's orbit is given by

Options

A. `E_(n)=-(2.18xx10^(-19)xxZ)/(2n^(2))J`
B. `E_(n)=-(2.179xx10^(-18)xxZ^(2))/(n^(2))J`
C. `E_(n)=-(21.79xx10^(-18)xxZ)/(2n^(2))J`
D. `E_(n)=-(21.8xx10^(-21)xxZ^(2))/(n^(2))J`

Answer

To find the electronic energy of a hydrogen atom in Bohr's orbit according to Bohr's theory, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: The energy of an electron in the nth orbit of a hydrogen atom is given by the formula:
\[ E_n = -\frac{2 \pi^2 m e^4}{n^2 h^2} \]
where:
- \(E_n\) = energy of the electron in the nth orbit
- \(m\) = mass of the electron
- \(e\) = charge of the electron
- \(n\) = principal quantum number (orbit number)
- \(h\) = Planck's constant

2. **Substitute Known Values**:
- Mass of electron, \(m = 9.1 \times 10^{-31} \, \text{kg}\)
- Charge of electron, \(e = 1.6 \times 10^{-19} \, \text{C}\)
- Planck's constant, \(h = 6.626 \times 10^{-34} \, \text{Js}\)

3. **Plug in the Values**:
Substitute these values into the formula:
\[ E_n = -\frac{2 \pi^2 (9.1 \times 10^{-31}) (1.6 \times 10^{-19})^4}{n^2 (6.626 \times 10^{-34})^2} \]

4. **Calculate the Energy**:
- Calculate \(e^4\):
\[ e^4 = (1.6 \times 10^{-19})^4 \]
- Calculate \(h^2\):
\[ h^2 = (6.626 \times 10^{-34})^2 \]
- Substitute these values back into the equation and simplify to find \(E_n\).

5. **Final Result**:
After performing the calculations, the energy of the electron in the nth orbit can be expressed as:
\[ E_n \approx -2.17 \times 10^{-18} \frac{1}{n^2} \, \text{J} \]
This indicates that the energy is negative, which is typical for bound systems.

6. **Identify the Correct Option**:
From the options provided, the correct expression for the energy of the hydrogen atom in Bohr's orbit is:
\[ E_n = -\frac{2.17 \times 10^{-18}}{n^2} \, \text{J} \]