In Bohr 's model of hydrogen when an electron jumps from n=1 to n=3 how much energy will be abosrbed

Options

A. `2.389 xx 10 ^(-12) ergs `
B. `0.239 xx 10 ^(-10) ergs`
C. `2.15 xx10 ^(-11) ergs`
D. `0.1936 xx 10^(-10) ergs`

Answer

To determine the energy absorbed when an electron jumps from n=1 to n=3 in the Bohr model of hydrogen, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Formula for Energy Change**:
The energy change (ΔE) for an electron transition in a hydrogen atom is given by the formula:
\[ \Delta E = -R_H \cdot Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
where:
- \( R_H \) is the Rydberg constant,
- \( Z \) is the atomic number,
- \( n_f \) is the final energy level,
- \( n_i \) is the initial energy level.

2. **Substitute Known Values**:
For hydrogen, the atomic number \( Z = 1 \) and the Rydberg constant \( R_H = 2.18 \times 10^{-18} \) joules. The initial state \( n_i = 1 \) and the final state \( n_f = 3 \).

Substituting these values into the formula:
\[ \Delta E = -2.18 \times 10^{-18} \cdot 1^2 \left( \frac{1}{3^2} - \frac{1}{1^2} \right) \]

3. **Calculate the Energy Change**:
Calculate \( \frac{1}{3^2} - \frac{1}{1^2} \):
\[ \frac{1}{3^2} = \frac{1}{9}, \quad \frac{1}{1^2} = 1 \]
Therefore:
\[ \frac{1}{3^2} - \frac{1}{1^2} = \frac{1}{9} - 1 = \frac{1 - 9}{9} = -\frac{8}{9} \]
Now substituting back into the energy change formula:
\[ \Delta E = -2.18 \times 10^{-18} \cdot \left( -\frac{8}{9} \right) \]
\[ \Delta E = 2.18 \times 10^{-18} \cdot \frac{8}{9} \]
\[ \Delta E = \frac{17.44}{9} \times 10^{-18} = 1.94 \times 10^{-18} \text{ joules} \]

4. **Convert Joules to Ergs**:
We know that \( 1 \text{ joule} = 10^7 \text{ ergs} \). Therefore:
\[ \Delta E = 1.94 \times 10^{-18} \text{ joules} \times 10^7 \text{ ergs/joule} \]
\[ \Delta E = 1.94 \times 10^{-11} \text{ ergs} \]

5. **Final Answer**:
The energy absorbed when the electron jumps from n=1 to n=3 is:
\[ \Delta E = 1.94 \times 10^{-11} \text{ ergs} \]