The ratio of the energy required to remove an electron from the first three Bohr's orbit of Hydrogen atom is

Options

A. `3:2:1`
B. `9:4:1`
C. `36 :9:4`
D. `1 :4 :9`

Answer

To find the ratio of the energy required to remove an electron from the first three Bohr orbits of a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Formula**:
The energy of an electron in the nth orbit of a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

2. **Calculate the Energy for Each Orbit**:
- For the first orbit (\( n = 1 \)):
\[ E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV} \]
- For the second orbit (\( n = 2 \)):
\[ E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6}{4} = -3.4 \, \text{eV} \]
- For the third orbit (\( n = 3 \)):
\[ E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -\frac{13.6}{9} \approx -1.51 \, \text{eV} \]

3. **Determine the Ionization Energy**:
The energy required to remove an electron (ionization energy) from each orbit is the negative of the energy in that orbit:
- Ionization energy from the first orbit:
\[ I_1 = -E_1 = 13.6 \, \text{eV} \]
- Ionization energy from the second orbit:
\[ I_2 = -E_2 = 3.4 \, \text{eV} \]
- Ionization energy from the third orbit:
\[ I_3 = -E_3 = \frac{13.6}{9} \approx 1.51 \, \text{eV} \]

4. **Set Up the Ratio**:
Now we can set up the ratio of the ionization energies:
\[ I_1 : I_2 : I_3 = 13.6 : 3.4 : \frac{13.6}{9} \]

5. **Simplify the Ratio**:
To simplify the ratio, we can express each term with a common denominator. The common denominator for 4 and 9 is 36:
- Multiply each term by 36:
\[ 13.6 \times 36 : 3.4 \times 36 : \frac{13.6}{9} \times 36 \]
- This gives:
\[ 489.6 : 122.4 : 54.4 \]
- Dividing each term by 13.6 gives:
\[ 36 : 9 : 4 \]

6. **Final Result**:
The final ratio of the energy required to remove an electron from the first three Bohr orbits of a hydrogen atom is:
\[ \text{Ratio} = 36 : 9 : 4 \]

### Answer:
The answer is option C: \( 36 : 9 : 4 \).