the centripetal acceleration of an electron in a Bohr's orbit is inversely proportional to

Options

A. principle quantum number
B. square of quantum number
C. cube of quantum number
D. forth power of its quantum number.

Answer

To find the centripetal acceleration of an electron in a Bohr's orbit and determine its relationship with the principal quantum number \( n \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Centripetal Acceleration**:
The centripetal acceleration \( a_c \) of an electron moving in a circular orbit is given by the formula:
\[ a_c = \frac{v^2}{r_n} \]
where \( v \) is the linear speed of the electron and \( r_n \) is the radius of the nth orbit.

2. **Centripetal Force and Coulomb Force**:
The centripetal force required to keep the electron in circular motion is provided by the Coulomb force between the electron and the nucleus. This can be expressed as:
\[ \frac{mv^2}{r_n} = \frac{k e^2}{r_n^2} \]
where \( m \) is the mass of the electron, \( k \) is Coulomb's constant, and \( e \) is the charge of the electron.

3. **Rearranging the Equation**:
From the above equation, we can express \( v^2 \) as:
\[ v^2 = \frac{k e^2}{m r_n} \]

4. **Substituting \( v^2 \) into the Centripetal Acceleration Formula**:
Substituting \( v^2 \) into the centripetal acceleration formula, we get:
\[ a_c = \frac{\frac{k e^2}{m r_n}}{r_n} = \frac{k e^2}{m r_n^2} \]

5. **Finding the Radius \( r_n \)**:
According to Bohr's model, the radius of the nth orbit is given by:
\[ r_n = \frac{n^2 h^2}{4 \pi^2 m k e^2} \]
where \( h \) is Planck's constant.

6. **Substituting \( r_n \) into the Centripetal Acceleration Formula**:
Now, substituting \( r_n \) into the expression for \( a_c \):
\[ a_c = \frac{k e^2}{m \left(\frac{n^2 h^2}{4 \pi^2 m k e^2}\right)^2} \]
Simplifying this gives:
\[ a_c \propto \frac{1}{r_n^2} \propto \frac{1}{\left(\frac{n^2 h^2}{4 \pi^2 m k e^2}\right)^2} \]

7. **Final Relationship**:
Since \( r_n \) is proportional to \( n^2 \), we find that:
\[ a_c \propto \frac{1}{n^4} \]
Therefore, the centripetal acceleration of an electron in a Bohr's orbit is inversely proportional to the fourth power of the principal quantum number \( n \).

### Conclusion:
The centripetal acceleration of an electron in a Bohr's orbit is inversely proportional to \( n^4 \).

---