In Bohr's model of the hydrogen atom, let R, V, T and E represent the radius of the orbit, speed of the electron, time period of revolution of electron and the total energy of the electron respectively. The quantity proportional to the quantum number n is

Options

A. VR
B. E
C. r
D. T

Explanation

To solve the problem, we need to analyze the relationships between the radius (R), speed (V), time period (T), and total energy (E) of the electron in a hydrogen atom according to Bohr's model. We will determine which of these quantities is proportional to the quantum number \( n \).

### Step-by-Step Solution:

1. **Radius of the Orbit (R)**:
- According to Bohr's model, the radius \( R \) of the orbit is given by the formula:
\[ R \propto \frac{n^2}{Z} \]
- Here, \( n \) is the principal quantum number and \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).
- Therefore, \( R \) is proportional to \( n^2 \).

2. **Speed of the Electron (V)**:
- The speed \( V \) of the electron in the orbit is given by:
\[ V \propto \frac{Z}{n} \]
- Thus, \( V \) is inversely proportional to \( n \).

3. **Total Energy (E)**:
- The total energy \( E \) of the electron in the orbit can be expressed as:
\[ E \propto -\frac{Z^2}{n^2} \]
- This indicates that the total energy is proportional to \( \frac{1}{n^2} \).

4. **Time Period of Revolution (T)**:
- The time period \( T \) can be calculated using the formula:
\[ T = \frac{2\pi R}{V} \]
- Substituting the proportionalities we found:
\[ T \propto \frac{R}{V} \propto \frac{\frac{n^2}{Z}}{\frac{Z}{n}} = \frac{n^3}{Z^2} \]
- Thus, \( T \) is proportional to \( n^3 \).

### Summary of Proportionalities:
- \( R \propto n^2 \)
- \( V \propto \frac{1}{n} \)
- \( E \propto \frac{1}{n^2} \)
- \( T \propto n^3 \)

### Conclusion:
From the analysis, we see that:
- The quantity that is proportional to the quantum number \( n \) is the radius \( R \), since \( R \propto n^2 \).

Thus, the answer to the question is:
**The quantity proportional to the quantum number \( n \) is \( R \) (the radius of the orbit).**