In Bohr model of the hydrogen atom, let R,v and E represent the radius of the orbit, speed of the electron and the total energy respectively. Which of the following quantities are directly proportional to the quantum number n?

Options

A. VR
B. RE
C. `R//E`
D. none of these

Explanation

To determine which quantities are directly proportional to the quantum number \( n \) in the Bohr model of the hydrogen atom, we will analyze the relationships between the radius \( r \), speed \( v \), and total energy \( E \) of the electron in the atom.

### Step-by-Step Solution:

1. **Understanding the Relationships**:
- In the Bohr model, the radius of the electron's orbit \( r \) is given by the formula:
\[ r \propto n^2 \]
- The speed of the electron \( v \) is inversely proportional to the quantum number \( n \):
\[ v \propto \frac{1}{n} \]
- The total energy \( E \) of the electron is also inversely proportional to the square of the quantum number \( n \):
\[ E \propto -\frac{1}{n^2} \]

2. **Analyzing Each Quantity**:
- **Radius \( r \)**: Since \( r \propto n^2 \), it is not directly proportional to \( n \) but to \( n^2 \).
- **Speed \( v \)**: Since \( v \propto \frac{1}{n} \), it is inversely proportional to \( n \), thus not directly proportional.
- **Total Energy \( E \)**: Since \( E \propto -\frac{1}{n^2} \), it is also inversely proportional to \( n^2 \), hence not directly proportional.

3. **Combining Quantities**:
- Now let's check the combinations:
- **Product \( v \cdot r \)**:
\[ v \cdot r \propto \left(\frac{1}{n}\right) \cdot (n^2) = n \]
This shows that \( v \cdot r \) is directly proportional to \( n \).
- **Product \( r \cdot E \)**:
\[ r \cdot E \propto (n^2) \cdot \left(-\frac{1}{n^2}\right) = -1 \]
This indicates that \( r \cdot E \) is independent of \( n \).
- **Ratio \( \frac{r}{E} \)**:
\[ \frac{r}{E} \propto \frac{n^2}{-\frac{1}{n^2}} = -n^4 \]
This shows that \( \frac{r}{E} \) is directly proportional to \( n^4 \).

4. **Conclusion**:
- Among the quantities analyzed, the only quantity that is directly proportional to the quantum number \( n \) is \( v \cdot r \).

### Final Answer:
The quantity that is directly proportional to the quantum number \( n \) is \( v \cdot r \).