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. v/E
D. R/E

Explanation

To solve the problem, we need to analyze the relationships between the radius \( r \), speed \( v \), and total energy \( E \) of the electron in a hydrogen atom according to the Bohr model, and how they relate to the principal quantum number \( n \).

### Step-by-step Solution:

1. **Determine the Formula for Speed \( v \)**:
The speed of the electron in the nth orbit of a hydrogen atom is given by:
\[ v = \frac{e^2}{2 \epsilon_0 h n} \]
From this formula, we can see that:
\[ v \propto \frac{1}{n} \]
This means that speed \( v \) is inversely proportional to the principal quantum number \( n \).

**Hint**: Remember that the speed decreases as the quantum number increases.

2. **Determine the Formula for Radius \( r \)**:
The radius of the nth orbit is given by:
\[ r = \frac{h^2 \epsilon_0}{\pi m e^2} n^2 \]
From this formula, we can see that:
\[ r \propto n^2 \]
This indicates that the radius \( r \) is directly proportional to the square of the principal quantum number \( n \).

**Hint**: The radius increases with the square of the quantum number.

3. **Determine the Formula for Total Energy \( E \)**:
The total energy of the electron in the nth orbit is given by:
\[ E = -\frac{m e^4}{8 \epsilon_0^2 h^2} \frac{1}{n^2} \]
From this, we can see that:
\[ E \propto \frac{1}{n^2} \]
This means that the total energy \( E \) is inversely proportional to the square of the principal quantum number \( n \).

**Hint**: The energy becomes less negative (increases) as \( n \) increases.

4. **Analyze the Given Quantities**:
Now we need to analyze the quantities \( vr \), \( re \), \( \frac{v}{E} \), and \( \frac{r}{E} \) to see which are directly proportional to \( n \).

- **For \( vr \)**:
\[ vr = v \cdot r = \left(\frac{1}{n}\right) \cdot (n^2) = n \]
Thus, \( vr \) is directly proportional to \( n \).

- **For \( re \)**:
\[ re = r \cdot E = (n^2) \cdot \left(-\frac{1}{n^2}\right) = -1 \]
Thus, \( re \) is a constant and not proportional to \( n \).

- **For \( \frac{v}{E} \)**:
\[ \frac{v}{E} = \frac{\left(\frac{1}{n}\right)}{\left(-\frac{1}{n^2}\right)} = -n \]
Thus, \( \frac{v}{E} \) is directly proportional to \( n \).

- **For \( \frac{r}{E} \)**:
\[ \frac{r}{E} = \frac{(n^2)}{\left(-\frac{1}{n^2}\right)} = -n^4 \]
Thus, \( \frac{r}{E} \) is directly proportional to \( n^4 \), but not directly proportional to \( n \).

### Conclusion:
The quantities that are directly proportional to the principal quantum number \( n \) are:
- \( vr \)
- \( \frac{v}{E} \)

### Final Answer:
The correct options are \( vr \) and \( \frac{v}{E} \).