Which of the following products in a hydrogen atom are independent of the principal quantum number n? The symbols have theirusual meanings

Options

A. un
B. Er
C. E pi
D. ur

Answer

To determine which products in a hydrogen atom are independent of the principal quantum number \( n \), we will analyze the relationships of velocity \( u \), energy \( E \), and radius \( r \) with respect to \( n \).

### Step-by-Step Solution:

1. **Understanding the Relationships**:
- The velocity \( u \) of an electron in a hydrogen atom is inversely proportional to the principal quantum number \( n \):
\[ u \propto \frac{1}{n} \]
- The energy \( E \) of the electron is given by:
\[ E \propto -\frac{1}{n^2} \]
- The radius \( r \) of the electron's orbit is directly proportional to \( n^2 \):
\[ r \propto n^2 \]

2. **Analyzing Each Product**:
- **Product \( U_n \)**:
\[ U_n = u \cdot n \]
Since \( u \propto \frac{1}{n} \):
\[ U_n \propto \frac{1}{n} \cdot n = \text{constant} \]
Thus, \( U_n \) is independent of \( n \).

- **Product \( E_r \)**:
\[ E_r = E \cdot r \]
Since \( E \propto -\frac{1}{n^2} \) and \( r \propto n^2 \):
\[ E_r \propto \left(-\frac{1}{n^2}\right) \cdot n^2 = \text{constant} \]
Thus, \( E_r \) is also independent of \( n \).

- **Product \( E_n \)**:
\[ E_n = E \cdot n \]
Since \( E \propto -\frac{1}{n^2} \):
\[ E_n \propto -\frac{1}{n^2} \cdot n = -\frac{1}{n} \]
Thus, \( E_n \) is dependent on \( n \).

- **Product \( U_r \)**:
\[ U_r = u \cdot r \]
Since \( u \propto \frac{1}{n} \) and \( r \propto n^2 \):
\[ U_r \propto \frac{1}{n} \cdot n^2 = n \]
Thus, \( U_r \) is also dependent on \( n \).

3. **Conclusion**:
The products that are independent of the principal quantum number \( n \) are:
- \( U_n \)
- \( E_r \)

### Final Answer:
The products in a hydrogen atom that are independent of the principal quantum number \( n \) are \( U_n \) and \( E_r \).