The product of linear momentum and angular momentum of an electron of the hydrogen atom is proportional to `n^(x)`, where x is

Options

A. 0
B. 1
C. `-2`
D. 2

Answer

To solve the problem, we need to determine the relationship between the product of linear momentum (P) and angular momentum (L) of an electron in a hydrogen atom, and how it relates to the principal quantum number \( n \).

### Step-by-Step Solution:

1. **Define Linear Momentum (P)**:
The linear momentum of the electron is given by:
\[ P = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

2. **Define Angular Momentum (L)**:
The angular momentum of the electron in a circular orbit is given by:
\[ L = mvr \]
where \( r \) is the radius of the orbit.

3. **Product of Linear and Angular Momentum**:
We need to find the product \( P \cdot L \):
\[ P \cdot L = (mv) \cdot (mvr) = m^2 v^2 r \]

4. **Relate Angular Momentum to Quantum Number (n)**:
According to Bohr's model, the angular momentum is quantized and can be expressed as:
\[ L = n \frac{h}{2\pi} \]
where \( h \) is Planck's constant and \( n \) is the principal quantum number.

5. **Substituting for L**:
Substitute \( L \) into the product:
\[ P \cdot L = P \cdot \left(n \frac{h}{2\pi}\right) \]

6. **Expressing r in terms of n**:
The radius \( r \) of the electron's orbit in a hydrogen atom is given by:
\[ r = 0.53 \frac{Z^2}{n} \text{ angstroms} \]
For hydrogen, \( Z = 1 \), thus:
\[ r \propto \frac{1}{n^2} \]

7. **Substituting r into the Product**:
Now, substituting \( r \) into the expression for \( P \cdot L \):
\[ P \cdot L = m^2 v^2 \cdot \left(0.53 \frac{1}{n^2}\right) \]

8. **Finding the Proportionality**:
Since \( v \) is proportional to \( \frac{1}{n} \) (from the energy levels), we can express \( v^2 \) as:
\[ v^2 \propto \frac{1}{n^2} \]
Therefore:
\[ P \cdot L \propto m^2 \cdot \frac{1}{n^2} \cdot \frac{1}{n^2} = \frac{m^2}{n^4} \]

9. **Final Expression**:
Thus, the product \( P \cdot L \) is proportional to:
\[ P \cdot L \propto \frac{1}{n^4} \]
This means that the product of linear momentum and angular momentum is proportional to \( n^{-4} \).

### Conclusion:
The exponent \( x \) in the expression \( n^x \) is \( -4 \).