Which of the following statements is true of hydrogen atom ?

Options

A. Angular momentum `prop (1)/(n)`
B. Linear momentum `prop (1)/(n)`
C. Radius `prop (1)/(n)`
D. Energy `prop (1)/(n)`

Explanation

To determine which statement is true regarding the hydrogen atom, we will analyze the properties of the hydrogen atom, particularly focusing on angular momentum, linear momentum, radius, and energy.

### Step-by-Step Solution:

1. **Understanding Angular Momentum (L)**:
- The angular momentum of an electron in a hydrogen atom is given by the formula:
\[ L = n \frac{h}{2\pi} \]
where \( n \) is the principal quantum number (1, 2, 3,...), and \( h \) is Planck's constant.
- This indicates that angular momentum is quantized and depends on the quantum number \( n \).

2. **Understanding Linear Momentum (P)**:
- The linear momentum \( P \) of an electron can be expressed as:
\[ P = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity.
- From the relationship of linear momentum and radius, we can derive:
\[ P = \frac{nh}{2\pi R} \]
where \( R \) is the radius of the electron's orbit.

3. **Understanding the Radius (R)**:
- The radius of the electron's orbit in a hydrogen atom is given by:
\[ R = \frac{n^2 h^2 \epsilon_0}{\pi m Z e^2} \]
where \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).
- This shows that the radius is proportional to \( n^2 \) and inversely proportional to \( Z \).

4. **Understanding Energy (E)**:
- The total energy of the electron in a hydrogen atom is given by:
\[ E = -\frac{Z e^2}{8\pi \epsilon_0 R} \]
- Substituting the expression for \( R \), we find that energy is proportional to:
\[ E \propto -\frac{Z^2}{n^2} \]
- This indicates that energy is quantized and inversely proportional to the square of the principal quantum number \( n \).

5. **Evaluating the Statements**:
- **Statement A (Angular Momentum)**: This is true as angular momentum is quantized and depends on \( n \).
- **Statement B (Linear Momentum)**: This is also true, as linear momentum is dependent on \( n \) and \( R \).
- **Statement C (Radius)**: This is true as radius is proportional to \( n^2 \) and inversely proportional to \( Z \).
- **Statement D (Energy)**: This is true as energy is inversely proportional to \( n^2 \).

After analyzing all the statements, we conclude that **Statement B** is the most directly true regarding the linear momentum of the hydrogen atom.