In an ordianary atom, as a first approximation, the motion of the nucleus can be ignored. In a positronium atom a positronreplaces the proton of hydrogen atom. The electron and positron masses are equal and , therefore , the motion of the positron cannot be ignored. One must consider the motion of both electron and positron about their center of mass. A detailed analyasis shows that formulae of Bohr's model apply to positronium atom provided that we replace `m_(e)` by what is known reduced mass is `m_(e)//2`.
The orbital radius of the first excited level of postronium atom is

Options

A. `4 a_(0)`
B. `a_(0)//2`
C. `8 a_(0)`
D. `2 a_(0)`

Explanation

To find the orbital radius of the first excited level of the positronium atom, we can follow these steps:

### Step 1: Understand the Concept of Reduced Mass
In a positronium atom, both the electron and the positron have the same mass, denoted as \( m_e \). The reduced mass \( \mu \) of the system can be calculated using the formula:
\[ \mu = \frac{m_1 m_2}{m_1 + m_2} \]
where \( m_1 \) and \( m_2 \) are the masses of the electron and positron, respectively. Since both masses are equal, we have:
\[ \mu = \frac{m_e \cdot m_e}{m_e + m_e} = \frac{m_e^2}{2m_e} = \frac{m_e}{2} \]

### Step 2: Recall the Bohr Model Formula for Radius
According to the Bohr model, the radius of the nth orbit in a hydrogen-like atom is given by:
\[ r_n = \frac{n^2 h^2 \epsilon_0}{\pi \mu e^2} \]
where:
- \( n \) is the principal quantum number,
- \( h \) is Planck's constant,
- \( \epsilon_0 \) is the permittivity of free space,
- \( e \) is the charge of the electron.

### Step 3: Substitute the Reduced Mass into the Formula
For the positronium atom, we replace \( \mu \) with \( \frac{m_e}{2} \):
\[ r_n = \frac{n^2 h^2 \epsilon_0}{\pi \left(\frac{m_e}{2}\right) e^2} \]
This simplifies to:
\[ r_n = \frac{2n^2 h^2 \epsilon_0}{\pi m_e e^2} \]

### Step 4: Calculate the Radius for the First Excited State
The first excited state corresponds to \( n = 2 \):
\[ r_2 = \frac{2(2^2) h^2 \epsilon_0}{\pi m_e e^2} = \frac{8 h^2 \epsilon_0}{\pi m_e e^2} \]

### Step 5: Relate it to the Bohr Radius
The Bohr radius \( a_0 \) is defined as:
\[ a_0 = \frac{h^2 \epsilon_0}{\pi m_e e^2} \]
Thus, we can express \( r_2 \) in terms of \( a_0 \):
\[ r_2 = 8 a_0 \]

### Final Answer
The orbital radius of the first excited level of the positronium atom is:
\[ \boxed{8 a_0} \]

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