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`.
If the Rydberg constant for hydrogen atom is `R` , then the Rydberg constant for positronium atom is

Options

A. `2 R`
B. `R`
C. ` R//2`
D. `4 R`

Explanation

To solve the problem, we need to determine the Rydberg constant for the positronium atom based on the information given about the hydrogen atom and the properties of positronium.

### Step-by-Step Solution:

1. **Understanding the Rydberg Constant for Hydrogen**:
The Rydberg constant \( R \) for hydrogen is given by the formula:
\[ R = \frac{2 \pi^2 m_e k^2 e^4}{h^3 c} \]
where \( m_e \) is the mass of the electron, \( k \) is Coulomb's constant, \( e \) is the charge of the electron, \( h \) is Planck's constant, and \( c \) is the speed of light.

2. **Identifying the Positronium Atom**:
In the positronium atom, we have an electron and a positron (which has the same mass as the electron). The reduced mass \( \mu \) of the system (electron and positron) is given by:
\[ \mu = \frac{m_e m_e}{m_e + m_e} = \frac{m_e^2}{2m_e} = \frac{m_e}{2} \]

3. **Rydberg Constant for Positronium**:
The formula for the Rydberg constant for positronium \( R' \) can be expressed similarly to that of hydrogen, but we replace \( m_e \) with the reduced mass \( \mu \):
\[ R' = \frac{2 \pi^2 \mu k^2 e^4}{h^3 c} \]
Substituting \( \mu = \frac{m_e}{2} \) into the equation:
\[ R' = \frac{2 \pi^2 \left(\frac{m_e}{2}\right) k^2 e^4}{h^3 c} \]

4. **Simplifying the Expression**:
This can be simplified to:
\[ R' = \frac{1}{2} \cdot \frac{2 \pi^2 m_e k^2 e^4}{h^3 c} = \frac{1}{2} R \]

5. **Conclusion**:
Therefore, the Rydberg constant for the positronium atom is:
\[ R' = \frac{R}{2} \]

### Final Answer:
The Rydberg constant for the positronium atom is \( \frac{R}{2} \).

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