The frequency of emission line for eny transition in positronium atom (consisting of a positron and an electron) is `x` times the frequency of the corresponding line in the case of `H` atom , where `x` is

Options

A. `sqrt2`
B. `1//2 sqrt2`
C. `1//2 sqrt2`
D. `1//2`

Explanation

To solve the problem of finding the ratio of the frequency of emission lines for transitions in a positronium atom compared to a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Levels**:
- The energy levels for positronium (E_n) can be expressed as:
\[ E_n = -\frac{6.8}{n^2} \text{ eV} \]
- The energy levels for hydrogen (E_n) can be expressed as:
\[ E_n = -\frac{13.6}{n^2} \text{ eV} \]

2. **Calculate the Energy Difference**:
- For positronium, the energy difference between the second and first energy levels (E_2 - E_1) is:
\[ E_{2p} - E_{1p} = E_2 - E_1 = -\frac{6.8}{2^2} - \left(-\frac{6.8}{1^2}\right) = -\frac{6.8}{4} + 6.8 = 6.8 \left(1 - \frac{1}{4}\right) = 6.8 \cdot \frac{3}{4} \]
- For hydrogen, the energy difference between the second and first energy levels (E_2 - E_1) is:
\[ E_{2h} - E_{1h} = E_2 - E_1 = -\frac{13.6}{2^2} - \left(-\frac{13.6}{1^2}\right) = -\frac{13.6}{4} + 13.6 = 13.6 \left(1 - \frac{1}{4}\right) = 13.6 \cdot \frac{3}{4} \]

3. **Relate Energy Difference to Frequency**:
- The frequency of the emitted photon (ν) is related to the energy difference by:
\[ E = h \nu \]
- Thus, for positronium:
\[ h \nu_p = 6.8 \cdot \frac{3}{4} \]
- And for hydrogen:
\[ h \nu_h = 13.6 \cdot \frac{3}{4} \]

4. **Find the Ratio of Frequencies**:
- Now, we can find the ratio of the frequencies:
\[ \frac{\nu_p}{\nu_h} = \frac{6.8 \cdot \frac{3}{4}}{13.6 \cdot \frac{3}{4}} = \frac{6.8}{13.6} \]
- Simplifying this gives:
\[ \frac{\nu_p}{\nu_h} = \frac{6.8}{13.6} = \frac{1}{2} \]

5. **Conclusion**:
- Therefore, the value of \( x \) is:
\[ x = \frac{1}{2} \]

### Final Answer:
The frequency of the emission line for any transition in the positronium atom is \( \frac{1}{2} \) times the frequency of the corresponding line in the case of the hydrogen atom.

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