In a mixture of `H-He^(+)` gas (`He^(+)` is singly ionized `He` atom), `H` atoms and `He^(+)` ions are excited to their respective first excited states. Subsequently, `H` atoms transfer their total excitation energy to `He^(+)` ions (by collisions). Assume that the Bohr model of atom is exctly valid.
The ratio of the kinetic energy of the `n=2` electron for the `H` atom to that of `He^(+)` ion is:

Options

A. `1//4`
B. `1//2`
C. `1`
D. `2`

Explanation

To find the ratio of the kinetic energy of the \( n=2 \) electron for the hydrogen atom (\( H \)) to that of the singly ionized helium ion (\( He^+ \)), we can use the Bohr model of the atom.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy in Bohr Model**:
The kinetic energy (\( KE \)) of an electron in a hydrogen-like atom can be expressed as:
\[ KE = \frac{Z^2 e^4 m}{2 \hbar^2 n^2} \]
where:
- \( Z \) is the atomic number,
- \( e \) is the charge of the electron,
- \( m \) is the mass of the electron,
- \( \hbar \) is the reduced Planck's constant,
- \( n \) is the principal quantum number.

2. **Kinetic Energy for Hydrogen Atom**:
For hydrogen (\( H \)), the atomic number \( Z = 1 \) and for the first excited state \( n = 2 \):
\[ KE_H = \frac{1^2 e^4 m}{2 \hbar^2 (2^2)} = \frac{e^4 m}{8 \hbar^2} \]

3. **Kinetic Energy for Helium Ion**:
For the singly ionized helium ion (\( He^+ \)), the atomic number \( Z = 2 \) and for the first excited state \( n = 2 \):
\[ KE_{He^+} = \frac{2^2 e^4 m}{2 \hbar^2 (2^2)} = \frac{4 e^4 m}{8 \hbar^2} = \frac{e^4 m}{2 \hbar^2} \]

4. **Finding the Ratio of Kinetic Energies**:
Now, we can find the ratio of the kinetic energy of the hydrogen atom to that of the helium ion:
\[ \text{Ratio} = \frac{KE_H}{KE_{He^+}} = \frac{\frac{e^4 m}{8 \hbar^2}}{\frac{e^4 m}{2 \hbar^2}} = \frac{1/8}{1/2} = \frac{1}{8} \times \frac{2}{1} = \frac{1}{4} \]

5. **Conclusion**:
Thus, the ratio of the kinetic energy of the \( n=2 \) electron for the hydrogen atom to that of the helium ion is:
\[ \frac{KE_H}{KE_{He^+}} = \frac{1}{4} \]

### Final Answer:
The ratio of the kinetic energy of the \( n=2 \) electron for the hydrogen atom to that of the \( He^+ \) ion is \( \frac{1}{4} \).

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