What is the ratio of time periods `(T_(1)//T_(2))` in second orbit of hydrogen atom to third orbit of `He^(+)` ion?

Options

A. `8//27`
B. `32//27`
C. `27//32`
D. None of these

Answer

To find the ratio of time periods \( \left( \frac{T_1}{T_2} \right) \) in the second orbit of the hydrogen atom to the third orbit of the \( He^+ \) ion, we can use Bohr's theory. According to this theory, the time period of revolution of an electron in a circular orbit is proportional to \( \frac{n^3}{Z^2} \), where \( n \) is the principal quantum number (the orbit number) and \( Z \) is the atomic number.

### Step-by-Step Solution:

1. **Identify the parameters for the hydrogen atom:**
- For the hydrogen atom (\( H \)), \( Z = 1 \) (since hydrogen has one proton).
- For the second orbit, \( n = 2 \).

Using the formula:
\[ T_1 \propto \frac{n^3}{Z^2} = \frac{2^3}{1^2} = \frac{8}{1} = 8 \]

2. **Identify the parameters for the \( He^+ \) ion:**
- For the helium ion (\( He^+ \)), \( Z = 2 \) (since helium has two protons).
- For the third orbit, \( n = 3 \).

Using the formula:
\[ T_2 \propto \frac{n^3}{Z^2} = \frac{3^3}{2^2} = \frac{27}{4} \]

3. **Calculate the ratio of time periods:**
\[ \frac{T_1}{T_2} = \frac{8}{\frac{27}{4}} = 8 \times \frac{4}{27} = \frac{32}{27} \]

### Final Answer:
The ratio of time periods \( \left( \frac{T_1}{T_2} \right) \) is \( \frac{32}{27} \).