Find the ratio of the time period of `2^(nd)` Bohr orbit of `He^(+)` and `4^(th)` Bohr orbit of `Li^(2+)`

Answer

To find the ratio of the time period of the 2nd Bohr orbit of \( \text{He}^+ \) and the 4th Bohr orbit of \( \text{Li}^{2+} \), we can follow these steps:

### Step 1: Understand the formula for the time period
The time period \( T \) of an electron in a Bohr orbit is given by the formula:
\[ T \propto \frac{n^3}{Z^2} \]
where \( n \) is the principal quantum number (the orbit number) and \( Z \) is the atomic number of the element.

### Step 2: Calculate the time period for \( \text{He}^+ \)
For \( \text{He}^+ \):
- The principal quantum number \( n = 2 \)
- The atomic number \( Z = 2 \)

Using the formula:
\[ T_1 \propto \frac{n^3}{Z^2} = \frac{2^3}{2^2} = \frac{8}{4} = 2 \]
So, we can denote:
\[ T_1 = k \cdot 2 \]
where \( k \) is a proportionality constant.

### Step 3: Calculate the time period for \( \text{Li}^{2+} \)
For \( \text{Li}^{2+} \):
- The principal quantum number \( n = 4 \)
- The atomic number \( Z = 3 \)

Using the formula:
\[ T_2 \propto \frac{n^3}{Z^2} = \frac{4^3}{3^2} = \frac{64}{9} \]
So, we can denote:
\[ T_2 = k \cdot \frac{64}{9} \]

### Step 4: Find the ratio of \( T_1 \) to \( T_2 \)
Now, we need to find the ratio \( \frac{T_1}{T_2} \):
\[ \frac{T_1}{T_2} = \frac{k \cdot 2}{k \cdot \frac{64}{9}} = \frac{2}{\frac{64}{9}} = 2 \cdot \frac{9}{64} = \frac{18}{64} = \frac{9}{32} \]

### Final Answer
The ratio of the time period of the 2nd Bohr orbit of \( \text{He}^+ \) to the 4th Bohr orbit of \( \text{Li}^{2+} \) is:
\[ \frac{T_1}{T_2} = \frac{9}{32} \]