The time period of revolution of electron in its ground state orbit in a hydrogen atom is `1.60 xx 10^(-16)` second. The time period of revolution of the electron in its first excited state in a `Li^(++)` ion is:

Options

A. `1.42xx10^(-16)s`
B. `1.20xx10^(-16)s`
C. `2.32xx10^(-16)s`
D. `2.42xx10^(-16)s`

Explanation

To find the time period of revolution of the electron in the first excited state of a lithium ion (Li²⁺), we can use the relationship between the time period, principal quantum number, and atomic number.

### Step-by-Step Solution:

1. **Understanding the Time Period Formula**:
The time period \( T \) of an electron in an orbit is given by:
\[ T = \frac{2\pi r}{v} \]
where \( r \) is the radius of the orbit and \( v \) is the velocity of the electron.

2. **Relating Radius and Velocity**:
The radius \( r \) of the orbit is proportional to the principal quantum number \( n \) and inversely proportional to the atomic number \( Z \):
\[ r \propto \frac{n^2}{Z} \]
The velocity \( v \) is proportional to the atomic number \( Z \) and inversely proportional to the principal quantum number \( n \):
\[ v \propto \frac{Z}{n} \]

3. **Combining the Relationships**:
We can express the time period \( T \) in terms of \( n \) and \( Z \):
\[ T \propto \frac{r}{v} \propto \frac{n^2/Z}{Z/n} = \frac{n^3}{Z^2} \]
Thus, we have:
\[ T \propto \frac{n^3}{Z^2} \]

4. **Setting Up the Ratio**:
We know the time period of the electron in the ground state of hydrogen (where \( n = 1 \) and \( Z = 1 \)):
\[ T_H = 1.6 \times 10^{-16} \text{ seconds} \]
For the first excited state of lithium ion (where \( n' = 2 \) and \( Z' = 3 \)):
\[ T_{Li^{++}} = T_H \cdot \frac{(n')^3}{(Z')^2} = T_H \cdot \frac{(2)^3}{(3)^2} \]

5. **Calculating the Time Period**:
Substituting the values:
\[ T_{Li^{++}} = 1.6 \times 10^{-16} \cdot \frac{8}{9} \]
\[ T_{Li^{++}} = 1.6 \times 10^{-16} \cdot 0.8889 \approx 1.42 \times 10^{-16} \text{ seconds} \]

### Final Answer:
The time period of revolution of the electron in its first excited state in a lithium ion (Li²⁺) is approximately:
\[ T_{Li^{++}} \approx 1.42 \times 10^{-16} \text{ seconds} \]