The time period of the electron in the ground state of hydrogen atom is two times the times period of the electon in the first excited state of a certain hydrongen like atom (Atomic number Z). The value of Z is

Options

A. `2`
B. `3`
C. `4`
D. None of these

Explanation

To solve the problem, we need to find the atomic number \( Z \) of a hydrogen-like atom given that the time period of the electron in the ground state of hydrogen is two times the time period of the electron in the first excited state of this hydrogen-like atom.

### Step-by-step Solution:

1. **Understand the time period formula for hydrogen:**
The time period \( T_H \) for the electron in the ground state (n=1) of a hydrogen atom is given by:
\[ T_H = 2\pi n^3 \]
For \( n = 1 \):
\[ T_H = 2\pi (1^3) = 2\pi \]

2. **Understand the time period formula for a hydrogen-like atom:**
For a hydrogen-like atom with atomic number \( Z \), the time period \( T_A \) for the electron in the \( n^{th} \) orbit is given by:
\[ T_A = \frac{2\pi n^3}{Z^2} \]
For the first excited state (where \( n = 2 \)):
\[ T_A = \frac{2\pi (2^3)}{Z^2} = \frac{16\pi}{Z^2} \]

3. **Set up the equation based on the problem statement:**
According to the problem, the time period of the electron in the ground state of hydrogen is two times the time period of the electron in the first excited state of the hydrogen-like atom:
\[ T_H = 2 \times T_A \]
Substituting the expressions we found:
\[ 2\pi = 2 \times \frac{16\pi}{Z^2} \]

4. **Simplify the equation:**
Dividing both sides by \( 2\pi \):
\[ 1 = \frac{16}{Z^2} \]

5. **Solve for \( Z^2 \):**
Rearranging gives:
\[ Z^2 = 16 \]

6. **Find \( Z \):**
Taking the square root of both sides:
\[ Z = 4 \]

### Final Answer:
The value of \( Z \) is \( 4 \).