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 based on the relationship between the time periods of the electron in the ground state of hydrogen and the first excited state of the hydrogen-like atom.

### Step-by-Step Solution:

1. **Understand the Time Period Formula**:
The time period \( T \) of an electron in the nth orbit of a hydrogen atom is given by:
\[ T_n = 2 \pi n^3 \]
For a hydrogen-like atom with atomic number \( Z \), the time period is modified to:
\[ T_n = \frac{2 \pi n^3}{Z^2} \]

2. **Calculate the Time Period for Hydrogen (Ground State)**:
For the ground state of hydrogen (where \( n = 1 \)):
\[ T_H = 2 \pi (1)^3 = 2 \pi \]

3. **Calculate the Time Period for Hydrogen-like Atom (First Excited State)**:
For the first excited state of the hydrogen-like atom (where \( n = 2 \)):
\[ T_X = \frac{2 \pi (2)^3}{Z^2} = \frac{2 \pi \cdot 8}{Z^2} = \frac{16 \pi}{Z^2} \]

4. **Set Up the Relationship Between Time Periods**:
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:
\[ T_H = 2 T_X \]
Substituting the values we found:
\[ 2 \pi = 2 \left(\frac{16 \pi}{Z^2}\right) \]

5. **Simplify the Equation**:
Divide both sides by \( 2 \):
\[ \pi = \frac{16 \pi}{Z^2} \]
Now, divide both sides by \( \pi \):
\[ 1 = \frac{16}{Z^2} \]

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

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

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