The time period of revolution of an electron in its ground state orbit in a hydrogen atom is `1.6 xx 10^(-16)` s. The frequency of the revoltuion in ( ` s^(-1)`). of the electron in its second exited state is

Options

A. `5.6 xx 10^(12)`
B. `7.8 xx 10^(16)`
C. `1.6 xx 10^(14)`
D. `2.3 xx 10^(14)`

Explanation

To find the frequency of revolution of an electron in its second excited state in a hydrogen atom, we can follow these steps:

### Step 1: Understand the relationship between time period and frequency
The frequency \( f \) is the reciprocal of the time period \( T \):
\[ f = \frac{1}{T} \]

### Step 2: Identify the time period in the ground state
The time period of revolution of an electron in its ground state (n=1) is given as:
\[ T_1 = 1.6 \times 10^{-16} \text{ s} \]

### Step 3: Determine the time period in the second excited state
The second excited state corresponds to \( n = 3 \). The time period \( T_n \) for an electron in the nth orbit is proportional to \( n^3 \):
\[ T_n \propto n^3 \]
Thus, we can write:
\[ \frac{T_1}{T_3} = \left(\frac{n_1}{n_3}\right)^3 \]
Where \( n_1 = 1 \) (ground state) and \( n_3 = 3 \) (second excited state).

### Step 4: Substitute the values into the equation
Substituting the values we have:
\[ \frac{T_1}{T_3} = \left(\frac{1}{3}\right)^3 = \frac{1}{27} \]
This implies:
\[ T_3 = 27 \times T_1 = 27 \times (1.6 \times 10^{-16}) \text{ s} \]

### Step 5: Calculate \( T_3 \)
Calculating \( T_3 \):
\[ T_3 = 27 \times 1.6 \times 10^{-16} = 43.2 \times 10^{-16} \text{ s} = 4.32 \times 10^{-15} \text{ s} \]

### Step 6: Calculate the frequency in the second excited state
Now, we can find the frequency \( f_3 \):
\[ f_3 = \frac{1}{T_3} = \frac{1}{4.32 \times 10^{-15}} \text{ s}^{-1} \]

### Step 7: Perform the calculation
Calculating \( f_3 \):
\[ f_3 = \frac{1}{4.32 \times 10^{-15}} \approx 2.31 \times 10^{14} \text{ s}^{-1} \]

### Final Answer
The frequency of the revolution of the electron in its second excited state is approximately:
\[ f_3 \approx 2.31 \times 10^{14} \text{ s}^{-1} \]