The time taken by the electron in one complete revolution in the `n^(th)` Bohr's orbit of the hydrogen atom is :

Options

A. Inversely proportional to ` n^2`
B. Directly proportional to ` n^3`
C. Directly proportional to ` (h)/(2 pi)`
D. Inversely proportional to `(n )/(h )`

Answer

To find the time taken by the electron in one complete revolution in the \( n^{th} \) Bohr's orbit of the hydrogen atom, we can follow these steps:

### Step 1: Understand the relationship between speed, distance, and time
The basic formula we will use is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
From this, we can rearrange it to find time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

### Step 2: Determine the distance traveled by the electron
The electron moves in a circular orbit, so the distance it travels in one complete revolution is the circumference of the circle. The formula for the circumference \( C \) is:
\[ C = 2\pi r \]
where \( r \) is the radius of the orbit.

### Step 3: Find the radius of the \( n^{th} \) Bohr orbit
According to Bohr's model, the radius \( r_n \) of the \( n^{th} \) orbit is given by:
\[ r_n = \frac{n^2 h^2}{4 \pi^2 k e^2 m} \]
where \( h \) is Planck's constant, \( k \) is Coulomb's constant, \( e \) is the charge of the electron, and \( m \) is the mass of the electron.

### Step 4: Determine the speed of the electron in the \( n^{th} \) orbit
The speed \( v \) of the electron in the \( n^{th} \) orbit can be expressed as:
\[ v = \frac{e^2 k Z}{n} \]
where \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

### Step 5: Substitute the values into the time formula
Now we can substitute the expressions for distance and speed into the time formula:
\[ \text{Time} = \frac{2\pi r_n}{v} \]
Substituting \( r_n \) and \( v \):
\[ \text{Time} = \frac{2\pi \left(\frac{n^2 h^2}{4 \pi^2 k e^2 m}\right)}{\frac{e^2 k Z}{n}} \]

### Step 6: Simplify the expression
After substituting and simplifying, we get:
\[ \text{Time} = \frac{n^3 h^2}{2 \pi k e^2 m Z} \]
For hydrogen (\( Z = 1 \)), this simplifies to:
\[ \text{Time} = \frac{n^3 h^2}{2 \pi k e^2 m} \]

### Step 7: Final result
The final expression for the time taken by the electron in one complete revolution in the \( n^{th} \) Bohr orbit of the hydrogen atom is:
\[ \text{Time} = 1.52 \times 10^{-16} n^3 \text{ seconds} \]