If in hydrogen atom, radius of `n^(th)` Bohr orbit is `r_n`., frequency of revolution of electron in `n^(th)` orbit is `f_n` choose the correct option.

Options

A.
B.
C.
D. Both (a) and (b)

Answer

To solve the problem regarding the radius and frequency of the electron in the nth Bohr orbit of a hydrogen atom, we will follow these steps:

### Step 1: Understanding the Radius of the nth Bohr Orbit
The radius \( r_n \) of the nth Bohr orbit is given by the formula:
\[ r_n = a_0 \cdot n^2 \]
where \( a_0 \) is the Bohr radius (approximately \( 0.529 \) Å for hydrogen), and \( n \) is the principal quantum number.

### Step 2: Analyzing the Relationship
From the formula \( r_n = a_0 \cdot n^2 \), we can see that the radius \( r_n \) is directly proportional to the square of the quantum number \( n \). Therefore, if we plot \( r_n \) against \( n \), we will get a parabolic graph.

### Step 3: Frequency of Revolution of the Electron
The frequency of revolution \( f_n \) can be derived from the time period \( T \) of one complete revolution:
\[ f_n = \frac{1}{T} \]
The time period \( T \) can be calculated as the distance traveled in one revolution divided by the speed of the electron. The distance traveled in one revolution is the circumference of the orbit:
\[ T = \frac{2 \pi r_n}{v} \]
where \( v \) is the speed of the electron.

### Step 4: Expressing Speed in Terms of n
The speed \( v \) of the electron in the nth orbit can be related to \( n \) as:
\[ v \propto \frac{Z}{n} \quad \text{(for hydrogen, } Z = 1\text{)} \]
Thus, we can express \( v \) as:
\[ v = k \cdot \frac{1}{n} \]
where \( k \) is a constant.

### Step 5: Substituting into the Time Period Formula
Substituting \( r_n \) and \( v \) into the time period formula:
\[ T = \frac{2 \pi (a_0 n^2)}{k \cdot \frac{1}{n}} = \frac{2 \pi a_0 n^3}{k} \]
Thus, the frequency becomes:
\[ f_n = \frac{k}{2 \pi a_0 n^3} \]

### Step 6: Finding the Ratio of Frequencies
To find the ratio of frequencies \( \frac{f_n}{f_1} \):
\[ \frac{f_n}{f_1} = \frac{f_n}{k/(2 \pi a_0)} = \frac{1}{n^3} \]

### Step 7: Taking Logarithm
Taking the logarithm of the ratio:
\[ \log\left(\frac{f_n}{f_1}\right) = -3 \log(n) \]
This indicates a linear relationship with a negative slope when plotted.

### Conclusion
Based on the analysis:
- The graph of \( r_n \) vs. \( n \) is parabolic.
- The graph of \( \log\left(\frac{f_n}{f_1}\right) \) vs. \( \log(n) \) is a straight line with a negative slope.

### Final Answer
The correct options are:
- A: The graph of \( r_n \) vs. \( n \) is parabolic.
- B: The graph of \( \log\left(\frac{f_n}{f_1}\right) \) vs. \( \log(n) \) is a straight line with a negative slope.

Thus, the answer is **D: Both A and B are correct.**