If, in a hydrogen atom, radius of nth Bohr orbit is `r_(n)` frequency of revolution of electron in nth orbit is `f_(n)` and area enclosed by the nth orbit is `A_(n)` , then which of the pollowing graphs are correct?

Options

A.
B.
C.
D.

Explanation

To solve the problem regarding the graphs related to the radius, frequency, and area of the nth Bohr orbit in a hydrogen atom, we will analyze each aspect step by step.

### Step 1: Analyze the Radius of the nth Bohr Orbit
The radius of the nth Bohr orbit in a hydrogen atom is given by the formula:
\[ r_n \propto n^2 \]
For hydrogen (where \(Z = 1\)), this can be expressed as:
\[ r_n = k \cdot n^2 \]
where \(k\) is a proportionality constant.

**Graph Analysis:**
- If we plot \(r_n\) (y-axis) against \(n\) (x-axis), the relationship is quadratic. Therefore, the graph will be a parabola opening upwards.

**Conclusion for Step 1:**
The graph between \(r_n\) and \(n\) is correct and should be a parabolic curve.

### Step 2: Analyze the Logarithmic Relationship of Radius
Next, we consider the logarithmic relationship:
\[ \frac{r_n}{r_1} \propto n^2 \]
Taking logarithms:
\[ \log\left(\frac{r_n}{r_1}\right) = \log(n^2) = 2 \log(n) \]
This indicates a linear relationship where the slope \(m = 2\).

**Graph Analysis:**
- If we plot \(\log\left(\frac{r_n}{r_1}\right)\) (y-axis) against \(\log(n)\) (x-axis), we will get a straight line with a positive slope of 2.

**Conclusion for Step 2:**
The graph between \(\log\left(\frac{r_n}{r_1}\right)\) and \(\log(n)\) is correct and is a straight line.

### Step 3: Analyze the Area of the nth Bohr Orbit
The area \(A_n\) enclosed by the nth orbit is given by:
\[ A_n \propto r_n^2 \propto (n^2)^2 = n^4 \]
Taking the ratio:
\[ \frac{A_n}{A_1} \propto n^4 \]
Taking logarithms:
\[ \log\left(\frac{A_n}{A_1}\right) = 4 \log(n) \]
This indicates a linear relationship with slope \(m = 4\).

**Graph Analysis:**
- If we plot \(\log\left(\frac{A_n}{A_1}\right)\) (y-axis) against \(n\) (x-axis), we will get a straight line with a positive slope of 4.

**Conclusion for Step 3:**
The graph between \(\log\left(\frac{A_n}{A_1}\right)\) and \(n\) is correct and is a straight line.

### Step 4: Analyze the Frequency of Revolution
The frequency \(f_n\) of the electron in the nth orbit is inversely proportional to \(n^3\):
\[ f_n \propto \frac{1}{n^3} \]
Thus, taking the ratio:
\[ \frac{f_n}{f_1} \propto \frac{1}{n^3} \]
Taking logarithms:
\[ \log\left(\frac{f_n}{f_1}\right) = -3 \log(n) \]
This indicates a linear relationship with a negative slope \(m = -3\).

**Graph Analysis:**
- If we plot \(\log\left(\frac{f_n}{f_1}\right)\) (y-axis) against \(n\) (x-axis), we will get a straight line with a negative slope.

**Conclusion for Step 4:**
The graph between \(\log\left(\frac{f_n}{f_1}\right)\) and \(n\) is incorrect if it is shown as a positive slope.

### Final Summary of Graphs:
1. **Graph of \(r_n\) vs. \(n\)**: Correct (Parabolic)
2. **Graph of \(\log\left(\frac{r_n}{r_1}\right)\) vs. \(\log(n)\)**: Correct (Straight Line, slope = 2)
3. **Graph of \(\log\left(\frac{A_n}{A_1}\right)\) vs. \(n\)**: Correct (Straight Line, slope = 4)
4. **Graph of \(\log\left(\frac{f_n}{f_1}\right)\) vs. \(n\)**: Incorrect (Should have negative slope)