In a hydrogen atom, the radius of `n^(th)` bohr orbit is `r_n`. The graph between log `(r_n//r_1)` and logn will be

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Explanation

To find the graph between log \((r_n/r_1)\) and log \(n\) for a hydrogen atom, we can follow these steps:

### Step 1: Understand the relationship between the radius of the nth orbit and the first orbit
The radius of the nth orbit \(r_n\) in a hydrogen atom is given by the formula:
\[ r_n = n^2 r_1 \]
where \(r_1\) is the radius of the first orbit.

### Step 2: Express \(r_n/r_1\)
From the formula above, we can express the ratio of the radii:
\[ \frac{r_n}{r_1} = n^2 \]

### Step 3: Take the logarithm of both sides
Now, we take the logarithm of both sides of the equation:
\[ \log\left(\frac{r_n}{r_1}\right) = \log(n^2) \]

### Step 4: Apply the logarithmic identity
Using the logarithmic identity \(\log(a^b) = b \cdot \log(a)\), we can simplify the right side:
\[ \log\left(\frac{r_n}{r_1}\right) = 2 \cdot \log(n) \]

### Step 5: Rearranging the equation
This can be rearranged to show the relationship between \(\log\left(\frac{r_n}{r_1}\right)\) and \(\log(n)\):
\[ \log\left(\frac{r_n}{r_1}\right) = 2 \log(n) \]

### Step 6: Identify the graph characteristics
This equation is in the form of \(y = mx\), where \(y = \log\left(\frac{r_n}{r_1}\right)\), \(x = \log(n)\), and \(m = 2\). This indicates that the graph will be a straight line passing through the origin with a slope of 2.

### Conclusion
Thus, the graph between \(\log\left(\frac{r_n}{r_1}\right)\) and \(\log(n)\) is a straight line passing through the origin with a slope of 2.

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