Let A_n be the area enclosed by the n^ th orbit in a hydrogen ato… - Vidyadip

MCQ

Let $A_n$ be the area enclosed by the $n^{th}$ orbit in a hydrogen atom. The graph of ln $\Big(\frac{\text{A}_\text{n}}{\text{A}_1}\Big)$ against in $(n):$

A
Will pass through the origin.
B
Will be a straight line with slope $4$.
✓
$A$ and $B$
D
Will be a circle.

✓

Answer

Correct option: C.

$A$ and $B$

$\text{r}_\text{n}=\text{n}^2\text{a}_0$
Area of the nth orbit is given by,
$\text{A}_\text{n}=\pi\text{r}^2_\text{n}=\pi\text{n}^4\text{a}^2_0$
$\text{A}_1=\pi\text{a}^{2}_0$
$\text{ln}\Big(\frac{\text{A}_\text{n}}{\text{A}_1}\Big)=\text{ln}\Big(\frac{\pi\text{n}^4\text{a}_0^2}{\pi\text{a}^2_0}\Big)$
$\text{ln}\Big(\frac{\text{A}_\text{n}}{\text{A}_1}\Big)=4\text{ln n}\ ...(\text{i})$
From the above expression, the graph of in $\Big(\frac{\text{A}_\text{n}}{\text{A}_1}\Big)$ against ln $(n)$ will be a straight line passing through the origin and having slope $4$.

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