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JEE Advanced 2016
Paper-1 2016
Multiple correct answers - Select all that apply
Question
Highly excited states for hydrogen-like atoms (also called Rydberg states) with nucles charge Ze are defined by their principal quantum number n, where n>>1. Which of the following statement(s) is(are) true?
Select ALL correct options:
A
Relative change in the radii of two consecutive orbitals does not depend on Z.
B
Relative change in the radii of two consecutive orbitals varies as 1/n.
C
Relative change in the energy of two consecutive orbitals varies as $1 / n^3$.
D
Relative change in the angular momenta of two consecutive orbitals varies as 1/n.
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
⚠ Partially correct. Some answers are missing.
Solution
orbital :
$ \begin{aligned} & \& \quad \frac{\Delta r}{r}=\text { relative change } \\ & \begin{aligned} r_n= & 0.529 \frac{n^2}{Z} \\ = & \frac{(n+1)^2-n^2}{n^2} \\ = & \frac{n^2+1+2 n-n^2}{n^2} \\ = & \frac{1}{\infty}+\frac{2 n}{n^2} \\ = & \frac{1}{n} \quad \therefore \text { dependent on } n . \\ E_n=-13.6 \frac{z^2}{n^2} \quad & \text { Relative change }=\frac{\Delta E}{E} \end{aligned} \end{aligned} $
$\begin{aligned} & =\frac{\frac{1}{(n+1)^2}-\frac{1}{n^2}}{\frac{1}{n^2}} \\ & =\frac{n^2-n^2-1-2 n}{n^2(n+1)^2} \times \frac{n^2}{1} \\ \Rightarrow \quad & \frac{-(1+2 n)}{(n+1)^2}\end{aligned}$
$\begin{aligned} & \Rightarrow \quad \frac{1+2 n}{n^2+1+2 n} \\ & =\frac{1}{n^2+1+2 n}+\frac{2 n}{n^2+1+2 n} \\ & \text { Doesn't match with 'C' } \\ & L_n=\frac{n h}{2 \pi} \\ & \text { Relative change }=\frac{\Delta L}{L}=\frac{n-(n-1)}{n}=\frac{1}{n}\end{aligned}$
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