Discussion Forum



1)

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principle quantum number n. where n >>1. Which of the following statement(s) is (are) true?


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

Answer:

Option A,B,D

Explanation:

As radius,

$r\propto \frac{n}{z}$

$\Rightarrow \frac{\delta r}{r}=\frac{\left(\frac{n+1}{z}\right)^{2}-\left(\frac{n}{z}\right)^{2}}{\left(\frac{n}{z}\right)^{2}}$

$\frac{2n+1}{n^{2}\approx \frac{2}{n}\propto \frac{1}{n}}$

Energy $E\propto \frac{z^{2}}{n^{2}}$

$\Rightarrow \frac{\triangle E}{E}=\frac{\frac{z^{2}}{n^{2}}-\frac{z^{2}}{(n+1)^{2}}}{\frac{z^{2}}{(n+1)^{2}}}$

$=\frac{n+1^{2}-n^{2}}{n^{2}.(n+1)^{2}}.(n+1)^{2}$

$\Rightarrow \frac{\triangle E}{E}=\frac{2n+1}{n^{2}}\simeq \frac{2n}{n^{2}}\propto \frac{1}{n}$

Angular momentum $L=\frac{nh}{2\pi}$

$\Rightarrow \frac{\triangle L}{L}=\frac{\frac{(n+1)h}{2\pi}-\frac{nh}{2\pi}}{\frac{nh}{2\pi}}$

$=\frac{1}{n} \propto \frac{1}{n}$