Highly excited states for hydrogen-like atoms | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) For a hydrogen-like atom,the radius of the $n$-th orbit is $r_n = a_0 \frac{n^2}{Z}$.The relative change in radius for two consecutive orbitals is

Vedclass · Accepted Answer

$A, B, D$

Vedclass · Answer

(A) For a hydrogen-like atom,the radius of the $n$-th orbit is $r_n = a_0 \frac{n^2}{Z}$.The relative change in radius for two consecutive orbitals is $\frac{\Delta r}{r} \approx \frac{dr}{r} = \frac{2n \, dn}{n^2} = \frac{2 \Delta n}{n}$. Since $\Delta n = 1$,$\frac{\Delta r}{r} \propto \frac{1}{n}$. This is independent of $Z$. Thus,$(A)$ and $(B)$ are correct.The energy of the $n$-th orbit is $E_n = -E_0 \frac{Z^2}{n^2}$.The relative change in energy is $\frac{\Delta E}{E} \approx \frac{dE}{E} = \frac{2n^{-3} \, dn}{n^{-2}} = \frac{2 \Delta n}{n}$. Since $\Delta n = 1$,$\frac{\Delta E}{E} \propto \frac{1}{n}$. Thus,$(C)$ is incorrect.The angular momentum is $L = \frac{nh}{2\pi}$.The relative change is $\frac{\Delta L}{L} = \frac{\Delta n}{n}$. Since $\Delta n = 1$,$\frac{\Delta L}{L} \propto \frac{1}{n}$. Thus,$(D)$ is correct.Therefore,the correct statements are $(A)$,$(B)$,and $(D)$.

$1$. Burning candle	$i$. Line spectrum
$2$. Sodium vapour	$ii$. Continuous spectrum
$3$. Bunsen flame	$iii$. Band spectrum
$4$. Dark lines in solar spectrum	$iv$. Absorption spectrum

Similar Questions

The collision of an electron with kinetic energy $5.5 eV$ and a hydrogen atom in its ground state can be described as

The spectrum of iodine gas under white light will be:

$A$ neutron collides head-on with a stationary hydrogen atom in its ground state.

$Assertion :$ $A$ tube light emits white light.
$Reason :$ Emission of light in a tube takes place at a very high temperature.

Match the following:
$1$. Burning candle $i$. Line spectrum
$2$. Sodium vapour $ii$. Continuous spectrum
$3$. Bunsen flame $iii$. Band spectrum
$4$. Dark lines in solar spectrum $iv$. Absorption spectrum

Vedclass Test Series

Exam Paper Generator

Online Exam Module