JEE Physics Atoms MCQs Set 17

Answer: (b) n = 5

Explanation:
1. 30.4 nm gives 40.8 eV, the 2 to 1 transition of He\(^+\) (\( 54.4(1-\tfrac14)=40.8 \)). 108.5 nm gives 11.43 eV.
2. \( 54.4\left(\dfrac14-\dfrac1{n^2}\right)=11.43\Rightarrow\dfrac1{n^2}=0.04\Rightarrow n=5 \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (c) 10.8 nm

Explanation:
1. Six lines means \( \dfrac{n(n-1)}2=6\Rightarrow n=4 \).
2. \( E=13.6\times9\left(1-\dfrac1{16}\right)=114.75 \) eV, so \( \lambda=\dfrac{1240}{114.75}=10.8 \) nm.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (a) 488.9 nm

Explanation:
1. \( \dfrac{1}{\lambda_1}=R\dfrac{5}{36} \), \( \dfrac1{\lambda_2}=R\dfrac3{16} \).
2. \( \lambda_2=660\times\dfrac{5/36}{3/16}=660\times\dfrac{20}{27}=488.9 \) nm.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (b) \( n=2\rightarrow n=4 \)

Explanation:
1. The photon energy is \( 13.6\left(1-\tfrac14\right)=10.2 \) eV.
2. For He\(^+\): \( 54.4\left(\tfrac14-\tfrac1{16}\right)=10.2 \) eV, so n = 2 to n = 4 absorbs it exactly. Other options need 7.55, 51 or 11.4 eV.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (d) \( \dfrac{20}{27}\lambda \)

Explanation:
1. M to L is 3 to 2: \( \dfrac1\lambda=RZ^2\dfrac5{36} \). N to L is 4 to 2: \( \dfrac1{\lambda'}=RZ^2\dfrac3{16} \).
2. \( \dfrac{\lambda'}{\lambda}=\dfrac{5/36}{3/16}=\dfrac{20}{27} \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (a) \( 16a_0 \)

Explanation:
1. Photon energy \( =\dfrac{12500}{980}=12.76 \) eV.
2. \( 13.6\left(1-\dfrac1{n^2}\right)=12.76\Rightarrow n=4 \), so \( r=n^2a_0=16a_0 \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (c) its kinetic energy increases but its potential energy and total energy decrease

Explanation:
1. \( K=-E=\dfrac{13.6Z^2}{n^2} \): K increases as n falls. \( U=2E \): U becomes more negative.
2. Total energy \( E=-K \) is lower, so it decreases (a photon is emitted).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (d) \( \lambda_1=\lambda_2=4\lambda_3=9\lambda_4 \)

Explanation:
1. \( \dfrac1\lambda=RZ^2\left(1-\tfrac14\right)\propto Z^2 \) (nuclear mass effects are tiny).
2. Z = 1, 1, 2, 3 gives \( \lambda_1=\lambda_2=4\lambda_3=9\lambda_4 \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (b) \( 1215\ Å \)

Explanation:
1. H first Balmer line: \( \dfrac1{6561}=R\left(\dfrac14-\dfrac19\right)=\dfrac{5R}{36} \). He\(^+\) second Balmer line (4 to 2): \( \dfrac1{\lambda}=4R\left(\dfrac14-\dfrac1{16}\right)=\dfrac{3R}4 \).
2. \( \lambda=6561\times\dfrac{5/36}{3/4}=6561\times\dfrac5{27}=1215\ Å \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (c) 823 nm

Explanation:
1. Largest UV wavelength: Lyman, n = 2 to 1: \( \dfrac1{122}=\dfrac{3R}{4} \Rightarrow \dfrac1R=91.5 \) nm.
2. The infrared starts with the Paschen series; its smallest wavelength is the series limit \( \lambda=\dfrac{9}{R}=9\times91.5=823 \) nm.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (a) \( r_n\propto n \)

Explanation:
1. Force on the electron: \( F=e\dfrac{dV}{dr}=\dfrac{eV_0}r \). So \( \dfrac{mv^2}r=\dfrac{eV_0}r\Rightarrow v^2=\dfrac{eV_0}m \), a constant.
2. Quantisation \( mvr=\dfrac{nh}{2\pi} \) with constant v gives \( r_n\propto n \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (b) \( l_H=l_{Li}\text{ and }|E_H|\lt |E_{Li}| \)

Explanation:
1. Second excited state: n = 3 for both, and \( L=\dfrac{nh}{2\pi} \) is the same.
2. \( |E|=\dfrac{13.6Z^2}{n^2} \): H: 1.51 eV; Li\(^{2+}\): 13.6 eV. So \( |E_H|\lt |E_{Li}| \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (d) \( \dfrac{18}{5R} \)

Explanation:
1. Energy levels scale with the orbiting mass, so the effective Rydberg constant is 2R. The first excited level is n = 2.
2. Longest wavelength: 3 to 2: \( \dfrac1\lambda=2R\left(\dfrac14-\dfrac19\right)=\dfrac{5R}{18}\Rightarrow\lambda=\dfrac{18}{5R} \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (a) \( 2\times10^7\ \text{m/s} \)

Explanation:
1. Doppler shift for a receding source: \( \dfrac{\Delta\lambda}\lambda=\dfrac vc \).
2. \( v=3\times10^8\times\dfrac{706-656}{656}=2.3\times10^7\approx2\times10^7 \) m/s.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (c) doubly ionised lithium

Explanation:
1. \( \dfrac1\lambda=RZ^2\left(1-\tfrac14\right)\propto Z^2 \) (nuclear mass has a negligible effect).
2. The largest Z, 3 for Li\(^{++}\), gives the shortest wavelength.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (d) \( n^2\left(\dfrac{h^2}{8\pi^2I}\right) \)

Explanation:
1. Quantised angular momentum: \( L=I\omega=\dfrac{nh}{2\pi} \).
2. \( E=\dfrac{L^2}{2I}=\dfrac{n^2h^2}{8\pi^2I} \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (b) \( 1.3\times10^{-10}\ \text m \)

Explanation:
1. Moment of inertia \( I=\mu d^2 \) with reduced mass \( \mu=\dfrac{12\times16}{28}=6.86 \) amu \( =1.143\times10^{-26} \) kg.
2. \( d=\sqrt{\dfrac{1.87\times10^{-46}}{1.143\times10^{-26}}}=1.28\times10^{-10}\approx1.3\times10^{-10} \) m.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (c) \( a^{-2} \)

Explanation:
1. Standing wave condition: \( a=\dfrac{n\lambda}2\Rightarrow\lambda=\dfrac{2a}n \), so \( p=\dfrac h\lambda=\dfrac{nh}{2a} \).
2. \( E=\dfrac{p^2}{2m}=\dfrac{n^2h^2}{8ma^2}\propto a^{-2} \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (a) \( n \)

Explanation:
1. \( p=\dfrac{nh}{2a} \).
2. \( v=\dfrac pm=\dfrac{nh}{2ma}\propto n \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (b) 4

Explanation:
1. The excitation energy of H in n = 2 is \( 13.6\times\tfrac34=10.2 \) eV. He\(^+\) in n = 2 has energy \( -13.6 \) eV.
2. Final energy \( -13.6+10.2=-3.4 \) eV \( =-\dfrac{54.4}{n^2}\Rightarrow n=4 \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Answer: (d) 1/4

Explanation:
1. \( K=\dfrac{13.6Z^2}{n^2} \).
2. \( \dfrac{K_H}{K_{He^+}}=\dfrac{1}{4} \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.