JEE Advanced 2022 Paper 1 · Physics · Atoms, Nuclei and Dual Nature of Radiation · medium

A Li^2+ ion in the n=3 state emits a photon on dropping to n=2. This photon falls on a metal of work function 4.0 eV. Take the hydrogen ground-state…

Numerical answer

\mathrm{Li^{2+}}

ion in the

n=3

state emits a photon on dropping to

n=2

. This photon falls on a metal of work function

4.0\ \mathrm{eV}

. Take the hydrogen ground-state energy as

-13.6\ \mathrm{eV}

h=6.63\times10^{-34}\ \mathrm{J\,s}

, electron mass

9.11\times10^{-31}\ \mathrm{kg}

and

1\ \mathrm{eV}=1.60\times10^{-19}\ \mathrm{J}

. The de Broglie wavelength (in nm) of the fastest emitted photoelectron is ______ . Give the answer correct to two decimal places.

Show answer and solution

Answer: 0.34

For a hydrogen-like ion

E_n=-13.6Z^2/n^2

. Photon energy

=13.6\times9\left(\tfrac14-\tfrac19\right)=17.0\ \mathrm{eV}

. Maximum kinetic energy

=17.0-4.0=13.0\ \mathrm{eV}=2.08\times10^{-18}\ \mathrm{J}

. Then

\lambda=\dfrac{h}{\sqrt{2mK}}=\dfrac{6.63\times10^{-34}}{\sqrt{2\times9.11\times10^{-31}\times2.08\times10^{-18}}}=3.41\times10^{-10}\ \mathrm{m}=0.34\ \mathrm{nm}

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