JEE Main 2024 · Physics · Atoms, Nuclei and Dual Nature of Radiation · hard

A Li^2+ ion in its n=3 state makes a transition to n=2. The emitted photon falls on a metal of work function 5.0 eV. The de Broglie wavelength of the…

Single correct option

\mathrm{Li^{2+}}

ion in its

n=3

state makes a transition to

n=2

. The emitted photon falls on a metal of work function

5.0\ \mathrm{eV}

. The de Broglie wavelength of the fastest photoelectrons is nearly (take the ground-state energy of hydrogen as

-13.6\ \mathrm{eV}

and

\lambda=\dfrac{12.27}{\sqrt{K}}\ \text{\AA}

for an electron with

K

in eV)

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Answer: (B) $3.5\ \text{\AA}$

For a hydrogen-like ion,

E_n=-13.6\dfrac{Z^2}{n^2}

eV, so the photon energy is

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

. Maximum kinetic energy

K=17.0-5.0=12\ \mathrm{eV}

. De Broglie wavelength

\lambda=\dfrac{12.27}{\sqrt{12}}=3.54\ \text{\AA}

. Using

Z=1

gives

1.9

eV, below the work function, so no emission would occur, which is a clear sign of the error.

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