Question

A gas containing hydrogen-like ions, with atomic number $Z$, emits photons in transition $n+2\to n$, where $n=Z$. These photons fall on a metallic plate and eject electrons having minimum de Broglie wavelength $\lambda$ of $5000\mathring{A}$. Find the value of $Z$, if the work function of metal is $4.2eV$.

Answer 1

Energy of electron corresponding to $5000A^{\circ }$ de-broglie wavelength=$\frac{hc}{\lambda }$

Energy in electron volts=$\frac{hc}{e\lambda }=2.48eV$

Total energy emitted due to transition=Work function+Energy of emitted electron=$4.2eV+2.48eV=6.68eV$

Therefore,$13.6\times Z^2(\frac{1}{(n{)}^2}-\frac{1}{(n+2{)}^2})eV=6.68eV$

Therefore, $Z=5$