What amount of energy should be added to an electron to reduce its de Broglie wavelength from $100$ to $50 \mathrm{pm}$?

Mass of the electron is $m=9.1\times {10}^{-31} \mathrm{kg}$, Planck's constant $h=6.626\times {10}^{-34} \mathrm{J} {\mathrm{Hz}}^{-1}$.

Mass of the neutron is $m_n=1.67\times {10}^{-27} \mathrm{kg}$, Planck's constant $h=6.626\times {10}^{-34} \mathrm{J} {\mathrm{Hz}}^{-1}$.

Mass of the hydrogen atom is $m_{\mathrm{H}}=1.67\times {10}^{-27} \mathrm{kg}$, Boltzmann constant $k=1.38\times {10}^{-23} {\mathrm{JK}}^{-1}$ , Planck's constant $h=6.626\times {10}^{-34} {\mathrm{JHz}}^{-1}$.

$R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), speed of light $c=3\times {10}^8 \mathrm{m} {\mathrm{s}}^{-1}$.

Proceeding from Mosley's law, find:

$\left(\mathrm{a}\right)$ the wavelength of the $K_{\mathrm{\alpha }}$ line in aluminium and cobalt;

$\left(\mathrm{b}\right)$ the difference in binding energies of $K$ and $L$ electrons in vanadium.

Atomic number for aluminium $Z=13$, for cobalt $Z=27$, for vanadium $Z=23$, $R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), Planck's constant $h=6.626\times {10}^{-34} \mathrm{J} {\mathrm{Hz}}^{-1}$, speed of light $c=3\times {10}^8 {\mathrm{ms}}^{-1}$.

$R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), speed of light $c=3\times {10}^8 {\mathrm{ms}}^{-1}$.

$R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), Planck's constant $h=6.626\times {10}^{-34} \mathrm{J} {\mathrm{Hz}}^{-1}$, speed of light $c=3\times {10}^8 {\mathrm{ms}}^{-1}$, charge of electron $e=1.66\times {10}^{-19} \mathrm{C}$

charge on electron is $e=1.6\times {10}^{-19} \mathrm{C}$, Plank's constant $h=6.626\times {10}^{-34} \mathrm{J} {\mathrm{Hz}}^{-1}$, speed of light $c=3\times {10}^8 {\mathrm{ms}}^{-1}$.