Find the binding energy of an electron in the ground state of hydrogen-like ions, in whose spectrum the third line of the Balmer series is equal to $108.5 \mathrm{nm}$.

$R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), Planck's constant $h=6.626\times {10}^{-34} {\mathrm{JHz}}^{-1}$, 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 $h=6.626\times {10}^{-34} {\mathrm{JHz}}^{-1}$, speed of light $c=3\times {10}^8 {\mathrm{ms}}^{-1}$.

Find the velocity of the photoelectrons liberated by electromagnetic radiation of wavelength $\mathrm{\lambda }=18.0 \mathrm{nm}$ from stationary ${\mathrm{He}}^{+}$ ions, in the ground state.

$R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), mass of electron, $m_{e^{-}}=0.911\times {10}^{-30} \mathrm{kg}$, Planck's constant $h=6.626\times {10}^{-34} {\mathrm{JHz}}^{-1}$, speed of light $c=3\times {10}^8 {\mathrm{ms}}^{-1}$.

Mass of the hydrogen atom is $m=1.67\times {10}^{-27} \mathrm{kg}$, $R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), Planck's constant $h=6.626\times {10}^{-34} {\mathrm{JHz}}^{-1}$.

From the conditions of the foregoing problem, find how much (in per cent) the energy of the emitted photon differs from the energy of the corresponding transition in a hydrogen atom.

Foregoing problem: A stationary hydrogen atom emits a photon corresponding to the first line of the Lyman series. What velocity does the atom acquire? (Answer: $3.25 \mathrm{m} {\mathrm{s}}^{-1}$)

Mass of the hydrogen atom is $m=1.67\times {10}^{-27} \mathrm{kg}$, $R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), Planck's constant $h=6.626\times {10}^{-34} {\mathrm{JHz}}^{-1}$, speed of light $c=3\times {10}^8 {\mathrm{ms}}^{-1}$.

Mass of the electron is $m=9.1\times {10}^{-31} \mathrm{kg}$, $R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), Planck's constant $h=6.626\times {10}^{-34} {\mathrm{JHz}}^{-1}$, speed of light $c=3\times {10}^8 {\mathrm{ms}}^{-1}$.

Mass of the electron is $m_e=9.1\times {10}^{-31} \mathrm{kg}$, mass of proton is $1.67\times {10}^{-27} \mathrm{kg}$, charge on electron is $e=1.6\times {10}^{-19} \mathrm{C}$, Planck's constant $h=6.626\times {10}^{-34} {\mathrm{JHz}}^{-1}$.

For atoms of light and heavy hydrogen ($\mathrm{H}$ and $\mathrm{D}$), find the difference
(a) between the binding energies of their electrons in the ground state;
(b) between the wavelengths of first lines of the Lyman series.

Mass of the electron is $m=9.1\times {10}^{-31} \mathrm{kg}$, charge on electron is $e=1.6\times {10}^{-19} \mathrm{C}$, $R$ is the Rydberg's constant ($R=1.097\times {10}^7 {\mathrm{m}}^{-1}$), Planck's constant $h=6.626\times {10}^{-34} {\mathrm{JHz}}^{-1}$, speed of light $c=3\times {10}^8 {\mathrm{ms}}^{-1}$.