What is the ratio of time periods $(T_1 / T_2)$ in the second orbit of a hydrogen atom to the third orbit of a $He^{+}$ ion?

Which of the following electronic transitions in a hydrogen atom results in the emission of a spectral line with the shortest wavelength?

The energy of an electron in the $n^{\text{th}}$ Bohr orbit of an $H$-atom is given by:

When a metal surface is irradiated with light of frequency $x \ Hz$,the kinetic energy of emitted photoelectrons is $z \ J$. When the same metal is irradiated with light of frequency $y \ Hz$,the kinetic energy of emitted electrons is $\frac{z}{3} \ J$. What is the threshold frequency (in $Hz$) of the metal?

$(i)$ What is the energy of an electron for hydrogen in $J \ atom^{-1}$ and $J \ mol^{-1}$?
$(ii)$ Calculate the energy of this electron per mole for the first transition ($n=1$ to $n=2$).
$(iii)$ Calculate the ionization energy of hydrogen.

The ionization energy of a hydrogen atom is $13.6 \, eV$. What is the energy required (in $eV$) to excite the electron from the ground state to the first excited state?