Calculate the energy of half a mole of photons of a radiation with a frequency of $3 \times 10^{12} \ Hz$.

For the $H$ spectrum,the limiting line in the $Balmer$ series has a frequency of ...... $(R_\infty = 3.29 \times 10^{15} \text{ cycles/s})$

The electron in a hydrogen atom,upon absorbing $12.75 \ eV$ of energy,will jump to the ....... orbit.

Calculate the wave number for the longest wavelength transition in the Balmer series of atomic hydrogen.

The radius of the second Bohr orbit for hydrogen atom is .......... $\mathring{A}$.
(Planck's constant $h = 6.626 \times 10^{-34} \, Js$; mass of electron $= 9.1091 \times 10^{-31} \, kg$; charge of electron $e = 1.60210 \times 10^{-19} \, C$; permittivity of vacuum $\epsilon_0 = 8.854185 \times 10^{-12} \, C^2 N^{-1} m^{-2}$)

$A$ single-electron species has a nuclear charge of $+Ze$,where $Z$ is a constant. The energy required to excite an electron from the $2^{nd}$ Bohr orbit to the $3^{rd}$ Bohr orbit is $47.2 \ eV$. Calculate the value of $Z$.