The wavelength of the radiation emitted,when in a hydrogen atom an electron falls from infinity to stationary state $n=1$,would be ............... $nm$ (Rydberg constant $= 1.097 \times 10^7 \ m^{-1}$)

Rutherford's $\alpha$-particle scattering experiment eventually led to the conclusion that

An electromagnetic radiation of wavelength $331.5 \ nm$ is made to strike the surface of a metal. Electrons are emitted with a kinetic energy of $1.2 \times 10^5 \ J \ mol^{-1}$. The work function (in $eV$) of the metal is (given: $h=6.63 \times 10^{-34} \ Js$, $N_{A}=6 \times 10^{23} \ mol^{-1}$, $1 \ eV = 1.6 \times 10^{-19} \ J$)?

The ratio of the shortest wavelength of two spectral series of the hydrogen spectrum is found to be about $9$. The spectral series are

$(i)$ The energy associated with the first orbit in the hydrogen atom is $-2.18 \times 10^{-18} \, J \, atom^{-1}$. What is the energy associated with the fifth orbit?
$(ii)$ Calculate the radius of Bohr's fifth orbit for the hydrogen atom.

The energy of an electron in the first orbit of $He^+$ is $-871.6 \times 10^{-20} \ J$. The energy of an electron in the first orbit of hydrogen is ...... .