Find out the amount of energy released (in joules) when a doubly ionized helium ion $\left({\mathrm{He}}^{2+}\right)$ takes up $2$ electrons to form a $\mathrm{He}$ atom in the ground state. The first ionization energy of a $\mathrm{He}$ atom is $3.4\times {10}^{-19}\mathrm{J}.$

A single-electron ion has a nuclear charge $+\mathrm{Ze}$ where $\mathrm{Z}$ is the atomic number and $\mathrm{e}$ is electronic charge. It requires $16.52 \mathrm{eV}$ to excite the electron from the second Bohr orbit to third Bohr orbit. Find the atomic number of the element.

A single-electron ion has a nuclear charge $+\mathrm{Ze}$ where $\mathrm{Z}$ is the atomic number and $\mathrm{e}$ is electronic charge. It requires $16.52 \mathrm{eV}$ to excite the electron from the second Bohr orbit to third Bohr orbit. Find the energy required for the transition of an electron from first to the third orbit.

A single-electron ion has a nuclear charge $+\mathrm{Ze}$ where $\mathrm{Z}$ is the atomic number and $\mathrm{e}$ is electronic charge. It requires $16.52 \mathrm{eV}$ to excite the electron from the second Bohr orbit to third Bohr orbit. Find wavelength required to remove an electron from first Bohr orbit to infinity.

A single-electron ion has a nuclear charge $+\mathrm{Ze}$ where $\mathrm{Z}$ is the atomic number and $\mathrm{e}$ is electronic charge. It requires $16.52 \mathrm{eV}$ to excite the electron from the second Bohr orbit to the third Bohr orbit. Find the kinetic energy of the electron in the first Bohr orbit.

For a beam of radiation coming out from a sample of ${\mathrm{He}}^{+}$ ions the following energy level diagram was observed (The beam only had photons corresponding to these transitions). If the above beam of radiations is used to excite a sample of $\mathrm{H}$-atoms in the ground state, then find the wavelengths of the photons subsequently emitted by the $\mathrm{H}$-atoms.