What is a hydrogen-like atom, and how are the energies and radii of its electron orbits related to those in hydrogen?.

Atoms having only one electron and Z protons in their nucleus are known as hydrogen-like atoms.

Consider the formula for the radii of hydrogen -like atom is as follows:

\[{{\bf{r}}_{\bf{n}}}{\bf{ = }}\frac{{{{\bf{n}}^{\bf{2}}}}}{{\bf{Z}}}{{\bf{a}}_{\bf{B}}}\]

Here,r_n radius of nth orbit, Z Number of protons in the nucleus, and a_B is the Bohr's radius

Energy of hydrogen-like atom is obtained as follows:

\[{E_n} = \frac{{ - {Z^2}}}{{{n^2}}}{E_o}\]

Here, E_n is the energy of nth orbit, Z is the number of protons in the nucleus and energy of ground state.

The radius of a hydrogen-like atom is as follows:

\[{r_{{\rm{Hydrogen - like }}}} = \frac{{{n^2}}}{{\rm{Z}}}{a_B}\]

For hydrogen atom, Z=1

Thus, radius is given as

\[\begin{array}{c}{r_{{\rm{Hydrogen }}}} = \frac{{{n^2}}}{1}{a_B}\\ = {n^2}{a_B}\end{array}\]

As a result, the radius of hydrogen and hydrogen-like atoms are related as follows:

\[{r_{{\rm{Hydrogen }}}} = \frac{1}{Z}{r_{{\rm{Hydrogen - like }}}}\]

Also, Energy of hydrogen-like atom is given as

\[{E_{{\rm{Hydrogen - like }}}} = \frac{{ - {{\rm{Z}}^2}}}{{{n^2}}}{E_o}\]

For hydrogen atom, Z=1

Thus, energy is given as

\[\begin{array}{c}{E_{{\rm{Hydrogen }}}} = \frac{{ - {{(1)}^2}}}{{{n^2}}}{E_o}\\ = \frac{{ - {E_o}}}{{{n^2}}}\end{array}\]

As a result, the energy of hydrogen and hydrogen-like atoms are related as follows:

Therefore, Radii of hydrogen atom and hydrogen-like atom are related as: \[{r_{Hydrogen}} = \frac{1}{Z}{r_{Hydrogen - like}}\]

Also,

Energy of hydrogen atom and hydrogen-like atom are related as:\[\left( {{E_{{\rm{Hydrogen }}}}} \right) = {Z^2}\left( {{E_{{\rm{Hydrogen - like }}}}} \right)\]