To determine the value of Z at which the relativistic effects might affect energies and whether it applies to all orbiting electrons or to some more than others, also guess if it is acceptable to combine quantum mechanical results.

The expression for the velocity of the electron v that orbits the nucleus of charge $+\mathrm{Ze}$is given by,

$\mathbf{v}\mathbf{=}\frac{{\mathbf{Ze}}^{\mathbf{2}}}{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{o}}\mathbf{hn}}$

Here, e is the unit charge, ${\mathbf{\epsilon }}_{\mathbf{o}}$is the permittivity of free space, h is the plank constant and n is the principal quantum numbers.

The expression for the Lorentz factor${\mathbf{\gamma }}_{\mathbf{v}}$ is given by,

${\mathbf{\gamma }}_{\mathbf{v}}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{-}{\left({\displaystyle \frac{v}{c}}\right)}^{\mathbf{2}}}}$

Here, c is the speed of light and v is the velocity of the object.

The expression for the Lorentz factor is evaluated as

$\begin{array}{rcl}{\mathrm{\gamma }}_{\mathrm{v}}& =& \frac{1}{\sqrt{1-{\left({\displaystyle \frac{\mathrm{v}}{\mathrm{c}}}\right)}^2}}\\ {\mathrm{\gamma }}_{\mathrm{v}}& =& \frac{1}{\sqrt{1-{\left({\displaystyle \frac{{\mathrm{Ze}}^2}{2{\mathrm{e}}_{\mathrm{o}}{0}^{2\mathrm{n}}}}\right)}^2}}\\ {\mathrm{\gamma }}_{\mathrm{v}}& =& \frac{1}{\sqrt{1-{\left({\displaystyle \frac{{\mathrm{Z}}^2{\mathrm{e}}^4}{4{\mathrm{c}}^2{0}^2{\mathrm{n}}^2{\mathrm{c}}^2}}\right)}^2}}\\ & & \end{array}$

Solve further as,

role="math" localid="1658483200126" $\begin{array}{rcl}\frac{{\mathrm{Z}}^2{\mathrm{e}}^4}{4{{\mathrm{\epsilon }}^2}_0{\mathrm{h}}^2{\mathrm{n}}^2{\mathrm{c}}^2}& =& 1-\frac{1}{{\mathrm{\gamma }}_{\mathrm{v}}}\\ \mathrm{Z}& =& \frac{2{\mathrm{\epsilon }}_0\mathrm{hcn}}{{\mathrm{e}}^2}\sqrt{1-\frac{1}{{\mathrm{\gamma }}_{\mathrm{v}}}}\end{array}$

The value of the Z is evaluated as,

Substitute all the values in the above equation,

$$\begin{gathered} \mathrm{Z}=\frac{2{\mathrm{\epsilon }}_0\mathrm{hcn}}{{\mathrm{e}}^2}\sqrt{1-\frac{1}{{\mathrm{\gamma }}_{\mathrm{v}}}} \\ \mathrm{z}=\frac{2\times \left(8.85\times {10}^{-12}{\mathrm{C}}^2/\mathrm{N}.{\mathrm{m}}^2\right)\left(6.63\times {10}^{-34}\mathrm{J}.\mathrm{s}\right)\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)\left(1\right)}{{\left(1.6\times {10}^{-19}\mathrm{C}\right)}^2}\sqrt{1-\frac{1}{{\left(1.01\right)}^2}} \\ \mathrm{z}=19.3 \end{gathered}$$

The value of Z for which the effects are produced that deflects from the classical explanations by 1% is 19.3.