Answer
if we call
- $E_H$: the ground-state energy of the hydrogen atom
- $Z_A$: the nuclear charge of A (A is a hydrogen-like system)
- $E_A$: the ground-state energy of A
we have the following relationship between the ground-state energy of the hydrogen-like systems and the nuclear charge: $$E_A=Z_A^2\times E_H$$
Work Step by Step
The nuclear charge is the total charge of all the protons in the nucleus of an atom. In the periodic table, it equals the atomic number of the element.
So, the nuclear charge of hydrogen $Z_H=1$
The nuclear charge of helium ion $Z_{He^+}=2$
The nuclear charge of lithium ion $Z_{Li^{2+}}=3$
Now we compare the ground-state energy $(E)$ of each atom / ion:
$\frac{E_{He^+}}{E_H}=\frac{-8.72\times10^{-18}J}{-2.18\times10^{-18}J}=4=2^2=Z_{He^+}^2$
$\frac{E_{Li^{2+}}}{E_H}=\frac{-1.96\times10^{-17}J}{-2.18\times10^{-18}J}\approx9=3^2=Z_{Li^{2+}}^2$
In other words,
$E_{He^+}=Z_{He^+}^2\times E_H$
$E_{Li^{2+}}=Z_{Li^{2+}}^2\times E_H$
Therefore, if we call
- $E_H$: the ground-state energy of the hydrogen atom
- $Z_A$: the nuclear charge of A (A is a hydrogen-like system)
- $E_A$: the ground-state energy of A
we have the following relationship: $$E_A=Z_A^2\times E_H$$
