(a) Let us approximate an electron in the $n^{\text {th }}$ shell (i.e., principal quantum number $n$ ) of an atom as being like an electron in the $n^{t h}$ shell of a hydrogen atom with an effective nuclear charge $Z$. Use your knowledge of the hydrogen atom to calculate the ionization energy of this electron (i.e., the energy required to pull the electron away from the atom) as a function of $Z$ and $n$.
(b) Consider the two approximations discussed in the text for estimating the effective nuclear charge:
- (Approximation a)
$$
Z=Z_{n u c}-N_{i n s i d e}
$$
- (Approximation b)
$$
Z=Z_{\text {nuc }}-N_{\text {inside }}-\left(N_{\text {same }}-1\right) / 2
$$
where $Z_{n u c}$ is the actual nuclear charge (or atomic number), $N_{\text {inside }}$ is the number of electrons in shells inside of $n$ (i.e., electrons with principal quantum numbers $\left.n^{\prime}<n\right)$, and $N_{s a m e}$ is the total number of electrons in the $n^{\text {th }}$ principal shell (including the electron we are trying to remove from the atom, hence the $-1$ ).
D. Explain the reasoning behind these two approximations.
D.Use these approximations to calculate the ionization energies for the atoms with atomic number
1 through 21. Make a plot of your results and compare them to the actual ionization energies (you will have to look these up on a table).
Your results should be qualitatively quite good. If you try this for higher atomic numbers, the simple approximations begin to break down. Why is this?