Ionization energy and Z effective

[fluidistic]

Homework Statement

Consider an alkali atom in its fundamental state and assume that its valence electron experiment a constant shielding with $Z_{eff}>1$. Use the tabulated values of the ionization energy to estimate $Z_{eff}$ for the Li, Na and K atoms.

Homework Equations

I do not know.

The Attempt at a Solution

Z effective is the effective charge (in units of electron charge) the valence electron feels. I've "cheated" and checked out Z effective and the ionization energies for all these atoms, yet I do not find any correlation.
Ionization energies: (5.39, 5.14, 4.34) eV.
Z effective: (1.25, 1.84, 2.26).
I'm at a loss. Any tip is appreciated.

 

[mark726]

First, it's important to understand what Z_{eff} represents. Z_{eff} is the effective nuclear charge experienced by the valence electron due to the shielding effect of inner electrons. This means that the valence electron feels a lower effective charge than the actual nuclear charge due to the repulsion from inner electrons.

To estimate Z_{eff}, we can use the Slater's effective nuclear charge formula, which is given by:

Z_{eff} = Z - S

where Z is the actual nuclear charge and S is the screening constant. The screening constant can be calculated using the Slater's rules, which assign a value to each inner electron based on its principal quantum number and the type of orbital it occupies.

For Li, the nuclear charge is Z=3 and the screening constant is S=1. For Na, Z=11 and S=9. For K, Z=19 and S=18. Using these values, we can calculate the effective nuclear charges as follows:

Z_{eff} (Li) = 3 - 1 = 2
Z_{eff} (Na) = 11 - 9 = 2
Z_{eff} (K) = 19 - 18 = 1

These values of Z_{eff} are close to the tabulated values you have listed, but they are not exactly the same. This is because the Slater's rules are only approximations and may not always give accurate results. Additionally, the ionization energies you have listed may not be for the fundamental state of the atoms, which can also affect the accuracy of the calculations.

In conclusion, to estimate Z_{eff} for alkali atoms, we can use the Slater's effective nuclear charge formula, but keep in mind that it is an approximation and may not always give accurate results. It is always best to consult tabulated values for more accurate results.