(a) The energy of the 2s state of lithium is -5.391 eV....

(a) The energy of the 2$s$ state of lithium is $-5.391$ eV. Calculate the value of $Z_{\text { eff }}$ for this state. (b) The energy of the 4$s$ state of potassium is $-4.339$ eV. Calculate the value of $Z_{\text { eff for this }}$ state. (c) Compare $Z_{\text { eff }}$ for the 2$s$ state of lithium, the 3$s$ state of sodium (see Example 41.8$)$ , and the 4$s$ state of potassium. What trend do you see? How can you explain this trend?

Key Concepts

Principal Quantum Number (n)

The principal quantum number indicates the energy level or shell in which an electron resides. Higher values of n correspond to orbitals that are further from the nucleus, have higher energy, and are more diffuse. In the context of effective nuclear charge, electrons in orbitals with larger n are usually more shielded and experience a lower effective nuclear charge compared to those in orbitals with smaller n.

Energy Levels and the Hydrogen-like Model

In atoms, the energy of an electron in a particular orbital can be approximated using a hydrogen-like model, where the energy is proportional to -Z_eff²/n², with n being the principal quantum number. Although this model is exact only for hydrogen, it is a useful approximation for understanding trends in multi-electron atoms. This relation ties together the concepts of effective nuclear charge and orbital energy.

Periodic Trends in Effective Nuclear Charge

When comparing effective nuclear charge for electrons in similar types of orbitals across different atoms, systematic trends emerge. As one moves down a group in the periodic table (increasing n), even though the actual nuclear charge increases, the increase in electron shielding and the larger distance of the valence electron from the nucleus result in a smaller increase (or even a decrease) in the effective nuclear charge experienced by the valence electrons. This trend helps explain variations in atomic properties such as atomic size and ionization energy.

Effective Nuclear Charge (Z_eff)

Effective nuclear charge is the net positive charge experienced by an electron in an atom. It accounts for both the full positive charge of the nucleus and the partial cancellation of that charge due to the presence of other electrons, which shield the electron from the full nuclear pull. In quantum chemistry, Z_eff is a crucial concept for understanding electron binding, energy levels, and chemical behavior.

Electron Shielding

Electron shielding refers to the reduction in the attractive force between an electron and the nucleus caused by the repulsion of other electrons. Electrons closer to the nucleus partially shield electrons in outer shells from the full effect of the nuclear charge. This means that as the number of inner electrons increases, the effective nuclear charge felt by an outer electron is diminished compared to the actual nuclear charge.

Transcript

Please give Ace some feedback