(a) The energy of the 2s state of lithium is -5.391 eV. Calculate the value of Zeff for this sta

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

Key Concepts

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.

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.

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.

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.

Transcript

00:01 Welcome to problem 34, chapter 41, quantum mechanics, atomic structure, and we are looking at many electron atoms.

00:11 This is the effective nuclear charge and an effect called screening.

00:18 Let's get started.

00:20 Our example is the sodium atom, a many electron atom, a atomic number of 11.

00:27 So that means we have 11 protons in the nucleus.

00:32 The nucleus is represented by the green pluses.

00:37 It's a little beaming, attractive pulse there.

00:41 To each of the electrons, which i have the distribution rate in here.

00:46 Some facts of the atom are that we have 10 core electrons and one valence electron.

01:00 Indicated in yellow.

01:04 The sodium atom has 11 electrons that fill the shells k -l -m.

01:14 Shels k and l are completely full.

01:18 They have the total amount of electrons allowed in their shells.

01:28 And m is where we have our single balance electron.

01:35 These shells are also represented by the quantum number n equal to 1, 2, 3, also as 1s, 2p, and so on.

02:00 But in our example, only 3s, because our valence electron is in the 3s orbital.

02:10 The final note would be that sodium is an alkali metal.

02:20 This problem asks us to discuss the screening effect, which is an effect that happens in our core electrons.

02:30 The 10 core electrons are in a spherically symmetric charge distribution around the nucleus.

02:37 Each of these electrons behave in a wave formation.

02:43 And we want to envision each individual electron, each of the 10 core electrons associated with one of the protons in the nucleus.

02:56 And the core takes up 10 protons.

03:00 However, the valence electron is still being attracted to the nucleus.

03:05 It is a part of the atom.

03:08 The last proton, the single proton that is a trance, and getting past this screening effect of the core electrons and attracting the valence electron...

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