(a) The energy of an electron in the 4 state of sodium is...

(a) The energy of an electron in the 4 state of sodium is $-1.947$ eV. What is the effective net charge of the nucleus "seen" by this electron? On the average, how many electrons screen the nucleus? (b) For an outer electron in the 4$p$ state of potassium, on the average 17.2 inner electrons screen the nucleus. (i) What is the effective net charge of the nucleus "seen" by this outer electron? (ii) What is the energy of this outer electron?

Key Concepts

Atomic Energy Levels and the Hydrogen-like Model

Atomic energy levels in a hydrogen-like atom are determined by the nuclear charge and the principal quantum number, with the energy given by an expression proportional to the square of the net effective nuclear charge divided by the square of the principal quantum number. When effective nuclear charge is used in such formulas, it accounts for the screening effects of other electrons, allowing for estimates of electron binding energies in multi-electron atoms.

Orbital Penetration

Orbital penetration describes the ability of an electron in a given orbital to spend time closer to the nucleus. Electrons in orbitals with higher penetration experience a greater effective nuclear charge because they encounter less shielding by inner electrons. This concept helps explain variations in effective nuclear charge among electrons in different types of orbitals.

Effective Nuclear Charge

Effective nuclear charge is the net positive charge experienced by an electron in a multi-electron atom. Due to the repulsion between electrons (screening), the full positive charge of the nucleus is partially canceled out, so the electron effectively ‘sees’ a lower nuclear charge than the actual atomic number. This concept is crucial in explaining trends in atomic properties such as ionization energy and atomic radii.

Electron Screening or Shielding

Electron screening, also known as shielding, refers to the phenomenon where inner-shell electrons reduce the full attractive force of the nucleus on outer-shell electrons. This reduction in nuclear attraction is quantified in the effective nuclear charge and explains why electrons in outer orbitals are less tightly bound than expected if they experienced the full nuclear charge.

Transcript

00:01 Is in the 4 -s state, n is equal to 4.

00:02 We're told that the energy of the sodium, e sub -n is equal to negative 1 .947 electron volts.

00:08 So what is the effective net charge in the nucleus seen by the electrons, and on average, how many electrons are screened by the nucleus? so to find the effective energy, that's z sub e, z effective, but the effective charge, i mean z sub e, we're going to use this equation.

00:24 The negative sign here comes from the fact they're dealing with electrons, then you have negative 13 .6 ev from the ground state of hydrogen here.

00:31 So we can rearrange this to solve for z effective.

00:34 And we find that z effective, which we're calling z sub e, is equal to the square root of n squared, which n is equal to four here, times 13 .6 electron volts, because the negative signs cancel, divided by, or excuse me, that's divided by negative 13 .6.

01:13 It's times the energy value.

01:20 So times e sub n, divided by 13 .6 eb.

01:30 And the electron volt values cancel because e subn is also an ev.

01:34 So we don't need to write electron bolt.

01:37 Z effective is unitless.

01:40 So carrying out this operation, we find that z effective here is equal to 1 .51.

01:46 So we can box it in as one of our answers for part a.

01:52 And then it also wants to know the amount of electrons that are screened by the nucleus.

01:58 So we're dealing with sodium, which normally has a charge of 11 electrons.

02:03 So we have 11 minus 1 .51 from z effective.

02:12 So 11 electrons minus 1 .51 equals 9 .49 electrons...