4th Quantum Number, Orbitals (severely in progress)

Three things happen to s orbitals as n increases:

Three things happen to s orbitals as n increases (Figure 4): They become larger, extending farther from the nucleus. They contain more nodes. This is similar to a standing wave that has regions of significant amplitude separated by nodes (points with zero amplitude). The number of radial nodes in an orbital is n - l - 1. For a given atom, the s orbitals also become higher in energy as n increases because of the increased distance from the nucleus.

where is probability density greatest? how is affected by distance?

greatest at r = 0 (at the nucleus) and decreases steadily with increasing distance. - At very large values of r, the electron probability density is tiny but not exactly zero.

The 1s orbital is spherically symmetrical, so the probability of finding a 1s electron at any given point depends only on what?

its distance from the nucleus.

azimuthal (or angular momentum) quantum number: symbol, allowed values, physical meaning

l 0 ≤ l ≤ n-1 - subshell, the shape of the orbital

magnetic quantum number: symbol, allowed values, physical meaning

ml - l ≤ ml ≤ l orientation of the orbital

spin quantum number: symbol, allowed values, physical meaning

ms +½, −½ - direction of the intrinsic quantum "spinning" of the electron

principal quantum number: symbol, allowed values, physical meaning

n 1, 2, 3, 4, .... - shell, the general region of an electron in the orbital (distance away from the nucleus)

If a shell contains a maximum of 32 electrons, what is the principal quantum number, n?

n = 4

In contrast to a simple circular orbit with a fixed radius, how are orbitals actually represented physically and mathematically?

orbitals are mathematically derived regions of space with different probabilities of having an electron in them

A dot density diagram displays black dots where there is a probability of finding an electron. How does this represent the probability of finding an electron in that region?

the higher density of dots, the higher the probability of finding the electron in that region. Taking the information from the dot density diagram, a plot of [ψ(r)]2 versus distance from the nucleus (r) can be constructed. This type of plot (shown in Figure 1b) is a plot of the probability density. This plot shows the probability of finding the electron near a particular point in space that lies a distance r from the nucleus.

wavefunction, ψ & probability of finding an electron at a given point

the probability density of finding an electron at a given point in space is [ψ(r)]2.

Maximum Number of Electrons: Calculate the maximum number of electrons that can occupy a shell with (a) n = 2, (b) n = 5, and (c) n as a variable. Note you are only looking at the orbitals with the specified n value, not those at lower energies.

(a) When n = 2, there are four orbitals (a single orbital with l = 0, and three orbitals labeled l = 1). These four orbitals can contain eight electrons. (b) When n = 5, there are five subshells of orbitals that we need to sum: 1 orbital with l = 0 3 orbitals with l = 1 5 orbitals with l = 2 7 orbitals with l = 3 + 9 orbitals with l = 4 ——————————————— 25 orbitals total Again, each orbital holds two electrons, so 50 electrons can fit in this shell. (c) The number of orbitals in any shell n will equal n2. There can be up to two electrons in each orbital, so the maximum number of electrons will be 2 × n2

Key Concepts and Summary

The azimuthal quantum number determines the general shape of the orbital. For l = 0 (an s orbital), the orbital is spherical. For l = 1 (p orbitals), there are three different p orbitals, but they all have a dumbbell shape. Different ml values are used to differentiate between the three orbitals. The d orbitals, with l = 2, generally have a four-lobed shape. As the n value increases, so does the complexity of the orbitals in each subshell. The first three quantum numbers describe these orbitals. The fourth quantum number is the spin quantum number. Each electron has a spin quantum number, ms, that can be equal to ±½. No two electrons in the same atom can have the same set of values for all the four quantum numbers, known as the Pauli exclusion principle. A consequence of the Pauli exclusion principle is that each orbital can hold at most two electrons.