General Chemistry/The Quantum Atom

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The Quantum Numbers[edit]

These four numbers are used to describe the location of an electron in an atom.

Number Symbol Possible Values
Principal Quantum Number
Angular Momentum Quantum Number
Magnetic Quantum Number
Spin Quantum Number

Principal Quantum Number (n)[edit]

Determines the shell the electron is in. The shell is the main component that determines the energy of the electron (higher n corresponds to higher energy), as well as size of the orbital, corresponding to maximum nuclear distance (higher n means further possible distance from the nucleus). The row that an element is placed on the periodic table tells how many shells there will be. Helium (n = 1), neon (n = 2), argon (n = 3), etc. Note that the shells will have different numbers, as described by the table above; for example, argon will contain the , , and subshells, for that total of 3.

Angular Momentum Quantum Number (l)[edit]

Also known as azimuthal quantum number. Determines the subshell the electron is in. Each subshell has a unique shape and a letter name. The s orbital is shaped like a sphere and occurs when l = 0. The p orbitals (there are three) are shaped like teardrops and occur when l = 1. The d orbitals (there are five) occur when l = 2. The f orbitals (there are seven) occur when l = 3. (By the way, when l = 4, the orbitals are "g orbitals", but they (and the l = 5 "h orbitals") can safely be ignored in general chemistry.). The numbers of the subshells in each shell can be calculated using the principal quantum number like so. For example, in the shell, the subshells are an subshell, and 3 subshells. You will learn how to determine the number of orbitals for each subshells in the next section.

This number also gives information as to what the angular node of an orbital is. A node is defined as a point on a standing wave where the wave has minimal amplitude. When applied to chemistry this is the point of zero-displacement and thus where no electrons are found. In turn angular node means the planar or conical surface in which no electrons are found or where there is no electron density. The models shown on this page show the most simple representations of these orbitals and their nodes. More accurate, but more complex depictions are not necessary for the scope of this book.

Here are pictures of the orbitals. Keep in mind that they do not show the actual path of the electrons, due to the Heisenberg Uncertainty Principle. Instead, they show the volume where the electron is most likely to occur, i.e. the probability amplitude is largest. The two colors represent two signs (phases) of the wave function (the choice is arbitrary). Each of the depicted orbitals is a superposition of two opposite m quantum numbers (see below).

ml 0 -1 and 1 -2 and 2 -3 and 3
S orbital → Hydrogen eigenstate n1 l0 m0.png
P orbitals → P2M0.png Px orbital.pngPy orbital.png
D orbitals → D3M0.png Dxz orbital.pngDyz orbital.png Dx2-y2 orbital.pngDxy orbital.png
F orbitals → F4M0.png Fyz2 orbital.pngFxz2 orbital.png Fz(x2-y2) orbital.pngFxyz orbital.png Fy(3x2-y2) orbital.pngFx(x2-3y2) orbital.png

Magnetic Quantum Number (ml)[edit]

ml -3 -2 -1 0 1 2 3
S orbital → Hydrogen eigenstate n1 l0 m0.png
P orbitals → Hydrogen eigenstate n2 l1 m-1.png Hydrogen eigenstate n2 l1 m0.png Hydrogen eigenstate n2 l1 m1.png
D orbitals → Hydrogen eigenstate n3 l2 m-2.png Hydrogen eigenstate n3 l2 m-1.png Hydrogen eigenstate n3 l2 m0.png Hydrogen eigenstate n3 l2 m1.png Hydrogen eigenstate n3 l2 m2.png
F orbitals → Hydrogen eigenstate n4 l3 m-3.png Hydrogen eigenstate n4 l3 m-2.png Hydrogen eigenstate n4 l3 m-1.png Hydrogen eigenstate n4 l3 m0.png Hydrogen eigenstate n4 l3 m1.png Hydrogen eigenstate n4 l3 m2.png Hydrogen eigenstate n4 l3 m3.png

Magnetic quantum number determines the orbital in which the electron lies. The number of orbitals in each subshell can be calculated like so: . ml determines how rapidly the complex phase increases around the z-axis. Without magnetic field, these orbitals all have the same energy, they are degenerate and can be combined into different shapes and spatial orientations. The orbitals in a subshell with degeneracy are called degenerate orbitals. This simply means that the orbitals in each p subshell all have the same energy level. The difference in shapes as well as orientation of higher subshells is not important during general chemistry, and the orbitals in the same higher subshells are still degenerate regardless of shape differences.due to of the fu king personality

Spin Quantum Number (ms)[edit]

Does not determines the spin on the electron. +½ corresponds to the up arrow in an electron configuration box. If there is only one electron in an orbital (one arrow in one box), then it is always considered +½. The second arrow, or down arrow, is considered -½. Every orbital can contain one "spin up" electron, and one "spin down" electron.

Some Examples[edit]

Let's examine the quantum numbers of electrons from a magnesium atom, 12Mg. Remember that each list of numbers corresponds to (n, l, ml, ms).

Two s electrons: (1, 0, 0, +½) (1, 0, 0, -½)
Two s electrons: (2, 0, 0, +½) (2, 0, 0, -½)
Six p electrons: (2, 1, -1, +½) (2, 1, -1, -½) (2, 1, 0, +½) (2, 1, 0, -½) (2, 1, 1, +½) (2, 1, 1, -½)
Two s electrons: (3, 0, 0, +½) (3, 0, 0, -½)

The Periodic Table[edit]

Notice a pattern on the periodic table. Different areas, or blocks, have different types of electrons. The two columns on the left make the s-block. The six columns on the right make the p-block. The large area in the middle (transition metals) makes the d-block. The bottom portion makes the f-block (Lanthanides and Actinides). Each row introduces a new shell (aka energy level). Basically, the row tells you how many shells of electrons there will be, and the column tells you which subshells will occur (and which shells they occur in). The value of ml can be determined by some of the rules we will learn in the next chapter. The value of ms doesn't really matter as long as there are no repeating values in the same orbital.