# Quantum Numbers

Submitted by ChemPRIME Staff on Thu, 12/09/2010 - 00:04

A characteristic of the diagram Fig. 1 in Electron Waves in the Hydrogen Atom is that it has been assigned an identifying label, namely, 1s. This enables us to distinguish it from other wave patterns the electron could possibly adopt if it moved about the nucleusThe collection of protons and neutrons at the center of an atom that contains nearly all of the atoms's mass. with a higher energyA system's capacity to do work.. Each of these three-dimensional wave patterns is different in shape, size, or orientation from all the others and is called an orbitalA mathematically defined region of electron density around one or more atoms; a wave function that defines the properties of a particular electron in an atom or molecule.. The word orbital is used in order to make a distinction between these wave patterns and the circular or elliptical orbits of the Bohr picture shown in The Wave Nature of the Electron.

Principal Quantum Number "n"

In the case of a particle in a one-dimensional box, the energy was determined by a positive whole number n. Much the same situation prevails in the case of the hydrogen atom. An integer called the principal quantum number, also designated by the symbol n, is used to label each orbital. The larger the value of n, the greater the energy of the electron and the larger the average distance of the electron cloud from the nucleus. The energy increases with n, in part, because the total number of nodes is n-1 for each wavefunction in shell n.

Angular Quantum Number "l"

The next quantum number, represented by l and called the "angular quantum number," can be any value in the range 0, 1, 2, ... n - 1. As we have seen in the case of 2-dimensional drum vibrations, l specifies the number of planar nodes in the wavefunction. This number represents the angular momentum of the orbital, and is important because it determines the shape of the orbital. This number is responsible for the s, p, d, f, etc., character of the orbital. l = 0 corresponds to an s orbital, l = 1 denotes a p orbital, and so forth.

Magnetic Quantum Number "ml"

The "magnetic quantum number" corresponds to the projection of the orbital along an axis, i.e. when in three-dimensional space, along the x, y, or z axis. This value falls in the range of -l, -l + 1, ... -1, 0, 1, ... l - 1, l.

Spin Quantum Number "ms"

The fourth quantum number, known as the "spin quantum number," refers to the intrinsic "spin" of the electron. This quantum number may hold only two values, either -1/2 or +1/2. The Pauli Exclusion PrincipleThe statement that no two electrons in an atom can have the same set of four quantum numbers; the principle leads to the rule that only two electrons (having opposite spin) can occupy an atomic orbital. states that each electron must have a unique set of four quantum numbers, so if two electrons are paired together in an orbital, they share three quantum numbers and must have opposite spin quantum numbers. This electron spin property is what causes a substanceA material that is either an element or that has a fixed ratio of elements in its chemical formula. to be paramagneticDescribes a substance containing unpaired electrons that is attracted into a magnetic field; the strength of attraction can be related to the number of unpaired electrons. or diamagneticDescribes a substance that is repelled very weakly by a magnetic field; a diamagnetic substance has no unpaired electrons., because a moving charge always creates a magnetic field.

Substances whose atoms, molecules, or ions contain unpaired electrons (which must be in different orbitals) are weakly attracted into a magnetic field, a property known as paramagnetism. This is because the Spin Quantum Number for the substance will not be zero since each electron will not have a partner to cancel. Paramagnetism is typically 0.1% as strong as the familiar "ferromagnetism" of common magnets.

Most substances have all their electrons paired. This means that each electron's spin number will be canceled by another electron (although they're usually in the same orbital, they need not be). The net spin will be zero for the substance, and it will not be attracted into a magnetic field, but actually repelled slightly. The repulsion is typically 0.1% as great as paramagnetic attraction. This property is known as diamagnetism.

Hence measurement of magnetic properties can tell us whether all electrons are paired or not.

There are several videos on YouTube showing

diamagnetism of grapes (mostly water),
diamagnetism of water,
diamagnetic levitation of frogs, bugs, fruit, and water,
as well as the surprising paramagnetism of liquid oxygen.

If you have access to J. Chem. Educ. Software, you will find links below to a series of videos which demonstrate how diamagnetic and paramagnetic materials respond to the presence of a strong magnet. The first video introduces the setup of the experiment, showing that various solidA state of matter having a specific shape and volume and in which the particles do not readily change their relative positions. compounds will be brought close to a strong magnet. The second video shows that sodium chloride is diamagnetic and lacks unpaired electrons, since it is not attracted to the magnet. The third video demonstrates that the paramagnetic manganese (II) sulfate monohydrate is attracted to the magnet due to its unpaired electrons. Finally, the fourth video brings a sample of liquid water toward the magnet. The water sample is not attracted, and is therefore diamagnetic.