# Quantum Numbers

**Topics:**Atomic orbital, Quantum mechanics, Quantum number

**Pages:**5 (1633 words)

**Published:**August 21, 2013

Quantum Numbers

The Bohr model was a one-dimensional model that used one quantum number to describe the distribution of electrons in the atom. The only information that was important was the size of the orbit, which was described by the n quantum number. Schrödinger's model allowed the electron to occupy three-dimensional space. It therefore required three coordinates, or three quantum numbers, to describe the orbitals in which electrons can be found. The three coordinates that come from Schrödinger's wave equations are the principal (n), angular (l), and magnetic (m) quantum numbers. These quantum numbers describe the size, shape, and orientation in space of the orbitals on an atom. The principal quantum number (n) describes the size of the orbital. Orbitals for which n = 2 are larger than those for which n = 1, for example. Because they have opposite electrical charges, electrons are attracted to the nucleus of the atom. Energy must therefore be absorbed to excite an electron from an orbital in which the electron is close to the nucleus (n = 1) into an orbital in which it is further from the nucleus (n = 2). The principal quantum number therefore indirectly describes the energy of an orbital. The angular quantum number (l) describes the shape of the orbital. Orbitals have shapes that are best described as spherical (l = 0), polar (l = 1), or cloverleaf (l= 2). They can even take on more complex shapes as the value of the angular quantum number becomes larger.

There is only one way in which a sphere (l = 0) can be oriented in space. Orbitals that have polar (l = 1) or cloverleaf (l = 2) shapes, however, can point in different directions. We therefore need a third quantum number, known as the magnetic quantum number (m), to describe the orientation in space of a particular orbital. (It is called the magnetic quantum number because the effect of different orientations of orbitals was first observed in the presence of a magnetic field.)

Rules Governing the Allowed Combinations of Quantum Numbers

* The three quantum numbers (n, l, and m) that describe an orbital are integers: 0, 1, 2, 3, and so on. * The principal quantum number (n) cannot be zero. The allowed values of n are therefore 1, 2, 3, 4, and so on. * The angular quantum number (l) can be any integer between 0 and n - 1. If n = 3, for example, l can be either 0, 1, or 2. * The magnetic quantum number (m) can be any integer between -l and +l. If l = 2, m can be either -2, -1, 0, +1, or +2.

Shells and Subshells of Orbitals

Orbitals that have the same value of the principal quantum number form a shell. Orbitals within a shell are divided into subshells that have the same value of the angular quantum number. Chemists describe the shell and subshell in which an orbital belongs with a two-character code such as 2p or 4f. The first character indicates the shell (n = 2 or n = 4). The second character identifies the subshell. By convention, the following lowercase letters are used to indicate different subshells. s:| | l = 0|

p:| | l = 1|

d:| | l = 2|

f:| | l = 3|

Although there is no pattern in the first four letters (s, p, d, f), the letters progress alphabetically from that point (g, h, and so on). Some of the allowed combinations of the n and l quantum numbers are shown in the figure below.

The third rule limiting allowed combinations of the n, l, and m quantum numbers has an important consequence. It forces the number of subshells in a shell to be equal to the principal quantum number for the shell. The n = 3 shell, for example, contains three subshells: the 3s, 3p, and 3d orbitals.

Possible Combinations of Quantum Numbers

There is only one orbital in the n = 1 shell because there is only one way in which a sphere can be oriented in space. The only allowed combination of quantum numbers for which n = 1 is the following. n| | l| | m| | |

1| | 0| | 0| | 1s|

There are four orbitals in...

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