Electron orbital

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An orbital (also electron orbital) is a mode of behavior of an electron in an atom or molecule.

According to quantum mechanics, an electron in an atom or molecule has two curious properties:

1. An electron's position in space is random, but described by a probability density function <math>\psi^2<math>. (As an object in space, <math>\psi^2<math> is a scalar-valued function of three variables, usually given as spherical coordinates.) The probability density function depends upon three so-called quantum numbers: <math>n<math>, <math>l<math>, and <math>m_l<math>. (These numbers appear in equations describing the behavior of the electron.)

2. An electron can only have certain discrete amounts of energy. The particular amount of energy that it has is determined by the probability density function in which it is bound.

The combination of an energy level and a probability density function is called an orbital.

The various types of orbitals

An orbital is uniquely identified by the values of the three quantum numbers, and each value of the three quantum numbers corresponds to exactly one orbital, but the quantum numbers only occur in certain combinations of values. The rules governing the possible values of the quantum numbers are as follows:

The quantum number <math>n<math> is always a positive integer. In fact, it can be any positive integer, but for reasons discussed below, small values occur more often than larger ones. Each atom has, in general, many orbitals associated with each value of <math>n<math>; these orbitals together are sometimes called a shell.

The quantum number <math>l<math> is always an integer. Within a shell where <math>n<math> is some integer <math>n_0<math>, <math>l<math> ranges thus: <math>0 \le l \le n_0-1<math>. For instance, the <math>n=1<math> shell has only orbitals with <math>l=0<math>, and the <math>n=2<math> shell has only orbitals with <math>l=-1<math>, <math>l=0<math>, and <math>l=1<math>. The set of orbitals associated with a particular value of <math>l<math> are sometimes collectively called a subshell.

The quantum number <math>m_l<math> is also always an integer. Within a subshell where <math>l<math> is some integer <math>l_0<math>, <math>m_l<math> ranges thus: <math>-l_0 \le m_l \le l_0<math>.

The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of <math>m_l<math> available in that subshell. Empty cells represent subshells that do not exist.

	<math>l=0<math>	1	2	3	4	...
<math>n=1<math>	<math>m_l=0<math>
2	0	-1, 0, 1
3	0	-1, 0, 1	-2, -1, 0, 1, 2
4	0	-1, 0, 1	-2, -1, 0, 1, 2	-3, -2, -1, 0, 1, 2, 3
5	0	-1, 0, 1	-2, -1, 0, 1, 2	-3, -2, -1, 0, 1, 2, 3	-4, -3, -2 -1, 0, 1, 2, 3, 4
...	...	...	...	...	...	...

Subshells are usually identified by their <math>n<math>- and <math>l<math>-values. <math>n<math> is represented by its numerical value, but <math>l<math> is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', 4 by 'g', with the other values following in alphabetical order. For instance, one may speak of the subshell with <math>n=2<math> and <math>l=0<math> as a '2s subshell'.

The shapes of orbitals

Any discussion of the shapes of electron orbitals is necessarily uncertain, because a given electron, regardless of which orbital it occupies, can at any moment be found at any distance from the nucleus and in any direction.

However, the electron is much more likely to be found in certain areas of the atom than in others. Given this, a boundary surface can be drawn so that all areas within the surface have high values of the probability density function and all areas outside the surface have low values. The precise placement of the surface is arbitrary, but any reasonably compact determination must follow a pattern specified by the behavior of <math>\psi^2<math>. This boundary surface is what is meant when the "shape" of an orbital is mentioned.

Generally speaking, the number <math>n<math> determines the size of the orbital, <math>l<math> its shape, and <math>m_l<math> its orientation. As <math>n<math> increases, the size of the orbital increases.

The relationship to <math>l<math> is more complex. <math>s<math>-orbitals (<math>l=0<math>) are shaped like spheres. <math>p<math>-orbitals have the form of two ellipsoids with a point of tangency at the nucleus. The three <math>p<math>-orbitals in each shell are oriented at right angles to each other, as determined by their respective values of <math>m_l<math>.

The number of electrons in a neutral atom increases with the atomic number. The electrons in the outermost shell, or valence electrons, tend to be responsible for an element's chemical behavior. Elements that contain the same number of valence electrons can be grouped together and display similar chemical properties.

No.	Element	1	2	3	4	5	6	7
1	Hydrogen	1
2	Helium	2
3	Lithium	2	1
4	Beryllium	2	2
5	Boron	2	3
6	Carbon	2	4
7	Nitrogen	2	5
8	Oxygen	2	6
9	Fluorine	2	7
10	Neon	2	8
11	Sodium	2	8	1
12	Magnesium	2	8	2
13	Aluminium	2	8	3
14	Silicon	2	8	4
15	Phosphorus	2	8	5
16	Sulfur	2	8	6
17	Chlorine	2	8	7
18	Argon	2	8	8
19	Potassium	2	8	8	1
20	Calcium	2	8	8	2
21	Scandium	2	8	9	2
22	Titanium	2	8	10	2
23	Vanadium	2	8	11	2
24	Chromium	2	8	13	1
25	Manganese	2	8	13	2
26	Iron	2	8	14	2
27	Cobalt	2	8	15	2
28	Nickel	2	8	16	2
29	Copper	2	8	18	1
30	Zinc	2	8	18	2
31	Gallium	2	8	18	3
32	Germanium	2	8	18	4
33	Arsenic	2	8	18	5
34	Selenium	2	8	18	6
35	Bromine	2	8	18	7
36	Krypton	2	8	18	8
37	Rubidium	2	8	18	8	1
38	Strontium	2	8	18	8	2
39	Yttrium	2	8	18	9	2
40	Zirconium	2	8	18	10	2
41	Niobium	2	8	18	12	1
42	Molybdenum	2	8	18	13	1
43	Technetium	2	8	18	14	1
44	Ruthenium	2	8	18	15	1
45	Rhodium	2	8	18	16	1
46	Palladium	2	8	18	18	0
47	Silver	2	8	18	18	1
48	Cadmium	2	8	18	18	2
49	Indium	2	8	18	18	3
50	Tin	2	8	18	18	4
51	Antimony	2	8	18	18	5
52	Tellurium	2	8	18	18	6
53	Iodine	2	8	18	18	7
54	Xenon	2	8	18	18	8
55	Caesium	2	8	18	18	8	1
56	Barium	2	8	18	18	8	2
57	Lanthanum	2	8	18	18	9	2
58	Cerium	2	8	18	19	9	2
59	Praseodymium	2	8	18	21	8	2
60	Neodymium	2	8	18	22	8	2
61	Promethium	2	8	18	23	8	2
62	Samarium	2	8	18	24	8	2
63	Europium	2	8	18	25	8	2
64	Gadolinium	2	8	18	25	9	2
65	Terbium	2	8	18	27	8	2
66	Dysprosium	2	8	18	28	8	2
67	Holmium	2	8	18	29	8	2
68	Erbium	2	8	18	30	8	2
69	Thulium	2	8	18	31	8	2
70	Ytterbium	2	8	18	32	8	2
71	Lutetium	2	8	18	32	9	2
72	Hafnium	2	8	18	32	10	2
73	Tantalum	2	8	18	32	11	2
74	Tungsten	2	8	18	32	12	2
75	Rhenium	2	8	18	32	13	2
76	Osmium	2	8	18	32	14	2
77	Iridium	2	8	18	32	15	2
78	Platinum	2	8	18	32	17	1
79	Gold	2	8	18	32	18	1
80	Mercury	2	8	18	32	18	2
81	Thallium	2	8	18	32	18	3
82	Lead	2	8	18	32	18	4
83	Bismuth	2	8	18	32	18	5
84	Polonium	2	8	18	32	18	6
85	Astatine	2	8	18	32	18	7
86	Radon	2	8	18	32	18	8
87	Francium	2	8	18	32	18	8	1
88	Radium	2	8	18	32	18	8	2
89	Actinium	2	8	18	32	18	9	2
90	Thorium	2	8	18	32	18	10	2
91	Protactinium	2	8	18	32	20	9	2
92	Uranium	2	8	18	32	21	9	2
93	Neptunium	2	8	18	32	22	9	2
94	Plutonium	2	8	18	32	24	8	2
95	Americium	2	8	18	32	25	8	2
96	Curium	2	8	18	32	25	9	2
97	Berkelium	2	8	18	32	27	8	2
98	Californium	2	8	18	32	28	8	2
99	Einsteinium	2	8	18	32	29	8	2
100	Fermium	2	8	18	32	30	8	2
101	Mendelevium	2	8	18	32	31	8	2
102	Nobelium	2	8	18	32	32	8	2
103	Lawrencium	2	8	18	32	32	9	2
104	Rutherfordium	2	8	18	32	32	10	2
105	Dubnium	2	8	18	32	32	11	2
106	Seaborgium	2	8	18	32	32	12	2
107	Bohrium	2	8	18	32	32	13	2
108	Hassium	2	8	18	32	32	14	2
109	Meitnerium	2	8	18	32	32	15	2
110	Darmstadtium	2	8	18	32	32	16	2
111	Unununium	2	8	18	32	32	17	2
112	Ununbium	2	8	18	32	32	18	2
113	Ununtrium	2	8	18	32	32	18	3
114	Ununquadium	2	8	18	32	32	18	4
115	Ununpentium	2	8	18	32	32	18	5
116	Ununhexium	2	8	18	32	32	18	6
117	Ununseptium	2	8	18	32	32	18	7
118	Ununoctium	2	8	18	32	32	18	8

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