The Location of Any Electron in an Atom Can Be Described by Unique Quantum Numbers
Quantum Numbers and Electron Configurations
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 merely information that was important was the size of the orbit, which was described by the northward quantum number. Schr�dinger'southward model allowed the electron to occupy iii-dimensional space. It therefore required three coordinates, or three quantum numbers, to draw the orbitals in which electrons can be establish.
The three coordinates that come from Schr�dinger's wave equations are the principal (northward), angular (50), and magnetic (m) quantum numbers. These quantum numbers draw the size, shape, and orientation in space of the orbitals on an atom.
The master quantum number (n) describes the size of the orbital. Orbitals for which northward = 2 are larger than those for which n = 1, for instance. Because they have opposite electrical charges, electrons are attracted to the nucleus of the atom. Energy must therefore exist absorbed to excite an electron from an orbital in which the electron is shut to the nucleus (north = 1) into an orbital in which it is further from the nucleus (n = two). The principal quantum number therefore indirectly describes the energy of an orbital.
The angular breakthrough number (l) describes the shape of the orbital. Orbitals accept shapes that are best described as spherical (50 = 0), polar (l = 1), or cloverleaf (l = ii). They can even take on more complex shapes every bit the value of the angular quantum number becomes larger.
There is only 1 style in which a sphere (l = 0) tin can be oriented in space. Orbitals that have polar (50 = 1) or cloverleaf (l = ii) shapes, notwithstanding, can point in different directions. We therefore demand a third quantum number, known as the magnetic quantum number (grand), to depict the orientation in space of a detail orbital. (It is called the magnetic breakthrough number because the effect of dissimilar orientations of orbitals was beginning observed in the presence of a magnetic field.)
Rules Governing the Immune Combinations of Breakthrough Numbers
- The iii breakthrough numbers (n, l, and m) that describe an orbital are integers: 0, 1, 2, 3, then on.
- The primary quantum number (north) cannot be zilch. The allowed values of northward are therefore 1, 2, 3, four, and and so on.
- The angular quantum number (l) tin exist whatsoever integer betwixt 0 and n - ane. If due north = 3, for example, l can be either 0, 1, or 2.
- The magnetic quantum number (m) can be any integer betwixt -l and +l. If fifty = ii, g tin be either -2, -1, 0, +one, or +2.
Shells and Subshells of Orbitals
Orbitals that have the aforementioned value of the principal quantum number form a shell. Orbitals inside a shell are divided into subshells that have the same value of the athwart quantum number. Chemists draw the shell and subshell in which an orbital belongs with a two-character code such as 2p or 4f. The first graphic symbol indicates the shell (north = two or northward = 4). The 2d character identifies the subshell. Past convention, the following lowercase letters are used to indicate different subshells.
| s: | l = 0 | |
| p: | l = i | |
| d: | l = 2 | |
| f: | 50 = 3 |
Although there is no pattern in the first four letters (s, p, d, f), the letters progress alphabetically from that indicate (thou, h, and so on). Some of the allowed combinations of the n and l breakthrough numbers are shown in the figure below.
The tertiary rule limiting immune combinations of the due north, fifty, and k breakthrough numbers has an important consequence. It forces the number of subshells in a shell to be equal to the chief quantum number for the shell. The n = 3 shell, for example, contains three subshells: the threes, 3p, and 3d orbitals.
Possible Combinations of Quantum Numbers
There is merely one orbital in the due north = one shell because in that location is only i mode in which a sphere can be oriented in infinite. The only immune combination of breakthrough numbers for which northward = 1 is the following.
There are four orbitals in the n = two shell.
| 2 | 1 | -i |  | |||
| 2 | 1 | 0 | 2p | |||
| 2 | 1 | 1 |
At that place is only one orbital in the 2s subshell. But, at that place are 3 orbitals in the 2p subshell because there are three directions in which a p orbital can point. One of these orbitals is oriented along the 10 axis, another along the Y axis, and the 3rd along the Z axis of a coordinate system, as shown in the effigy below. These orbitals are therefore known as the 2pten , twopy , and iipz orbitals.
There are nine orbitals in the n = 3 shell.
In that location is one orbital in the iiisouth subshell and iii orbitals in the 3p subshell. The n = 3 shell, still, also includes threed orbitals.
The five different orientations of orbitals in the 3d subshell are shown in the figure below. One of these orbitals lies in the XY plane of an XYZ coordinate organisation and is called the 3d xy orbital. The 3d xz and 3d yz orbitals have the aforementioned shape, only they lie between the axes of the coordinate arrangement in the XZ and YZ planes. The fourth orbital in this subshell lies along the Ten and Y axes and is called the 3dx 2 -y 2 orbital. Most of the infinite occupied by the fifth orbital lies along the Z centrality and this orbital is called the 3dz 2 orbital.
The number of orbitals in a shell is the square of the principal quantum number: itwo = 1, 22 = 4, 3two = nine. At that place is one orbital in an s subshell (fifty = 0), three orbitals in a p subshell (50 = 1), and v orbitals in a d subshell (l = 2). The number of orbitals in a subshell is therefore 2(l) + ane.
Earlier nosotros tin can employ these orbitals nosotros need to know the number of electrons that can occupy an orbital and how they tin can be distinguished from one another. Experimental evidence suggests that an orbital can concord no more than two electrons.
To distinguish betwixt the ii electrons in an orbital, we need a fourth quantum number. This is called the spin quantum number (s) because electrons acquit as if they were spinning in either a clockwise or counterclockwise fashion. One of the electrons in an orbital is arbitrarily assigned an s quantum number of +1/2, the other is assigned an s quantum number of -1/two. Thus, it takes three quantum numbers to ascertain an orbital but four breakthrough numbers to identify one of the electrons that can occupy the orbital.
The allowed combinations of n, l, and m breakthrough numbers for the get-go four shells are given in the table below. For each of these orbitals, there are ii allowed values of the spin quantum number, s.
Summary of Allowed Combinations of Quantum Numbers
| northward | l | m | Subshell Annotation | Number of Orbitals in the Subshell | Number of Electrons Needed to Fill up Subshell | Total Number of Electrons in Subshell | |||||
| ���������������������������������������������������������������� | |||||||||||
| 1 | 0 | 0 | 1s | 1 | 2 | 2 | |||||
| ���������������������������������������������������������������� | |||||||||||
| ii | 0 | 0 | 2s | 1 | 2 | ||||||
| 2 | one | one,0,-ane | 2p | 3 | 6 | 8 | |||||
| ���������������������������������������������������������������� | |||||||||||
| iii | 0 | 0 | 3s | i | ii | ||||||
| 3 | ane | 1,0,-ane | 3p | three | 6 | ||||||
| iii | two | ii,one,0,-i,-ii | 3d | five | 10 | 18 | |||||
| ���������������������������������������������������������������� | |||||||||||
| 4 | 0 | 0 | 4s | 1 | 2 | ||||||
| 4 | one | one,0,-1 | 4p | iii | 6 | ||||||
| four | 2 | 2,one,0,-one,-two | 4d | 5 | x | ||||||
| four | 3 | three,2,i,0,-1,-ii,-3 | 4f | 7 | fourteen | 32 | |||||
The Relative Energies of Atomic Orbitals
Because of the force of attraction betwixt objects of opposite accuse, the virtually of import factor influencing the free energy of an orbital is its size and therefore the value of the principal quantum number, n. For an atom that contains but one electron, at that place is no difference between the energies of the different subshells inside a vanquish. The 3south, 3p, and 3d orbitals, for example, accept the same energy in a hydrogen atom. The Bohr model, which specified the energies of orbits in terms of naught more than the distance between the electron and the nucleus, therefore works for this atom.
The hydrogen atom is unusual, however. As shortly as an atom contains more than than one electron, the different subshells no longer have the same energy. Within a given shell, the s orbitals ever have the lowest free energy. The free energy of the subshells gradually becomes larger every bit the value of the athwart quantum number becomes larger.
Relative energies: s < p < d < f
As a outcome, two factors control the energy of an orbital for most atoms: the size of the orbital and its shape, as shown in the figure below.
A very simple device tin be constructed to approximate the relative energies of diminutive orbitals. The immune combinations of the due north and fifty breakthrough numbers are organized in a tabular array, as shown in the figure beneath and arrows are drawn at 45 caste angles pointing toward the bottom left corner of the tabular array.
The order of increasing energy of the orbitals is then read off past following these arrows, starting at the acme of the first line and and then proceeding on to the 2d, third, fourth lines, and then on. This diagram predicts the following order of increasing energy for diminutive orbitals.
ones < iis < twop < 3south < threep <foursouthward < 3d <4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < sevens < 5f < 6d < 7p < 8due south ...
Electron Configurations, the Aufbau Principle, Degenerate Orbitals, and Hund'southward Rule
The electron configuration of an atom describes the orbitals occupied by electrons on the cantlet. The basis of this prediction is a rule known as the aufbau principle, which assumes that electrons are added to an cantlet, ane at a time, starting with the lowest energy orbital, until all of the electrons have been placed in an appropriate orbital.
A hydrogen atom (Z = ane) has just 1 electron, which goes into the everyman energy orbital, the 1due south orbital. This is indicated by writing a superscript "1" after the symbol for the orbital.
H (Z = 1): 1southward 1
The next chemical element has two electrons and the second electron fills the 1s orbital because in that location are only ii possible values for the spin quantum number used to distinguish betwixt the electrons in an orbital.
He (Z = 2): 1s 2
The 3rd electron goes into the next orbital in the free energy diagram, the 2south orbital.
Li (Z = 3): idue south two 2southward i
The fourth electron fills this orbital.
Be (Z = 4): 1s ii 2s 2
Later the 1due south and 2s orbitals have been filled, the side by side lowest free energy orbitals are the three iip orbitals. The fifth electron therefore goes into one of these orbitals.
B (Z = v): 1s ii 2s 2 twop 1
When the fourth dimension comes to add a sixth electron, the electron configuration is obvious.
C (Z = vi): 1s ii twos two twop ii
However, there are three orbitals in the twop subshell. Does the second electron go into the same orbital as the first, or does it go into one of the other orbitals in this subshell?
To reply this, we demand to empathise the concept of degenerate orbitals. Past definition, orbitals are degenerate when they have the same energy. The energy of an orbital depends on both its size and its shape because the electron spends more than of its time farther from the nucleus of the atom as the orbital becomes larger or the shape becomes more complex. In an isolated atom, however, the energy of an orbital doesn't depend on the direction in which information technology points in space. Orbitals that differ but in their orientation in space, such as the 2p10 , 2py , and twopz orbitals, are therefore degenerate.
Electrons fill degenerate orbitals according to rules first stated by Friedrich Hund. Hund's rules can be summarized equally follows.
- 1 electron is added to each of the degenerate orbitals in a subshell before 2 electrons are added to any orbital in the subshell.
- Electrons are added to a subshell with the aforementioned value of the spin quantum number until each orbital in the subshell has at to the lowest degree one electron.
When the time comes to place two electrons into the iip subshell we put one electron into each of two of these orbitals. (The choice between the 2px , 2py , and 2pz orbitals is purely arbitrary.)
C (Z = 6): ones 2 iis 2 iipten i 2py i
The fact that both of the electrons in the 2p subshell take the same spin breakthrough number tin can be shown by representing an electron for which due south = +1/two with an
arrow pointing upward and an electron for which south = -1/2 with an arrow pointing downwards.
The electrons in the 2p orbitals on carbon tin therefore be represented every bit follows.
When we become to N (Z = seven), we have to put 1 electron into each of the three degenerate iip orbitals.
| North (Z = seven): | 1s ii 2south 2 2p iii |  |
Considering each orbital in this subshell now contains one electron, the next electron added to the subshell must accept the contrary spin quantum number, thereby filling i of the twop orbitals.
| O (Z = 8): | 1southward ii 2s 2 2p 4 |  |
The ninth electron fills a second orbital in this subshell.
| F (Z = nine): | ones 2 iis 2 2p 5 |  |
The tenth electron completes the twop subshell.
| Ne (Z = 10): | 1south two 2south two 2p half-dozen |  |
There is something unusually stable most atoms, such as He and Ne, that have electron configurations with filled shells of orbitals. By convention, we therefore write abbreviated electron configurations in terms of the number of electrons beyond the previous element with a filled-shell electron configuration. Electron configurations of the next two elements in the periodic table, for instance, could be written every bit follows.
Na (Z = 11): [Ne] 3s 1
Mg (Z = 12): [Ne] 3southward 2
The aufbau process can be used to predict the electron configuration for an chemical element. The actual configuration used by the element has to be determined experimentally. The experimentally determined electron configurations for the elements in the get-go four rows of the periodic table are given in the table in the following section.
The Electron Configurations of the Elements
(1st, 2nd, 3rd, and quaternary Row Elements)
| Diminutive Number | Symbol | Electron Configuration | ||
| ���������������������������������������������������������������� | ||||
| ane | H | 1due south 1 | ||
| 2 | He | 1s two = [He] | ||
| 3 | Li | [He] 2s ane | ||
| 4 | Be | [He] 2s 2 | ||
| v | B | [He] twosouthward 2 2p 1 | ||
| 6 | C | [He] 2southward two 2p 2 | ||
| 7 | N | [He] 2due south two 2p 3 | ||
| viii | O | [He] iis two twop 4 | ||
| nine | F | [He] 2due south 2 2p five | ||
| 10 | Ne | [He] twosouthward 2 2p vi = [Ne] | ||
| xi | Na | [Ne] 3southward 1 | ||
| 12 | Mg | [Ne] 3due south 2 | ||
| thirteen | Al | [Ne] 3south 2 3p 1 | ||
| 14 | Si | [Ne] 3s 2 3p 2 | ||
| 15 | P | [Ne] 3southward 2 threep 3 | ||
| xvi | S | [Ne] 3due south 2 threep four | ||
| 17 | Cl | [Ne] 3southward 2 iiip 5 | ||
| xviii | Ar | [Ne] threesouthward two 3p six = [Ar] | ||
| 19 | 1000 | [Ar] 4south 1 | ||
| 20 | Ca | [Ar] ivsouth 2 | ||
| 21 | Sc | [Ar] 4s two 3d 1 | ||
| 22 | Ti | [Ar] ivdue south 2 3d ii | ||
| 23 | 5 | [Ar] 4south 2 3d 3 | ||
| 24 | Cr | [Ar] 4s 1 3d 5 | ||
| 25 | Mn | [Ar] 4south 2 3d v | ||
| 26 | Fe | [Ar] 4s two 3d vi | ||
| 27 | Co | [Ar] 4s 2 3d 7 | ||
| 28 | Ni | [Ar] 4s 2 3d 8 | ||
| 29 | Cu | [Ar] 4s 1 3d x | ||
| 30 | Zn | [Ar] ivs 2 iiid 10 | ||
| 31 | Ga | [Ar] 4s two 3d 10 4p 1 | ||
| 32 | Ge | [Ar] 4s ii 3d 10 4p 2 | ||
| 33 | As | [Ar] 4southward ii 3d x fourp three | ||
| 34 | Se | [Ar] 4s 2 3d x 4p 4 | ||
| 35 | Br | [Ar] 4s 2 3d 10 ivp 5 | ||
| 36 | Kr | [Ar] 4s 2 3d x 4p vi = [Kr] | ||
Exceptions to Predicted Electron Configurations
There are several patterns in the electron configurations listed in the table in the previous section. One of the most hit is the remarkable level of understanding between these configurations and the configurations we would predict. At that place are merely 2 exceptions amidst the offset 40 elements: chromium and copper.
Strict adherence to the rules of the aufbau process would predict the following electron configurations for chromium and copper.
| predicted electron configurations: | Cr (Z = 24): [Ar] 4s two 3d 4 | |
| Cu (Z = 29): [Ar] 4s 2 3d 9 |
The experimentally determined electron configurations for these elements are slightly different.
| actual electron configurations: | Cr (Z = 24): [Ar] foursouthward one 3d 5 | |
| Cu (Z = 29): [Ar] 4southward ane 3d ten |
In each case, one electron has been transferred from the ivsouthward orbital to a 3d orbital, even though the 3d orbitals are supposed to be at a higher level than the foursouth orbital.
Once we get beyond atomic number 40, the deviation between the energies of adjacent orbitals is small enough that it becomes much easier to transfer an electron from one orbital to some other. Most of the exceptions to the electron configuration predicted from the aufbau diagram shown earlier therefore occur among elements with diminutive numbers larger than 40. Although it is tempting to focus attention on the handful of elements that accept electron configurations that differ from those predicted with the aufbau diagram, the astonishing thing is that this unproblematic diagram works for so many elements.
Electron Configurations and the Periodic Tabular array
When electron configuration information are arranged and so that nosotros tin compare elements in one of the horizontal rows of the periodic table, we find that these rows typically represent to the filling of a trounce of orbitals. The 2nd row, for instance, contains elements in which the orbitals in the n = 2 shell are filled.
| Li (Z = 3): | [He] 2southward ane | |
| Be (Z = 4): | [He] iisouth ii | |
| B (Z = 5): | [He] 2s 2 2p 1 | |
| C (Z = vi): | [He] iis ii 2p two | |
| N (Z = vii): | [He] twos ii 2p three | |
| O (Z = 8): | [He] twos two 2p 4 | |
| F (Z = ix): | [He] 2s ii twop v | |
| Ne (Z = 10): | [He] 2s 2 iip 6 |
There is an obvious pattern within the vertical columns, or groups, of the periodic table every bit well. The elements in a group accept like configurations for their outermost electrons. This relationship tin can be seen by looking at the electron configurations of elements in columns on either side of the periodic tabular array.
| Group IA | Group VIIA | |||||
| H | 1south 1 | |||||
| Li | [He] 2s 1 | F | [He] 2s 2 2p five | |||
| Na | [Ne] 3s 1 | Cl | [Ne] 3s 2 3p 5 | |||
| Chiliad | [Ar] 4s 1 | Br | [Ar] ivs 2 3d ten 4p 5 | |||
| Rb | [Kr] fives 1 | I | [Kr] vs 2 4d 10 fivep 5 | |||
| Cs | [Xe] 6s one | At | [Xe] sixs ii 4f 14 vd 10 6p 5 |
The figure below shows the relationship betwixt the periodic table and the orbitals being filled during the aufbau process. The two columns on the left side of the periodic table represent to the filling of an due south orbital. The next 10 columns include elements in which the v orbitals in a d subshell are filled. The 6 columns on the right represent the filling of the 3 orbitals in a p subshell. Finally, the fourteen columns at the bottom of the table represent to the filling of the 7 orbitals in an f subshell.
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Source: https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/quantum.html
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