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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 only information that was important was the size of the orbit, which was described by the n quantum number. Schrodinger'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 Schrodinger'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 2 p or
4 f. 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 3 s , 3 p , and 3 d 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 the n = 2 shell.
n l m 2 0 0 2s 2 1 - 2 1 0 2p
2 1 1
There is only one orbital in the 2 s subshell. But, there are three orbitals in the 2 p subshell because there are three directions in which a p orbital can point. One of these orbitals is oriented along the X axis, another along the Y axis, and the third along the Z axis of a coordinate system, as shown in the figure below. These orbitals are therefore known as the 2 px , 2 py , and 2 pz orbitals.
There are nine orbitals in the n = 3 shell.
n l m
Summary of Allowed Combinations of Quantum Numbers
n l m
Subshell Notation
Number of Orbitals in the Subshell
Number of Electrons Needed to Fill Subshell
Total Number of Electrons in Subshell
1 0 0 1s 1 2 2
2 0 0 2s 1 2 2 1 1,0,-1 2p 3 6 8
3 0 0 3s 1 2 3 1 1,0,-1 3p 3 6
3 2
3d 5 10 18
4 0 0 4s 1 2 4 1 1,0,-1 4p 3 6
4 2 2,1,0,-1,- 2
4d 5 10
1,-2,-3 4f^7 14
The Relative Energies of Atomic Orbitals
Because of the force of attraction between objects of opposite charge, the most important factor influencing the energy of an orbital is its size and therefore the value of the principal quantum number, n. For an atom that contains only one electron, there is no difference between the energies of the different subshells within a shell. The 3 s , 3 p , and 3 d orbitals, for example, have the same energy in a hydrogen atom. The Bohr model, which specified the energies of orbits in terms of nothing more than the distance between the electron and the nucleus, therefore works for this atom.
The hydrogen atom is unusual, however. As soon as an atom contains more than one electron, the different subshells no longer have the same energy. Within a given shell, the s orbitals always have the lowest energy. The energy of the subshells gradually becomes larger as the value of the angular quantum number becomes larger.
Relative energies: s < p < d < f
As a result, 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 can be constructed to estimate the relative energies of atomic orbitals. The allowed combinations of the n and l quantum numbers are organized in a table, as shown in the
figure below and arrows are drawn at 45 degree angles pointing toward the bottom left corner of the table.
The order of increasing energy of the orbitals is then read off by following these arrows, starting at the top of the first line and then proceeding on to the second, third, fourth lines, and so on. This diagram predicts the following order of increasing energy for atomic orbitals.
1 s < 2 s < 2 p < 3 s < 3 p <4 s < 3 d <4 p < 5 s < 4 d < 5 p < 6 s < 4 f < 5 d < 6 p < 7 s < 5 f < 6 d < 7 p < 8 s ...
Electron Configurations, the Aufbau Principle, Degenerate Orbitals, and Hund's Rule
The electron configuration of an atom describes the orbitals occupied by electrons on the atom. The basis of this prediction is a rule known as the aufbau principle , which assumes that electrons are added to an atom, one 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 = 1) has only one electron, which goes into the lowest energy orbital, the 1 s orbital. This is indicated by writing a superscript "1" after the symbol for the orbital.
H ( Z = 1): 1 s^1
The next element has two electrons and the second electron fills the 1 s orbital because there are only two possible values for the spin quantum number used to distinguish between the electrons in an orbital.
He ( Z = 2): 1 s^2
The third electron goes into the next orbital in the energy diagram, the 2 s orbital.
Li ( Z = 3): 1 s^2 2 s^1
The fourth electron fills this orbital.
Be ( Z = 4): 1 s^2 2 s^2
After the 1 s and 2 s orbitals have been filled, the next lowest energy orbitals are the three 2 p orbitals. The fifth electron therefore goes into one of these orbitals.
B ( Z = 5): 1 s^2 2 s^2 2 p^1
When the time comes to add a sixth electron, the electron configuration is obvious.
C ( Z = 6): 1 s^2 2 s^2 2 p^2
However, there are three orbitals in the 2 p 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 answer this, we need to understand the concept of degenerate orbitals. By 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 of its time further 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 it points in space. Orbitals that differ only in their orientation in space, such as the 2 px , 2 py , and 2 pz orbitals, are therefore degenerate.
Mg ( Z = 12): [Ne] 3 s^2
Practice Problem 8:
Predict the electron configuration for a neutral tin atom (Sn, Z = 50).
The aufbau process can be used to predict the electron configuration for an element. The actual configuration used by the element has to be determined experimentally. The experimentally determined electron configurations for the elements in the first 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 4th Row Elements)
Atomic Number Symbol Electron Configuration
1 H 1 s^1 2 He 1 s^2 = [He] 3 Li [He] 2 s^1 4 Be [He] 2 s^2 5 B [He] 2 s^2 2 p^1 6 C [He] 2 s^2 2 p^2 7 N [He] 2 s^2 2 p^3 8 O [He] 2 s^2 2 p^4 9 F [He] 2 s^2 2 p^5 10 Ne [He] 2 s^2 2 p^6 = [Ne] 11 Na [Ne] 3 s^1 12 Mg [Ne] 3 s^2 13 Al [Ne] 3 s^2 3 p^1 14 Si [Ne] 3 s^2 3 p^2 15 P [Ne] 3 s^2 3 p^3 16 S [Ne] 3 s^2 3 p^4 17 Cl [Ne] 3 s^2 3 p^5 18 Ar [Ne] 3 s^2 3 p^6 = [Ar] 19 K [Ar] 4 s^1 20 Ca [Ar] 4 s^2 21 Sc [Ar] 4 s^2 3 d^1 22 Ti [Ar] 4 s^2 3 d^2 23 V [Ar] 4 s^2 3 d^3 24 Cr [Ar] 4 s^1 3 d^5
25 Mn [Ar] 4 s^2 3 d^5 26 Fe [Ar] 4 s^2 3 d^6 27 Co [Ar] 4 s^2 3 d^7 28 Ni [Ar] 4 s^2 3 d^8 29 Cu [Ar] 4 s^1 3 d^10 30 Zn [Ar] 4 s^2 3 d^10 31 Ga [Ar] 4 s^2 3 d^10 4 p^1 32 Ge [Ar] 4 s^2 3 d^10 4 p^2 33 As [Ar] 4 s^2 3 d^10 4 p^3 34 Se [Ar] 4 s^2 3 d^10 4 p^4 35 Br [Ar] 4 s^2 3 d^10 4 p^5 36 Kr [Ar] 4 s^2 3 d^10 4 p^6 = [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 striking is the remarkable level of agreement between these configurations and the configurations we would predict. There are only two exceptions among the first 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] 4 s^2 3 d^4 Cu ( Z = 29): [Ar] 4 s^2 3 d^9
The experimentally determined electron configurations for these elements are slightly different.
actual electron configurations: Cr ( Z = 24): [Ar] 4 s^1 3 d^5 Cu ( Z = 29): [Ar] 4 s^1 3 d^10
In each case, one electron has been transferred from the 4 s orbital to a 3 d orbital, even though the 3 d orbitals are supposed to be at a higher level than the 4 s orbital.
Once we get beyond atomic number 40, the difference between the energies of adjacent orbitals is small enough that it becomes much easier to transfer an electron from one orbital to another. Most of the exceptions to the electron configuration predicted from the aufbau diagram shown earlier therefore occur among elements with atomic numbers larger than 40. Although it is tempting to focus attention on the handful of elements that have electron configurations that differ from those predicted with the aufbau diagram, the amazing thing is that this simple diagram works for so many elements.
Electron Configurations and the Periodic Table
