**Quantum Numbers and ElectronConfigurations**

Quantum Numbers | Rules Governing Quantum Numbers | Shells and Subshells of Orbitals | Possible Combinations of Quantum Numbers | Table: Summary of Allowed Combinations of Quantum Numbers |

Relative Energies of Atomic Orbitals | Electron Configurations, the Aufbau Principle, Degenerate Orbitals, and Hund's Rule | Table: The Electron Configurations of the Elements | Exceptions to Predicted Electron Configurations | Electron Configurations and the Periodic Table |

The Bohr model was a one-dimensional model that used one quantum number to describe thedistribution 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 modelallowed the electron to occupy three-dimensional space. It therefore required threecoordinates, or three **quantum numbers**, to describe the orbitals in which electronscan 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 numbersdescribe 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 ofthe atom. Energy must therefore be absorbed to excite an electron from an orbital in whichthe electron is close to the nucleus (*n* = 1) into an orbital in which it is furtherfrom the nucleus (*n* = 2). The principal quantum number therefore indirectlydescribes 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 valueof 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, canpoint in different directions. We therefore need a third quantum number, known as the **magneticquantum number** (*m*), to describe the orientation in space of a particularorbital. (It is called the *magnetic* quantum number because the effect of differentorientations of orbitals was first observed in the presence of a magnetic field.)

**Rules Governing the Allowed Combinations ofQuantum 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.

**Practice Problem 7:**Describe the allowed combinations of the *n*, *l*, and *m* quantum numbers when *n* = 3.

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**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 theangular quantum number. Chemists describe the shell and subshell in which an orbitalbelongs with a two-character code such as 2*p* or 4*f*. The first characterindicates the shell (*n* = 2 or *n* = 4). The second character identifies thesubshell. By convention, the following lowercase letters are used to indicate differentsubshells.

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). Someof the allowed combinations of the *n* and *l* quantum numbers are shown in thefigure 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 shellto be equal to the principal quantum number for the shell. The *n* = 3 shell, forexample, 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 inwhich a sphere can be oriented in space. The only allowed combination of quantum numbersfor 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 | -1 | ||||

2 | 1 | 0 | 2p | |||

2 | 1 | 1 |

There is only one orbital in the 2*s* subshell. But, there are three orbitals inthe 2*p* subshell because there are three directions in which a *p* orbital canpoint. 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 figurebelow. These orbitals are therefore known as the 2*p _{x}*, 2

*p*,and 2

_{y}*p*orbitals.

_{z}There are nine orbitals in the *n* = 3 shell.

n | l | m | ||||

3 | 0 | 0 | 3s | |||

3 | 1 | -1 | ||||

3 | 1 | 0 | 3p | |||

3 | 1 | 1 | ||||

3 | 2 | -2 | ||||

3 | 2 | -1 | 3d | |||

3 | 2 | 0 | ||||

3 | 2 | 1 | ||||

3 | 2 | 2 |

There is one orbital in the 3*s* subshell and three orbitals in the 3*p*subshell. The *n* = 3 shell, however, also includes 3*d* orbitals.

The five different orientations of orbitals in the 3*d* subshell are shown in thefigure below. One of these orbitals lies in the *XY* plane of an *XYZ*coordinate system and is called the 3*d*_{xy} orbital. The 3*d*_{xz}and 3*d*_{yz} orbitals have the same shape, but they lie between the axes ofthe coordinate system in the *XZ* and *YZ* planes. The fourth orbital in thissubshell lies along the *X* and *Y* axes and is called the 3*d _{x}*

^{2}

_{-y}

^{2}orbital. Most of the space occupied by the fifth orbital lies along the

*Z*axis andthis orbital is called the 3

*d*

_{z}^{2}orbital.

The number of orbitals in a shell is the square of the principal quantum number: 1^{2}= 1, 2^{2} = 4, 3^{2} = 9. There is one orbital in an *s* subshell (*l*= 0), three orbitals in a *p* subshell (*l* = 1), and five orbitals in a *d*subshell (*l* = 2). The number of orbitals in a subshell is therefore 2(*l*) +1.

Before we can use these orbitals we need to know the number of electrons that canoccupy an orbital and how they can be distinguished from one another. Experimentalevidence suggests that an orbital can hold no more than two electrons.

To distinguish between the two electrons in an orbital, we need a fourth quantumnumber. This is called the **spin quantum number** (*s*) because electrons behaveas if they were spinning in either a clockwise or counterclockwise fashion. One of theelectrons in an orbital is arbitrarily assigned an *s* quantum number of +1/2, theother is assigned an *s* quantum number of -1/2. Thus, it takes three quantum numbersto define an orbital but four quantum numbers to identify one of the electrons that canoccupy the orbital.

The allowed combinations of *n*, *l*, and *m* quantum numbers for thefirst four shells are given in the table below. For each of these orbitals, there are twoallowed values of the spin quantum number, *s*.

*Summary of Allowed Combinations of QuantumNumbers*

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 | 2,1,0,-1,-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 | ||||||

4 | 3 | 3,2,1,0,-1,-2,-3 | 4f | 7 | 14 | 32 |

**The Relative Energies of Atomic Orbitals**

Because of the force of attraction between objects of opposite charge, the mostimportant factor influencing the energy of an orbital is its size and therefore the valueof 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. The3*s*, 3*p*, and 3*d* orbitals, for example, have the same energy in ahydrogen atom. The Bohr model, which specified the energies of orbits in terms of nothingmore than the distance between the electron and the nucleus, therefore works for thisatom.

The hydrogen atom is unusual, however. As soon as an atom contains more than oneelectron, 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 graduallybecomes larger as the value of the angular quantum number becomes larger.

Relative energies: *s* < *p* < *d*< *f*

A very simple device can be constructed to estimate the relativeenergies of atomic orbitals. The allowed combinations of the *n* and *l* quantumnumbers are organized in a table, as shown in the figure below and arrows are drawn at 45degree angles pointing toward the bottom left corner of the table.

The order of increasing energy of the orbitals is then read off by following thesearrows, 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 energyfor 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* ...

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**Electron Configurations, the Aufbau Principle,Degenerate Orbitals, and Hund's Rule**

The **electron configuration** of an atom describes the orbitals occupied byelectrons on the atom. The basis of this prediction is a rule known as the **aufbauprinciple**, which assumes that electrons are added to an atom, one at a time, startingwith the lowest energy orbital, until all of the electrons have been placed in anappropriate orbital.

A hydrogen atom (*Z* = 1) has only one electron, which goes into the lowest energyorbital, 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* orbitalbecause there are only two possible values for the spin quantum number used to distinguishbetween 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 energyorbitals are the three 2*p* orbitals. The fifth electron therefore goes into one ofthese 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 electrongo into the same orbital as the first, or does it go into one of the other orbitals inthis subshell?

To answer this, we need to understand the concept of **degenerate orbitals**. Bydefinition, orbitals are *degenerate* when they have the same energy. The energy ofan orbital depends on both its size and its shape because the electron spends more of itstime further from the nucleus of the atom as the orbital becomes larger or the shapebecomes more complex. In an isolated atom, however, the energy of an orbital doesn'tdepend on the direction in which it points in space. Orbitals that differ only in theirorientation in space, such as the 2*p _{x}*, 2

*p*, and 2

_{y}*p*orbitals, are therefore degenerate.

_{z}Electrons fill degenerate orbitals according to rules first stated by Friedrich Hund. **Hund'srules** can be summarized as follows.

- One electron is added to each of the degenerate orbitals in a subshell before two electrons are added to any orbital in the subshell.
- Electrons are added to a subshell with the same value of the spin quantum number until each orbital in the subshell has at least one electron.

When the time comes to place two electrons into the 2*p* subshell we put oneelectron into each of two of these orbitals. (The choice between the 2*p _{x}*,2

*p*, and 2

_{y}*p*orbitals is purely arbitrary.)

_{z}C (*Z* = 6): 1*s*^{2} 2*s*^{2} **2 p_{x}^{1}2p_{y}^{1}**

The fact that both of the electrons in the 2*p* subshell have the same spinquantum number can be shown by representing an electron for which *s* = +1/2 with an

arrow pointing up and an electron for which *s* = -1/2 with an arrow pointingdown.

The electrons in the 2*p* orbitals on carbon can therefore be represented asfollows.

When we get to N (*Z* = 7), we have to put one electron into each of the threedegenerate 2*p* orbitals.

N (Z = 7): | 1s^{2} 2s^{2} 2p^{3} |

Because each orbital in this subshell now contains one electron, the next electronadded to the subshell must have the opposite spin quantum number, thereby filling one ofthe 2*p* orbitals.

O (Z = 8): | 1s^{2} 2s^{2} 2p^{4} |

The ninth electron fills a second orbital in this subshell.

F (Z = 9): | 1s^{2} 2s^{2} 2p^{5} |

The tenth electron completes the 2*p* subshell.

Ne (Z = 10): | 1s^{2} 2s^{2} 2p^{6} |

There is something unusually stable about atoms, such as He and Ne, that have electronconfigurations with filled shells of orbitals. By convention, we therefore writeabbreviated electron configurations in terms of the number of electrons beyond theprevious element with a filled-shell electron configuration. Electron configurations ofthe next two elements in the periodic table, for example, could be written as follows.

Mg (*Z* = 12): [Ne] 3*s*^{2}

**Practice Problem 8:**Predict the electron configuration for a neutral tin atom (Sn, *Z* = 50).

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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. Theexperimentally determined electron configurations for the elements in the first four rowsof the periodic table are given in the table in the following section.

*Learning Activity*

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*The Electron Configurations of the Elements *

*(1st, 2nd, 3rd, and 4th Row Elements)*

Atomic Number | Symbol | Electron Configuration | ||

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ | ||||

1 | H | 1s^{1} | ||

2 | He | 1s^{2} = [He] | ||

3 | Li | [He] 2s^{1} | ||

4 | Be | [He] 2s^{2} | ||

5 | B | [He] 2s^{2} 2p^{1} | ||

6 | C | [He] 2s^{2} 2p^{2} | ||

7 | N | [He] 2s^{2} 2p^{3} | ||

8 | O | [He] 2s^{2} 2p^{4} | ||

9 | F | [He] 2s^{2} 2p^{5} | ||

10 | Ne | [He] 2s^{2} 2p^{6} = [Ne] | ||

11 | Na | [Ne] 3s^{1} | ||

12 | Mg | [Ne] 3s^{2} | ||

13 | Al | [Ne] 3s^{2} 3p^{1} | ||

14 | Si | [Ne] 3s^{2} 3p^{2} | ||

15 | P | [Ne] 3s^{2} 3p^{3} | ||

16 | S | [Ne] 3s^{2} 3p^{4} | ||

17 | Cl | [Ne] 3s^{2} 3p^{5} | ||

18 | Ar | [Ne] 3s^{2} 3p^{6} = [Ar] | ||

19 | K | [Ar] 4s^{1} | ||

20 | Ca | [Ar] 4s^{2} | ||

21 | Sc | [Ar] 4s^{2} 3d^{1} | ||

22 | Ti | [Ar] 4s^{2} 3d^{2} | ||

23 | V | [Ar] 4s^{2} 3d^{3} | ||

24 | Cr | [Ar] 4s^{1} 3d^{5} | ||

25 | Mn | [Ar] 4s^{2} 3d^{5} | ||

26 | Fe | [Ar] 4s^{2} 3d^{6} | ||

27 | Co | [Ar] 4s^{2} 3d^{7} | ||

28 | Ni | [Ar] 4s^{2} 3d^{8} | ||

29 | Cu | [Ar] 4s^{1} 3d^{10} | ||

30 | Zn | [Ar] 4s^{2} 3d^{10} | ||

31 | Ga | [Ar] 4s^{2} 3d^{10} 4p^{1} | ||

32 | Ge | [Ar] 4s^{2} 3d^{10} 4p^{2} | ||

33 | As | [Ar] 4s^{2} 3d^{10} 4p^{3} | ||

34 | Se | [Ar] 4s^{2} 3d^{10} 4p^{4} | ||

35 | Br | [Ar] 4s^{2} 3d^{10} 4p^{5} | ||

36 | Kr | [Ar] 4s^{2} 3d^{10} 4p^{6} = [Kr] |

**Exceptions to Predicted Electron Configurations **

There are several patterns in the electron configurations listed in the table in theprevious section. One of the most striking is the remarkable level of agreement betweenthese configurations and the configurations we would predict. There are only twoexceptions among the first 40 elements: chromium and copper.

Strict adherence to the rules of the aufbau process would predict the followingelectron configurations for chromium and copper.

predicted electron configurations: | Cr (Z = 24): [Ar] 4s^{2} 3d^{4} | |

Cu (Z = 29): [Ar] 4s^{2} 3d^{9} |

The experimentally determined electron configurations for these elements are slightlydifferent.

actual electron configurations: | Cr (Z = 24): [Ar] 4s^{1} 3d^{5} | |

Cu (Z = 29): [Ar] 4s^{1} 3d^{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 the4*s* orbital.

Once we get beyond atomic number 40, the difference between the energies of adjacentorbitals is small enough that it becomes much easier to transfer an electron from oneorbital to another. Most of the exceptions to the electron configuration predicted fromthe aufbau diagram shown earlier therefore occur among elementswith atomic numbers larger than 40. Although it is tempting to focus attention on thehandful of elements that have electron configurations that differ from those predictedwith the aufbau diagram, the amazing thing is that this simple diagram works for so manyelements.

*Electron Configurations and the Periodic Table*

When electron configuration data are arranged so that we can compare elements in one ofthe horizontal rows of the periodic table, we find that these rows typically correspond tothe filling of a shell of orbitals. The second row, for example, contains elements inwhich the orbitals in the *n* = 2 shell are filled.

Li (Z = 3): | [He] 2s^{1} | |

Be (Z = 4): | [He] 2s^{2} | |

B (Z = 5): | [He] 2s^{2} 2p^{1} | |

C (Z = 6): | [He] 2s^{2} 2p^{2} | |

N (Z = 7): | [He] 2s^{2} 2p^{3} | |

O (Z = 8): | [He] 2s^{2} 2p^{4} | |

F (Z = 9): | [He] 2s^{2} 2p^{5} | |

Ne (Z = 10): | [He] 2s^{2} 2p^{6} |

There is an obvious pattern within the vertical columns, or groups, of the periodictable as well. The elements in a group have similar configurations for their outermostelectrons. This relationship can be seen by looking at the electron configurations ofelements in columns on either side of the periodic table.

Group IA | Group VIIA | |||||

H | 1s^{1} | |||||

Li | [He] 2s^{1} | F | [He] 2s^{2} 2p^{5} | |||

Na | [Ne] 3s^{1} | Cl | [Ne] 3s^{2} 3p^{5} | |||

K | [Ar] 4s^{1} | Br | [Ar] 4s^{2} 3d^{10} 4p^{5} | |||

Rb | [Kr] 5s^{1} | I | [Kr] 5s^{2} 4d^{10} 5p^{5} | |||

Cs | [Xe] 6s^{1} | At | [Xe] 6s^{2} 4f^{14} 5d^{10} 6p^{5} |

The figure below shows the relationship between the periodic table and the orbitalsbeing filled during the aufbau process. The two columns on the left side of the periodictable correspond to the filling of an *s* orbital. The next 10 columns includeelements in which the five orbitals in a *d* subshell are filled. The six columns onthe right represent the filling of the three orbitals in a *p* subshell. Finally, the14 columns at the bottom of the table correspond to the filling of the seven orbitals inan *f* subshell.

**Practice Problem 9:**Predict the electron configuration for calcium (*Z* = 20) and zinc (*Z* = 30) from their positions in the periodic table.

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