Quantum Number Help:

Quantum number help

 

Three quantum numbers (n, l, and ml) are used to specify a particular spatial orbital (Ψ).  Only certain Ψs are allowed, and only certain values and combinations of quantum numbers are allowed.

 

Quantum numbers:

         Symbol                              Name                                                             Relates to                                            Allowed values

1)               n                principal quantum number                       energy and average radius                         1, 2, 3, …

 

                  As n increases, the energy becomes less negative and the average radius increases.

                  Sometimes you see these energy levels called ‘shells’ or ‘levels’.

                  For a given n, there are n2 orbitals in that level.

 

2)               l                 angular quantum number                          shape and ‘how fast’                             0, 1, 2, …, n – 1

 

                  Within an n, there are ‘sublevels’ or ‘subshells’ specified by an l value.

                  For a given n, there are n – 1 possible values of l.  For example, if:

                           n = 1                          l can be 0

                           n = 2                          l can be 0 or 1

                           n = 3                          l can 0, 1, or 2

                           n = 4                          l can 0, 1, 2, or 3                     and so on

 

                  By convention, certain letters are used to denote a particular l.  Regardless of n, if:      

l = 0                           s

                           l = 1                           p

                           l = 2                           d

                           l = 3                           f                                                     and so on

 

3)               ml              magnetic quantum number                       orientation                                                 0, ±1, ±2, …, ± l

 

                  The particular ml value specifies the orbital within a given subshell (within a given shell)

                  For a given l value, there are 2l +1 possible values of ml.

                 

Regardless of n, if:      

l = 0         (s)              ml = 0 only                               

There is only one component (s orbital) in any s-type subshell (l = 0).

One ‘hotel room’ for electrons.

l = 1         (p)             ml = +1, 0, or -1           

There are three components (p orbitals) in any p-type subshell (l = 1).

Three ‘hotel rooms’ for electrons.

l = 2         (d)             ml = +2,+1,0,-1,or -2          

There are five components (d orbitals) in any d-type subshell (l = 2).

Five ‘hotel rooms’ for electrons.

l = 3         (f)              ml = +3, +2,+1,0,-1,-2, -3

There are seven components (f orbitals) in any f-type subshell (l = 3).

Seven ‘hotel rooms’ for electrons.

 

 

So, if n = 4, what orbitals (possible combinations of quantum numbers) exist?

For n = 4, we know there should be n2 = 42 = 16 orbitals total.

 

First, what types of orbitals are there (types of sublevels)?  We need the possible l values.

n = 4                                   says           l can range from 0 up to n – 1 = 4 – 1 = 3

So, l can be 0, 1, 2, or 3

                                                                        The 4s, 4p, 4d, and 4f sublevels exist (n = 4; l = 0, 1, 2, 3, respectively)

 

Next, in each sublevel, which orbitals exist?  We need the possible ml values for each l.

n = 4, l = 0               4s                       ml = 0                                         

There is one 4s orbital (2l +1 = 1):    4s

                  Note:  any s-type sublevel has one orbital only

                 

                  Possible combination of quantum numbers to indicate 4s:

                           n = 4, l = 0, ml = 0

 

n = 4, l = 1               4p                       ml = +1, 0, or -1           

There are three 4p orbitals (2l +1 = 3): 4px, 4py, 4pz

                  Note:  any p-type sublevel has three orbitals

                 

                  Possible combinations of quantum numbers to indicate which 4p orbital:

n = 4, l = 1, ml = +1                               

                                             n = 4, l = 1, ml =   0                                

                                             n = 4, l = 1, ml = -1                                 The different ml values indicate different orientations.

 

 

n = 4, l = 2               4d                       ml = +2, +1, 0, -1, -2  

There are five 4d orbitals (2l +1 = 5): 4dx2-y2, 4dz2, dxy, dxz, dyz

                  Note:  any d-type sublevel has five orbitals

                 

                  Possible combinations of quantum numbers to indicate which 4d orbital:

                                             n = 4, l = 2, ml = +2                               

                                             n = 4, l = 2, ml = +1                               

                                             n = 4, l = 2, ml =   0                                

                                             n = 4, l = 2, ml = -1                                

                                             n = 4, l = 2, ml = -2                                 The different ml values indicate different orientations.

 

 

n = 4, l = 3               4f                        ml = +3, +2, +1, 0, -1, -2, -3    

There are seven 4f orbitals (2l +1 = 7): don’t worry about labels – there are seven

                  Note:  any f-type sublevel has seven orbitals

                  Possible combinations of quantum numbers to indicate which 4f orbital:

                                             n = 4, l = 3, ml = +3                               

                                             n = 4, l = 3, ml = +2                               

                                             n = 4, l = 3, ml = +1                               

                                             n = 4, l = 3, ml =   0                                

                                             n = 4, l = 3, ml = -1                                

                                             n = 4, l = 3, ml = -2     

                                             n = 4, l = 3, ml = -3                                 The different ml values indicate different orientations.

 

 

1 s orbital +  3  p orbitals +  5  d orbitals +  7  f orbitals = 16 total orbitals, as expected for n = 4.