Quantum numbers and Atomic orbitals

Quantum Numbers:

Quantum number are a set of numbers which contain all the information about an electron, i.e., its energy , type of orbital etc.They are:

i. Principal quantum number(n): It specifies the position and energy of the electron, i.e, the energy level or the shell the electron belongs to and the its energy. It is denoted by letter singlequotensinglequote and takes integral values starting from 1.It gives the information about the average distance of the electron from the nucleus (the size of the electron cloud ) and energy of the electron in hydrogen like atom and hydrogen like entities.For nth shell the energy is given by the expression:

               En = -2π²me⁴ / n²h²  .where the symbols have their usual meanings. In case , of multi electron entities the energy given by this expression is only approximate.

 ii.Azimuthal quantum number(l) ( also called angular momentum quantum number or subsidiary quantum number) : It denotes the subshell present within any shell.These subshells have different energy and angular momentum and they explian the presence of fine lines in the spectra of elements.

 l can take values starting from 0 to n-1, i.e., l = 0,1,2,3,................ (n-1).

Subshells are denoted by letter s,p,d,f,g etc corresponding to the values of l = 0,1,2,3 .. respectively.The letters s,p ,d,f, g stand for sharp, principle, diffused, fundamental, degenerate spectral lines respectively.Thus, the azimuthal quantum number gives the following information:

           a.The number of subshells present.

           b. The angular momentum of the electrons present in these subshells.

           c. The shapes of the subshells.

• For l = 0 , the subshell is called s subshell. Its shape is spherical.

• For l = 1 , the subshell is called the p subshell, its shape is like a dumb-bell.

• For l = 2, the the subshell is called the p subshell, its shape is like a dumb-bell.

• For, l = 3, the subshell is called the f subshell.

In order of increasing energy they can be arranged as :

          s < p < d < f.

Any subshell is represented by the combination of numbers of qunatum numbers singlequotenlsinglequote .

For example, for n = 1 and l = 0 the subshell is 1s ( note here that we use the name instead of the values of the number l ) .

For n = 3 and l = 2 the subshell is 3d..

A subshell is only possible when l < n .

Total number of subshells present is equal to the numerical value of the principal quantum number n.

Orbital angular momentum (L)  depends upon l and is given by: 

                                                        L  =  [ l(l+1) ]^1/2 h / 2π

And , the magnitude of the magnetic moment (μ_L )  is: 

                                                                    μ_L = eh/ 4πmc X  [ l(l+1) ]^1/2

Magnetic Quantum number ( m or m_l) : This quantum number denotes the spatial orientations of atomic orbitals.It also explains Zeeman effect (the splitting of spectral lines in the presence of magnetic field) and Starksinglequotes effect (the splitting of spectral lines in presence of magnetic field).

 This is because any charge in motion (electron in our case) produces electric field , which in turn produces magnetic field which interacts with the external field and the elliptical orbitals orient themselves in certain directions according to the field.So, the magnetic quantum numbers determines the number of preferred orientations of the electrons iun the subshell.

Each orientation consitutes a different orbital, so the magnetic quantum number decides the number of orbitals in a subshell.Also, m ( or m_l) can take values ranging from -l to +l (l being the azimuthal quantum number).So, for each value of l there are 2l + 1  values of m.

i. For l = 0 (s subshell), m = 2l + 1 = 1 , therefore, s subshell has only one orbital called the s - orbital

ii. For l = 1 ( p subshell ) m = 2l + 1 = 3 , p - orbitals are three in number.These are oriented along the x, y and z axes and are denoted as p_x , p_y and p_z.

iii. Similarly d orbitals ( l = 2) are five in number . They are denoted as d_z², d_xz, d_yz, d_x²-y² and d_xy.

Also, all the orbitals within the same subshell have the same energy. So , all the p orbitals have the same energy .Similarly, all the d orbitals have the same energy.So,the orbitals within the same subshell having identical energy levels (in absence of any external field) are called degenerate orbitals.

The absence of  external field is important in the definition because in the presence of external field the orbitals do not remain degenerate.

Spin Quantum number(s or m_s): Since an orbitals contains two electrons, another quantum number comes into play to differentiate between the two electrons in the same orbitals (becuase they have opposite spins).

 The two electrons move in clockwise and anti clockwise directions , and they have a magnetic moment associated with them (because of their spin) and the magnetic moments of two electrons cancel each other out.

This explains why some substances are diamagnetic while some are paramagnetic.Diamagnetic substances are repelled by external magnetic field, it happens when the magnetic moment of a substance is zero or all the orbitals of the atom are completely filled.

While, paramagnetic subtances are those which are attracted by the external magnetic field due to a net magnetic moment because some of the orbitals are not completely filled.

Their magnetic moment (μ) is given by the formula [N(N+2)]^1/2  BM (Bohr Magneton).

The two quantum numbers are denoted by ½ and -½  or by ? and ? ( for clockwise and anti clockwise spins respectively)

Since , an orbital can only contain 2 electrons , so there are two electrons in s sub-shell , 6 electrons in p sub-shell, 10 in d sub-shell and 14 in f sub-shell.

Spin angular momentum = [S(S+ 1)]^1/2 (h / 2π)  where S = 1/2

Spin magnetic moment  (μ_s)  = [S(S+ 1)]^1/2 eh / 2 π m c.

[SolvedExample]

Using s,p,d,f notations , describe the orbital with the following quantum numbers.

a) n = 2, l = 1  b) n = 4 l = 0   c) n = 5 , l = 3  d) n = 3, l = 2

Solution:

a) 2p

b) 4s

c) 5f

d) 3d

[/SolvedExample]

Pauli Exclusion Principle:

Wolfgang Pauli, a german physicist , gave a quamtum mechanical principle that for electrons in an atom, no two electrons can have same set of quantum numbers i.e., if n , l , m_l are same, so m_s should be different so that they have opposite spins.

In other words , an orbitals can have at most two electrons and they must have opposite spins.

[SolvedExample]

Example:What is the total number of orbitals associated with the principal quantum number n = 3 ?

Solution:For n = 3, l = 0,1,2 , i.e.,the subshells are 3s, 3p and 3d. 

Now, ssubshell contains one orbital, p subshell contains three orbitals and d subshell contains 5 orbitals.

Therfore, total number of orbitals = 9.

[/SolvedExample] 

  Remember: Number of subshell in nth shell = n.  (i.e., l = n)

Number of orbitals in a subshell = 2l + 1. 

No. of electrons possible in a subshell = 2(2l + 1). 

Number of orbitals in a nth shell = n2.

Maximum possible electrons in nth shell = 2n². 

Shapes of atomic orbitals:

The shape of atomic orbitals depends upon the probability density or the wave function of the electron.The wave function doesnsinglequotet vary uniformly in the space surounding the nucleus,but keeps on varying.The region where the probability of finding the electron becomes zero is called a node. In other words, the point (or set of points) where the graph of wave function changes its sign from +ve to -ve represents a node.

There are two types of nodes: 

                                                     i. Spherical / radical node.

                                                      ii. Planar node.

S - orbitals have only spherical nodes whereas p and d orbitals have both spherical and planar nodes.

S orbitals are spherical in shape because the probability of finding the electron is constant in all directions.Also, for s orbitals , azimuthal quantum number l = 0 so magnetic moment = zero.Therefore, s orbitals have only one possible orientation.This also explains their spherical shape.

P orbitals : In p orbitals the probability of finding the electron is high in two identical lobes on the opposite side of each other .This orientation gives them the shape of a dumb-bell.And, the plane perpendicular to the axis of the lobes is a nodal plane because the probability of finding the electron along that plane is zero.

For, p orbital l = 1 , therefore , m is -1, 0 and +1.Therefore, p orbitals have three possible orientations namely , p_x , p_y and p_z.

The shapes of s and  p orbitals. Please  note that the p orbitals  are like dumb - bells ( Source : http://course1.winona.edu)

D orbitals:

 For d orbitals, l = 2.

Therefere, the magnetic quantum number has values -2 , -1, 0, 1 and 2. Therefore, there are five d orbitals, Except d_z² other d orbitals have a double dumb bell shape.

Shapes of d orbitals Source (http://www.chemistry.ucsc.edu)

Formula for the number of nodes:

Number of spherical nodes = n - l - 1.

Number of planar nodes = l .

Total number of nodes in any orbital = n-1.

Note: All d orbitals have nodal planes except d_z² which has none.