topic-3: electrons in atoms(summer course). quantum numbers and electron orbitals in the wave...

TOPIC-3: ELECTRONS IN ATOMS(Summer course)

Quantum Numbers and Electron Orbitals

In the wave mechanics the electrons in an atom composed of more than one electron are distributed in the shells. The shells are composed of one subshell or many subshells,the subshells are made up of one orbital or many orbitals. Each electron of an atom is defined through three quantum numbers referring to the shell, subshell and orbital.

Quantum Numbers and Electron Orbitals

• Principal Quantum number, n: The energy levels in atom are divided into the shells represented by the principle quantum number, “n”. As in the Bohr quantum theory, it may have only positive, nonzero (n = 1, 2, 3, …..) integral values. In addition to the numbers, to indicate the layers, some letters are also used. The shells are the regions where electrons are more likely to be found. The greater the n value, the farer the shell from the nucleus.

• 1 2 3 45...

• K L M N O …

Quantum Numbers and Electron Orbitals

Angular momentum quantum number, l: Energy levels include sub-energy levels. Consequently, shells are seperated into subshells each of which is represented with angular momentum quantum number “l” .This determines the geometrical shape of the electron probability distribution. The number “l” can have all values ranging from 0, 1, 2 to n-1. For n=1 the maximum and unique value of “l” is 0 which means that the level K contains one sub-level. For n=2 , “l” will have 0 and 1 values. Thus, L level is composed of two sub-levels. The total number of sub-levels in a level is equal to the principal quantum number. The sub-shells are indicated as below:

0 1 2 3 4 5 6 …s p d f g h i …

Quantum Numbers and Electron Orbitals

• To indicate a sub-shell in a shell, the principal quantum number “n” and the angular momentum quantum number are written next to each other . For the second shell (L), the subshells s and p are indicated as 2s (n = 2, l = 0) and 2p (n = 2, l =1 ) .

• Magnetic quantum number, ml: Each subshell is composed of one or more orbitals and each orbit in a sub-shell is defined as magnetic quantum number “ml”. This number may be a positive or negative integer including zero and ranging from – l to +l.

Principal quantum number

n

Orbital quantum number

l

Sub-shell

Magnetic quantum number

ml

The number

of orbitals in the sub-shell

3 0 3s 0 1

  1 3p -1,0,1 3

  2 3d -2,-1,0,1,2 5

Quantum numbers and Electron Orbitals

The shells and sub-shells of Hydrogen atom

s orbitalss orbital: Spherically symmetric

p orbitals

p orbital: Electron density is in form of a dumbbell.Two lobes are seperated by a nodal plane in which charge density drops to zero.

d orbitald orbital: There are 5 different type of d orbitals. Their orientations vary respectively.

Electron Spin-The fourth Quantum NumberStern-Gerlach experimentAg atoms vaporized in the oven are collimated into a beam by the slit and the beam is passed through a non-uniform magnetic field. The beam splits in two with two opposite directions ( A spinning unpaired electron behaves as two magnets with opposite pole directions.

Electron SpinSpin magnetic quantum number, ms: An electron generates a magnetic field because of its spin on its axis. As a result of this action ( spin at one direction, and at the opposite direction) the spin magnetic quantum number may have values : ms=+1/2 ve ms=-1/2.

Multielectron Atoms• Schrödinger developed his wave equation for the hydrogen atom. For multielectron atoms a new factor arises: mutual repulsions between electrons. Because exact electron positions are not known,electron repulsions can only be approximated.

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Multielectron Atoms• In multielectron atoms the attractive force of nucleus for a given electron increases as the nuclear charge rises, which leads to a decrease of the energy level of an orbital. Hence, multielectron atoms have different orbital energies

The orbitals with different energy levels

The orbital energy level decreases with rising nucleus charge

The electron configuration of an atom is a designation of how electrons are distributed among various orbitals.

Rules for Assigning Electrons in Orbitals

1. Electrons occupy orbitals in a way that minimizes the energy of the atom.

Electron Configurations

Electron Configurations• The diagram shows the order in which electrons

occupy orbitals in these shells, first 1s then 2s and 2p and so on. The order of the filling of orbitals has been established by experiment, principally through spectroscopy and it is the order that we must follow in assigning electron configurations to the element. Except for a few elements the order in which the orbitals fill in:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

Electron Configuration

2. No two electrons in an atom may have all four quantum numbers alike (Pauli exclusion principle).

3. When orbitals of identical energy are available, electrons initially occupy these orbitals singly. As a result of this rule, known as Hund’s rule an atom tends to have as many unpaired electrons as possible. The electrons do this by seeking out empty orbitals of similar energy in preference to pairing up with other electrons in half-filled orbitals.

Notation of Electron Configuration

Since the atomic number of Carbon element is 6, in all of three indications there are 6 electrons. The electrons are those with parallel spins which occupy different orbitals in the same sub-shell singly.

spdf notation (condensed)

spdf notation (expanded)

orbital diagram

The Aufbau ProcessAufbau means “constructing or building” and what we do is assign electron configurations to the elements in order of increasing atomic number.

The Electron Configuration of some elements(C, N, Ne, Na)

Valence Electrons• Electrons that are added to the electronic shell of highest

principal quantum number(the outermost or valence shell) are called valence electrons. The electron configuration of Na is written below with the neon core ( 1s2s2p6 ) and for the other third period elements only the valence-shell electron configuration is shown.

• Na Mg Al Si P S Cl Ar• [Ne]3s13s2 3s23p13s23p23s23p3 3s23p4 3s23p5 3s23p6

• 6C: [He]2s22p224Cr: [Ar]4s13d5

53I: [Kr]4d105s25p5

The elements of the third period end with Argon. After argon instead of 3d the next sub-shell to fill is 4s.

The 19. electron of potassum occupies 4s instead of 3d orbital since 4s has lower energy level.

Example: Write out the electron configuration of 38Sr, [38Sr]+2 and 26Fe ,[26Fe]+2 in the condensed spdf notation ?

38Sr: 1s22s22p63s23p64s23d104p65s2 (according to the order of orbital energy levels)

Solution• 38Sr: 1s22s22p63s23p63d104s24p65s2 (according to the

increasing principal quantum number “n” )

• 38Sr: [Kr]5s2 (according to the order with the indication of noble gas core electron configuration )

• [38Sr]+2: 1s22s22p63s23p63d104s24p6

• 26Fe: 1s22s22p63s23p64s23d6

• [26Fe]+2: 1s22s22p63s23p63d6