Notes and Study Materials -Atomic Structure

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About this unit

Atomic number, isotopes and isobars. Concept of shells and subshells, dual nature of matter and light, de Broglie’s relationship, Heisenberg uncertainty principle, concept of orbital, quantum numbers, shapes of s,p and d orbitals, rules for filling electrons in orbitals- Aufbau principle, Pauli exclusion principles and Hund’s rule, electronic configuration of atoms, stability of half filled and completely filled orbitals.

ATOM

John Dalton proposed in 1808 that atom is the smallest indivisible particle of matter.

ATOMIC RADII

Atomic radii are of the order of 10–8 cm (1Å) and radii of nuclei are nearly 10–13 cm.
Radius of the nucleus is thus th of the radius of atom.
Radius of atom Radius of nucleus

ELECTRON

  1. It was discovered through the study of Cathode rays (discovered by Zulius Plucker) and the name was proposed by Stoney.
  2. Charge : It was determined by Mullikan by oil drop method as –1.602 × 10–19 coulombs or 4.803 × 10–10 e.s.u.
  3. Mass : It was found by J. J. Thomson as
    9.11 × 10–28 g.
  4. Specific charge : e/m ratio is called specific charge and was determined by Thomson as 1.76 × 108 coulombs/gm.
  5. Radius : It is of the order 10–15 cm.
  6. Density : 2.17 × 1017 g/cc.
  7. Mass of electron at speed v is m =
  8. Atomic mass unit : It is 0.0005486 amu.
  9. Mass of one mole of electron : It is 0.55 mg.

CATHODE RAYS

Originate from cathode. Electrons were discovered by cathode ray experiment.

SOME PROPERTIES OF CATHODE RAYS

  1. They cast shadow of the object in their path
  2. Rotate a mica wheel
  3. Deflected by electric and magnetic fields in a direction showing negative charge.

PROTON (H+)

Discovered by Goldstein (1886) through perforated cathode rays experiment which showed the presence of anode or canal rays.
  1. Mass : It was found to be 1.672 × 10–24 g or 1.672 × 10–27 kg or 1.00728 amu. It is about 1837 times heavier than an electron.
  2. Charge : It carries unit positive charge 1.602 × 10–19 coulombs or 4.803 × 10–10 esu.
  3. Specific charge : It is 9.58 × 104 coulomb/gm. It varies with nature of gas and is maximum if H2 is used.
  4. Charge on 1 mole of proton is 96500 coulomb or 1 Faraday.
  5. Volume : The volume  for proton is approximately 1.5 × 10–38 cm3.

NEUTRON (0N1)

Discovered by Chadwick by bombarding Be or B atoms (sheet) with high speed -particles
  1. Mass : Its mass is 1.675 × 10–24 gm or 1.675 × 10–27 kg or 1.00866 amu.
  2. It is heavier than proton by 0.18%.
  3. Density : Its density is 1.5 × 1014 g/cm3.
  4. Specific Charge : It is zero.
  5. Stability : It is least stable of all elementary particles present in an atom.
  6. Disintegration : Isolated neutron is unstable and disintegrates into electron, proton and neutrino.
  7. Among all elementary particles neutron is the heaviest and least stable.

OTHER SUBATOMIC PARTICLES

  1. Positron (Positive electron +1e0). Discovered by Dirac (1930) and C. D. Anderson (1932). They are highly unstable and produce -rays on combining with electrons.
  2. Neutrino and Antineutrino are particles of small mass and no charge as stated by Fermi (1934). Anti-neutrino spin clockwise and neutrino spin anticlockwise.
  3. Meson : They are unstable particles and include pions  (+, or 0) Kaons (K+, K, K0, K–0) and eta meson ().
Mass : They have mass intermediate of electron and proton.
Discovery : By Yukawa (1935) and Kemmer.
  1. Anti proton (–1p1) : Negative proton produced by Segre and Weigland (1955) by proton-proton and proton-neutron collisions.
  2. v-particles : They may be positive, negative or neutral. Discovered by G. D. Rochester and C C. Butler v and v0 are 2200 times heavier than electron. Heavier disintegrate into pions and lighter into mesons.

THOMSON’S ATOMIC MODEL

Atom is a sphere of positive electricity with a number of electrons distributed within the sphere.

RUTHERFORD’S NUCLEAR MODEL

It is based upon -particles scattering experiment. Only a few (one in 10,000) -particles were returned back from the Au-foil (10–4 mm thick).


Conclusion – Atom consists of two parts – (a) Nucleus (b) Extra nuclear part.
Drawbacks – Model fails to explain the stability of the atoms and line spectrum of hydrogen.

NUCLEUS

  1. Nucleus : It is small heavy  and positively charged material located in the centre of atom and electrons are distributed in extra nuclear part of atom and revolve around the nucleus.
  2. Radius : It is of the order 1.5 × 10–13 cm to
    6.5 × 10–13 cm (1.5 – 6.5 Fermi). In general
Where r0 is a proportionality constant with value 1.4 × 10–13 cm. and A is mass number.
  1. Volume : It is about 10–39 cm3. and that of atom is 10–24 cm3
  2. Density : It is about 1014 g/cm3.   
  3. Diameter : It is about 10–15 m or 1 fm (1 fm = 10–15 m)
  4. Nucleus contains neutrons and protons, collectively known as nucleons.

ATOMIC NUMBER/MOSELEY’S POSTULATES

The number of protons present in an atom is called the atomic number, denoted by Z. Moseley postulated that the frequency of X-rays produced when beam of strong electrons fall on metal target, called anti-cathode is related to the charge present on the nucleus of an atom of the element used as anti cathode. Mathematically
; where = frequency of X-rays, Z = nuclear  charge, a and b = constants

MASS NUMBER

It is sum of number of protons and neutrons present in the nucleus (nucleons as a whole) and denoted by A. It is always a whole number.

AVERAGE ATOMIC MASS

It is the average mass of all existing isotopes and not necessarily a whole number.

ISOTOPES

Isotopes are atoms of the same element having same atomic number but different mass number. e.g. 8O16, 8O17 and 8O18. They were discovered by Soddy (1911).

ISOBARS

Atoms of different elements having same mass number but different atomic numbers e.g. 19K40, 20Ca40.

ISOTONES

Atoms of different elements with different atomic and mass numbers but same number of neutrons e.g. 14Si30, 15P31, 16S32.

ISODIAPHERS

Atoms having same Isotopic number.

ISOELECTRONIC SPECIES

Species having same number of electrons e.g. CO and CN (both contain 14 electrons each) Na+ and Ne (both contain 10 electrons each).
The ionic size decreases with increasing effective nuclear charge  of iso-electronic species.
Ionic size of isoelectronic species depend on (effective nuclear charge).
Ionic size

 

Species   C4– N3– O2– 
Nuclear Charge 6 7 8 9
Total electrons  10 10 10 10
  .6 .7 .8 .9
Ionic Radius (Å) 2.60 1.70 1.40 1.36
Species  Na+ Mg2+ Al3+ Si4+
Nuclear Charge 11 12 13 14
Total electrons  10 10 10 10
(effective
Nuclear charge) 1.1 1.2 1.3 1.4
Ionic radius (Å) 0.95 0.65 0.50 0.41

 

FAILURE OF RUTHERFORD’S MODEL

According to classical theory of electromagnetism whenever a charge is subjected to an acceleration around an opposite charge, it emits radiations continuously. Therefore the electron while moving around nucleus in circular path must lose energy, go into spiral motion and ultimately fall into the nucleus. Practically it does not happen.

PLANCK QUANTUM THEORY

According to Max Planck (1901) radiant energy is emitted or absorbed only in discrete units in form of bundle or packets of energy called photon (quantum). Photon is not a material body. It is massless bundle of energy

 

Energy associated with each photon (quantum) :
E = h =h
h = Planck’s constant = 6.626 × 10–34 Js in S.I. units
(or 6.6726 × 10–27 ergs in c.g.s. units).
= frequency of radiation (each photon).
c = velocity of light,
= wavelength of radiation.

 

Thus, a body can radiate energy in multiples of quantum h, 2h, 3h …. nh where n is an integer.

INTENSITY OF LIGHT

It is defined as number of photons falling per unit area per sec. and depends upon wavelength of photons.
or
It is defined as amount of energy falling per unit area per sec and depends upon wavelength of photons.

ELECTROMAGNETIC RADIATION

Electromagnetic radiation by James maxwell (1870). An electrically charged particles moving under acceleration produces alternating electrical and magnetic fields mutually perpendicular to each other. These fields are transmitted in the form of waves having same wavelengths, frequency, speed and amplitude and are called electromagnetic waves or electromagnetic radiations. In vacuum all types of electromagnetic radiations travel at the same speed (3.0 × 108 ms–1) regardless of wavelengths.

WAVELENGTH

It is the distance between two neighbouring crests or troughs of wave.

FREQUENCY

It is the number of waves which pass through a particular point in one second. Unit is Hertz (Hz) or cycles per second. 1 Cps = 1 Hz.

VELOCITY

It is the distance travelled by wave in one second. Unit is
m sec–1 and denoted by c.
c = .

WAVE NUMBER

It is the number of wavelengths per cm. It is equal to the inverse of wavelength. Unit is cm–1 and is denoted by .  

AMPLITUDE

It is the height of crest or trough. Square of amplitude determines the amount of energy carried by the wave.

ELECTROMAGNETIC SPECTRUM

Arrangement of all electromagnetic radiations in the increasing order of their wavelengths or decreasing order of frequencies is called electromagnetic spectrum.

 

Rays  Wavelength ( in Å) Frequency (in Hz)
Cosmic Rays  3×1021 to infinity
Rays  0.01  3×1019 to 3×1020
X Rays  1.0  2×1016 to 3×1019
UV Rays  150  7.9×1014 to 2×1016
Visible Light  3800  3.9×1014 to 7.9×1014
Infra Red  7600  1×1011 to 3.95×1014
Micro Waves  6×106  1×109  to 5×1011
Radio Waves  3×109  1×105  to 1×109

ATOMIC SPECTRUM

Atoms of different elements emit electromagnetic radiations of definite frequencies when excited by heating, passing current or electric discharge. Arrangement of these radiations in decreasing order of frequencies is called atomic spectrum.

DISPERSION

Phenomenon of splitting of beam of light into radiations of different frequencies after passing through a prism is called dispersion.

CONTINUOUS SPECTRUM

It is obtained by passing sunlight (white light) through a prism. The light is dispersed or resolved into continuous spectra of colours from Violet to Red. It contains radiations of all the frequencies.

LINE SPECTRUM

It is an atomic spectrum of an element which consists of a number of bright lines separated by dark bands. Atomic Spectra of most elements is line spectrum.

ABSORPTION SPECTRUM

It is obtained by passing white light through solutions or vapours of chemical substance and then is analysed by spectroscope. It has few dark lines in otherwise continuous spectrum.

EMISSION SPECTRUM

It is obtained by passing radiations from the atoms through prism. It has few bright lines against a dark background.

HYDROGEN SPECTRUM

It is obtained by passing light being emitted from discharge tube containing hydrogen at low pressure through spectrograph.

 

Hydrogen Spectrum has five Series

 

Spectral Line  Region n1 n2
Lyman Series  U.V.  1 2, 3, 4…..
Balmer Series  Visible 2 3, 4, 5…..   
Paschen Series I.R.  3 4, 5, 6…..
Brackett Series I.R.  4 5, 6, 7….
Pfund Series  I.R.  5 6, 7, 8….

 

 

Wavelength of line in spectrum is given by the expression
RH = Rydberg Constant, Z = charge on nucleus,
n1, n2 = electronic levels involved in transition, = Wave number
also for hydrogen  where is frequency.
  1. For calculation of longest wavelength line use n2 nearest and for shortest wavelength line use n2 infinity e.g. value of longest wavelength in Balmer Series of hydrogen spectrum use n1 = 2 and n2 = 3.
  2. Last line of spectrum is called Series limit. Last line is the line of shortest wavelength and high energy when n2 =  we get last wavelength
(series limit) = ,
  1. Number of Lines in a Transition : Mathematical formula for number of lines is follows as
No. of lines =

BLACK BODY RADIATION

The radiation emitted by a body when heated is called black body radiation. The frequency of radiation increases with temperature. At a given temperature the intensity of radiation emitted increases with decrease of wavelength, reaches a maximum value and then starts decreasing with further decrease of wavelength. A black body can emit and absorb all frequencies.

PHOTOELECTRIC EFFECT

Phenomenon of ejection of electrons from the surface of a metal when light of suitable frequency strikes on it is called photoelectric effect.
  1. Threshold frequency (v0) : The minimum frequency of incident radiation to cause the photoelectric effect is called threshold frequency.
  2. Work function : A part of the photons energy that is absorbed by the metal surface to release the electron is known as work function of the surface denoted by . The remaining part of the energy of photons goes into the Kinetic energy of the electron emitted.
If 0 is the threshold frequency and the frequency of incident light then  and E = h.
Note:
  • K.E. is independent of the intensity of light.
  • Number of photoelectrons Intensity of light
  • K.E. is directly proportional to frequency of incident light.
  • is known as Einstein’s photoelectric equation.
  • Energy required to stop the ejection of electrons is given by eV0 where e is the electric charge and V0 is stopping potential.

BOHR’S MODEL OF ATOM

Proposed by Niel Bohr to overcome the drawbacks of Rutherford’s model.
  1. Electrons revolve around nucleus only in certain selected circular orbits. These orbits are associated with definite energies and are called energy shells or levels.
  2. Electrons can move only in those circular orbits where angular momentum is a whole number and multiple of h/2. i.e. mvr =. or simply an integral number of wavelengths should fit in given electron orbit of radius r i.e. n=2r.
  3. Electrons energy in a particular orbit is constant.
  4. Lowest energy state is called ground state and when electron absorbs energy and jumps to higher state, it is called excited state        
  5. Electronic energy is negative because at infinite distance there is no interaction between electron and nucleus thus energy is zero. While when close to nucleus, attraction takes place, energy is  released and it becomes negative as it was already zero. The energy of electron increases with the value of n, but the difference of energy between two successive orbits decreases. Thus
E2 – E1 > E3 – E2 > E4 – E3 ………. etc.
  1. Energy of electron in nth orbit

 

where m = Mass of the electron,
e = Charge on the electron,
h = Planck’s constant
n = Principal quantum number,  
k = A universal constant = 9.0 × 109 J.m/C2
The constant k is inverse of permitivity factor 40 of the medium . The numerical value of permitivity factor is 40 = 1.11264 × 10–10 C2N–1m–2. In C.G.S. system k = 1.
  1. Radius of nth orbit
rn = Å
  1. Velocity of electron in nth orbit,  
cm/sec.
The velocity of electron in first orbit of hydrogen is   of the velocity of light.
  1. Kinetic energy of electron in nth orbit,
  1. Potential energy of electron in nth orbit,
  1. Total energy of electron in nth orbit,
  1. Number of revolutions per second in nth orbit,
  1. Angular velocity
  2. Angular momentum = mvr
  3. Number of spectral lines when electron jumps from the nth to ground level =
  4. The electrons energy is generally expressed in kcal or kJ mol–1 or in electron volts eV.
1 erg mol–1 = 1.44×1013 kcal mol–1 = 6.022×1013 kJ mol–1
1eV = 1.602×10–19J
  1. Some important values :
 
In c.g.s. system,
m = 9.109×10–28g
e = 4.803×10–10 esu,
h = 6.626×10–27 ergs,
k = 1
 RH

 

In S I system,
m = 9.109×10–31kg
e = 1.602×10–19C,
h = 6.626×10–34 J.s,
k = 9.0 × 109 Jm/C2
RH =
In SI system the charge e is replaced by

LIMITATIONS OF BOHR’S MODEL

  1. Explains the spectrum of elements having only one electron
  2. Does not explain splitting of spectral lines under magnetic field (Zeeman effect)  and electric field (stark effect)
  3. Does not explain quantisation of angular momentum.
  4. It goes against the Heisenberg’s uncertainity principle.

SOMMERFIELD MODEL

  1. Motion of electrons is in closed elliptical paths of definite energy levels having nucleus on either of the focii.
  2. Angular momentum is quantized
  3. where k = 1, 2 ———n.
  4. It does not explain distribution of electrons in extranuclear part of atom and also does not explain for de Broglie concept.

QUANTUM MECHANICS

It was developed independently by Warner Heisenberg and Erwin Schrodinges and takes into account the dual behaviour (particle and wave nature) of matter proposed by de Broglie.
Planck’s Quantum theory successfully explains.
  1. Photoelectric effect
  2. Black-body radiation
  3. Line spectra of H-atom
  4. Variation of heat capacity of solids with temperature.

DE- BROGLIE PRINCIPLE (1924)

  1. Proposes that just as radiations have particle nature, the material particles are also associated with wave nature.
  2. de Broglie wavelength is  h =  Planck’s constant m = mass of object ; v= velocity and this equation is called the de Broglie equation.

DAVISSON AND GERMER’S EXPERIMENT

Confirms the wave nature of electrons.

SCINTILLATION METHOD AND PHOTOELECTRIC EFFECT

Confirm the particle nature.

HEISENBERG’S UNCERTAINITY PRINCIPLE

“It is not possible to determine simultaneously the position and momentum of small moving sub-atomic e.g., particle , such as electron with entire certainty”.
  1. Mathematically
where, x = uncertainity in position
p = uncertainity in momentum and h = Planck’s constant
  1. As the mass of particle increases, the uncertainity decreases

QUANTUM MECHANICAL MODEL OF ATOM

  1. Based on de Broglie’s and Heisenberg’s principle.
  2. Put forward by Schrodinger (1920). Behaviour of electron was described in terms of equation known as Schrodinger wave equation
where is amplitude of electron wave and is also called wave function. x, y, z, are space coordinates, m is mass of electron, h is Planck’s constant, E is total energy and V is potential energy of the electron.
  1. Many solutions for this equation are possible for hydrogen but only certain solutions are permissible and are called eigen values
  2. The solution must be single valued, should satisfy the relation and must be finite and continuous.
  3. has no physical significance but gives intensity of electrons and thus gives probability of electron in a particular region.

ORBITALS

Orbitals are the regions in space around nucleus where probability of finding the electron is maximum.
  1. Probability does not become zero even at infinity and is given by  .
  2. Electron orbitals in atoms are called atomic orbitals while those in molecules are called molecular orbitals.
  3. Orbitals have definite energy and momentum and are quantized. i.e,  En = –E1/n2 thus Bohr’s concept of well defined orbits is ruled out.   

QUANTUM NUMBERS

  1. Four quantum numbers (n, l, m, s) help in providing complete information about an electron in an atom.
  2. Principal quantum number (n) determines the energy and average distance of electron. It has whole number values also denoted as K, L, M, N. etc. As n increases, distance of  electron from nucleus increases and energy increases.
  3. Azimuthal quantum number (l) determines angular momentum of the electron. It also determines the shape of orbitals and it may have all possible whole number values from 0 to n–1 for each principal energy level. The sublevels are:
Value of  l 0    1 2 3
Sub-shell  s    p d f
Magnitude of angular momentum of an electron in orbital, mvr =
Angular momentum of an electron in any orbit,
mvr
  1. Magnetic quantum number (m) defines the orientation of electrons cloud in a particular sub shell. Values of m are the number of orbitals associated with a particular sub shell in main shell. Values of m lie from  0 to l. Total values of ‘m’ for a given n is n2. Total values of ‘m’ for a given l is 2l +1. The table shows a clear relation between quantum numbers.  
Shell (n) Sub -shells (l) Orbitals (m)
n=1K shell  l= 01s m = 0
n=2L shell  l= 02s m = 0 
l= 12p m = –1,0, +1

 

n=3M shell  l= 0 3s m = 0
l= 1 3p m = –1, 0, +1
l= 2 3d      m = –2, –1, 0,+1, +2
  1. Spin quantum number (s) tells the spin of the electron. It can have two value   (clockwise) and   (anticlockwise). Mathematically where s is amplitude of spin quantum angular momentum.

SHAPE OF ORBITALS

  1. s orbitals are spherically symmetrical.
  2. p orbitals are dumbbell shaped.
  3. d orbitals have five different orientation. Three of them dxy, dyz, dxz are identical in shape but have different orientation.
  4. The plane passing through nucleus where probability of finding the electron is zero is called a nodal plane. Number of nodal planes in an orbital = l. Number of nodal planes increases with increasing value of n. e.g. 1s has no nodal plane. 2s has one nodal plane. For e.g. : s orbitals (l=0) have no nodal plane, p orbital (l=1) have one nodal plane, d orbitals (l=2) have two nodal planes.
Nodal plane = n-l-1
  1. Orbitals of a sub shell having same energy are called degenerated orbitals.
  2. Spherical surface within an orbital where probability of finding an electron is zero is called spherical or radial node. Number of spherical nodes = (n–l–1). Angular or non spherical nodes = (l). Thus total nodes = (n – 1).
Shape of s-orbital :
Shape of p-orbital :

 

for and for,  

 

Shape of d-orbitals :

 

PAULI’S EXCLUSION PRINCIPLE

No two electrons in an atom can have same values for all the four quantum numbers.
  1. It is not possible to accomodate more than two electrons in an orbital. In other words. s sub shell can have maximum of 2 electrons p sub shell can have maximum of 6 electrons. Thus max. no. of electrons in a shell can be 2n2.
  2. Maximum number of electrons in a sub shell can be 2, 6, 10, 14 in s, p, d, f respectively and max. electrons in an atomic orbital can be 2.
Maximum number of electrons in a sub shell is equal to
where
Note : Maximum numbers of electrons in an orbital = 2

AUFBAU’S RULE

Electrons are added to orbitals in increasing order of energies. The order of energies for orbitals  is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s.
  1. The order of energies can be calculated by (n + l) rule. i.e. orbitals are filled in order of increasing (n+l) values the one with lower n value is filled first.
  2. The energy of atomic orbitals for H-atom depends on the value of n only.
1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f

HUND’S RULE OF MAXIMUM MULTIPLICITY

The pairing of electrons in orbitals of a subshell does not take place until all orbitals of sub shell are singly occupied.
  1. This arrangement leads to lower energy level.
  2. Singly occupied orbitals should have same spins giving rise to lower energies.

RADIAL PROBABILITY DISTRIBUTION CURVES

The electron density is directly proportional to . The larger the value of  more is the probability of finding the electrons. Schrodinger wave equation may be separated into a product of three functions dependent on
  
R (r) = Radial wave function, it may be 0, or
R2 = Radial density in per unit volume of spherical shell.
It is always positive.
Radial probability. It is defined as maximum density of electrons in the volume of spherical shell having small thickness dr.
Note : is the volume of spherical shell having small thickness dr.
    
   
In case of s orbitals the number of peaks is equal to n,
In case of p orbitals the number of peaks is equal to (n–1),
In case of d, orbitals the number of peaks is equal to (n–2)
The point at which the probability of finding the electrons is zero is called nodal point.
The distance of maximum probability increases with increase in the value of n. hence 2s, 2p electrons are greater distance than 1s. and have greater energy also.

ANGULAR PROBABILITY DISTRIBUTION CURVES

The total angular (). () depends only on the direction and remain independent of the distance electrons from the nucleus
Angular probability distribution curves for s and p orbitals. The length of the line OP is proportional to the probability of finding the electrons. The length of the line OP is the same in all directions for s orbital Hence there are equal chances for finding electrons in all directions from the nucleus.

The length of line decreases with increasing in the value of angle . Hence there are more chances for finding the electrons along the axes for p orbitals.

RITZ. COMBINATION  PRINCIPLE

It states that the wave number (reciprocal of wavelength) of any line in hydrogen spectrum of a particular series can be represented as a difference of two terms, one of which is constant and other varies throughout the series. Mathematically,  .
RH = Rydberg constant

COMPTON EFFECT

 

The decrease in energy (or increase in wavelength) of X-rays after the scattering from the surface of carbon or light element is known as Compton effect.

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