Atomic theories

1

First Atomic Theory

The atomic theory of matter was first proposed by John Dalton, known as Dalton's atomic theory. Dalton regarded that atom is the ultimate particle of matter.

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Fundamental Particles

Matter is made up of molecules and molecules are made up of atoms. Dalton's atomic theory proposed that atoms were indivisible. But modern discoveries showed that atom is not indivisible and has a complex structure. Electrons, protons and neutrons are the fundamental particles of atom. Nucleons: Protons and Neutrons are present in the nucleus and are called Nucleons. Protons are positively charged with unit mass. Neutrons are neutral with unit mass.

Electrons:

They are negatively charged with negligible mass.

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Cathode Ray Tube

Atomic structure was obtained from the experiments on electrical discharge through gases. During the discharge tube experiment, Crookes observed that rays were found to pass from negatively charged filament (cathode) to positively charged plate (anode). Cathode ray tube is made of glass containing two thin pieces of metal, called electrodes, sealed in it. The electrical discharge through the gases could be observed only at very low pressures and at very high voltages. By maintaining low pressure and high voltage in discharge tube, current or stream of particles move in the tube from cathode to anode. Those rays are known as cathode rays or cathode ray

particles.

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Properties of Cathode Rays

Cathode rays starts from cathode and move towards anode. These rays themselves are not visible but their behaviour can be observed with the help of certain kind of materials (fluorescent or phosphorescent) which glow when hit by them. Rays travel in straight lines in the absence of electric and magnetic field. In the presence of electric and magnetic field, they are deflected which indicates that cathode rays contain negatively charged particles known as electrons. Cathode rays found to be independent of nature of the cathode material and

nature of the gas in the tube.

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Charge to Mass Ratio of an Electron

J.J.Thomson measured e/m ratio of the electron based on following points. Greater the magnitude of the charge on the particle, greater is the deflection when electric and magnetic field is applied. Lighter the mass of the particle, greater will be the deflection. The deflection of electrons from its original path increases when voltage increases from the above points, Thomson was able to determine the value of

charge to mass ratio as 1.758820 X 1011 Ckg−1

6

Charge of an Electron

Mullikan determined the charge of the electron by an oil drop experiment By carefully measuring the effects of the electrical field on the movement of many droplets. Charge on the oildrops was always an integral multiple of 1.60 X 10−19 c

me = ee/me = 1.60×10−191.758820×1011ckg−1 = 9.1094×10−31kg

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Properties of Protons

Anode rays travel in straight line, and these are material particles. Charge: Anode rays are positively charged, and get deflected by external magnetic field and affect the photographic plate. em of Anode Rays: em value of these rays is smaller than that of electrons. em value of anode rays depends upon nature of the gas. em value of anode rays is maximum when the gas present in the tube is hydrogen. Material in Anode Rays: By the dissociation and ionisation of hydrogen under low pressure discovered with charge +1 and mass 1, particles are called proton.

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Atomic Models

Thomson's Model - J.J.Thomson, in 1898, proposed that an atom possesses a spherical shape radius) approximately 10−10m) in which the positive charge is uniformly distributed. According to Thomson, atom is like watermelon and electrons are embedded like seeds in watermelon.The positive charge is distributed like fibrous material of water melon. An important feature of this model is that the mass of the atom is assumed to be uniformly distributed over

the atom. It cannot explain electrical neutrality of the atom.

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Rutherford's Model of an Atom

Rutherford proposed atomic model based on α - ray scattering experiment.

Scattering of a narrow beam of α - particles as they passed through a thin

gold foil and it is covered with fluorescent ZnS screen. When α -particles

struck the screen, then flash of light was produced at that point.

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Observations of Rutherford's Model

Most of the α - particles passes through the foil undeflected.

A small fraction of α - particles were deflected by small angles.

A very few α - particles bounced back were deflected by 180∘.

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Rutherford conclusions from his experiment

Rutherford drew the following conclusions from his experiment: 1. Most of the space in the atom is empty. 2. A few positively charged alpha particles were deflected. The

deflection must be due to enormous repulsive forces showing that the positive charge of the atom is not spread out of the atom.

3. Main postulates in Rutherfords model. All the positive charge and mass of the atom is present in a very small

region at the centre of the atom. It is called nucleus. The size of the nucleus is very small in comparison to the size of the atom. Most of the space outside the nucleus is empty. The electrons revolve round the nucleus like planets revolve round the sun. The centrifugal force arising due to fast moving electrons balances the

coulombic force of attraction of the nucleus and the electrons. Rutherford's atomic model is comparable with the solar system. So, it is

called planetary model.

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DIAGRAM

Defects of Rutherford's Atomic Model

It is against the law of electrodynamics. It fails to explain the atomic spectrum or line spectrum.

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Atomic Number

A neutral atom contains equal number of electrons and protons. The number of electrons or protons present in an atom of an element is called its atomic number. Atomic number is denoted by Z. Atomic number is equal to

the nuclear positive charge of an element.

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Mass Number

The sum of protons and neutrons in the atom of an element is called its mass number. It is denoted by A. Mass number is always a whole number. Number

of neutrons = A - Z

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Isotopes

Atoms with identical atomic number but different mass numbers are known as isotopes. Isotopes exhibit similar chemical properties Eg : Isotopes of hydrogen: Protium (1H1)

Deuterium (1H2) or D

Tritium (1H3) or T

Isotopes of chlorine:

17Cl35 and 17Cl37

It is evident that difference between the isotopes is due to the presence of different number of neutrons present in the nucleus. For example, considering of hydrogen atom again, 99.985% of hydrogen

atoms contain only one proton. This isotope is called protium (11H). Rest of the percentage of hydrogen atom contains two other isotopes, the one

containing 1 proton and 1 neutron is called deuterium (D, 0.015% ) and the

other one possessing 1 proton and 2 neutrons is calledtritium (T). The

latter isotope is found in trace amounts on the earth. Isotopes of an element have the same number of protons and electrons but

differ in the number of neutrons.

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Isobars

Atoms with same atomic mass number with different atomic number are known as isobars.

Eg : 146Cl , 147N

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Wave Length( λ)

The distance between two neighbouring troughs or crests in wave is known as wave length.

The units of wave length are m, cm, A∘ , nm,mμ.

1A∘ =10−8 cm =10−10 m

1nm = 1mμ = 10 A∘

Electromagnetic radiation

1

Nature of Electromagnetic Radiaition

Cosmic rays, γ - rays, X - rays, UV light,visible light, Infrared light, micro

waves, TV waves and radio waves are called electromagnetic radiationbecause they are made up of electric and magnetic fields propagating in perpendicular directions in one another. Electromagnetic radiations have wave characteristics and no medium is required for their propagation. They can travel through the vacuum. All

electro-magnetic radiations have same velocity.

2

Frequency(v)

The number of waves that pass through a given point in one second is called frequency. The units of frequency are sec−1, cycles per second (cps) or Hertz (Hz).

1 cps = 1 Hz= sec−1

3

Wave Number(v¯)

The number of wavelengths per centimeter or the reciprocal of wavelength is called wave number. The unit of wave number is cm−1 or m−1

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Amplitude(A)

The height of the crest or depth of the trough of a wave is called amplitude.

Amplitude is a measure of the intensity or brightness of a beam of light.

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Velocity(v)

The distance travelled by a wave in one second is called its velocity. The units of velocity are m/sec or cm/sec. All types of electromagnetic radiations have the same velocity which is equal

to 3x1010 cm/ sec or 3x108 m/ sec

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FORMULA

Relationship between Wave Characteristics

v = Cλ or λ = Cv .......... (1)

v¯ = 1λ = vc ...........(2)

Where v = frequency in sec−1

λ = wavelength in cm

C = velocity of light = 3x1010 cm/ sec

v¯ = wave number in cm−1

The wave length of UV light is 1800 - 3800 A∘ The wave length of visible light is 3800 - 7600 A∘ The wave length of IR radiation is 7600 - 31060 A∘

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Compton Effect

The increase in wave length or decrease in energy of the X - rays after

scattering from an object is called the Compton effect.

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FORMULA

Planck's Quantum Theory

Substances absorb or emit light discontinuously in the form of small packets or bundles. The smallest packet of energy is called quantum. The smallest particle of energy is called is called photon. The radiation is propagated in the form of waves. The energy of a quantum is directly proportional to the frequency of the

radiation - E∝v

The energy of a quantum is E = hv = hcλ = hcv ̄

Where E = Energy in ergs h = Planck's constant

= 6.625 x 10−27 erg−sec

= 6.625 x 10−34 Joule−sec

C = Velocity of light = 3 x 1010 cm/ sec = 3 x 108 m/ sec

v = Frequency of radiation in sec−1

λ = Wave length in cm

v¯ = Wave number in cm−1

A body can absorb or emit in whole number of quantum (E=n(hv))

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Photo-Electric Effect

In 1887, H.Hertz performed a very interesting experiment that is photo electric effect.

E = 12375λ Where E = Energy in eV

λ = wavelength in A∘

The radiation is propagated in the form of photons. Planck's equation determines both wave nature and particle nature of light. When light is exposed to clean metallic surface, electrons are ejected from the surface. This effect is called photo electric effect. The electrons are ejected from the metal surface as soon as the beam of light strikes the surface,i.e., there is no time lag between the striking of light beam and the ejection of electrons. Conditions for Photo-electric Effect:. Ejection of electrons from the surface of a metal by irradiating it with light of suitable frequency. The photo electric effect is readily exhibited by alkali metals like K and Cs. A part of the energy of photon is used to escape the electron from the attractive forces and the remaining energy is used in increasing the kinetic

energy of electron.

hv = W+KE

K.E. = 12meV2 , W = hvo

∴hv = hvo + 12meV2

me= mass of the electron, V = velocity of the ejected electron, vo=Threshold

frequency Effect of Intensity and Frequency of Light: In photo electric effect, the number of photo electrons emitted is proportional to intensity of incident light. Kinetic energy of photo electrons depends only on the frequency of incident light and not on the intensity of light. The minimum energy required for emission of photo electrons is called threshold energy or work function.

For each metal, there is a characteristic minimum frequency vo (also known

as threshold frequency) below which photoelectric effect is not observed. At

a frequency v > vo then photoelectric effect is observed.

Bohr's atomic model

1

Spectra

Spectrum formed by rays and its properties: 1. Sun light or light from an incandescent filament lamp gives a continuous

spectrum. 2. When a gas or a vapour of a metal is kept in a discharge tube and higher

potential is applied, a line spectrum is formed. 3. Each element has its own characteristic line spectrum. 4. The characteristic lines in atomic spectra can be used in chemical analysis

to identify unknown atoms in the same way as finger prints are used to identify people.

5. The spectra obtained by the emission of energy by the excited atoms are called emission spectra.

6. These spectra consist of bright lines on the dark background. 7. When white light is passed through a gas and the emergent beam of light is

allowed to fall on a photographic plate, the spectrum obtained is called absorption spectrum.

8. As the substance absorbs certain portion of white light, dark lines appear on bright background.

9. For a given element, dark lines in the absorption spectrum coincides with the bright lines in the emission spectrum.

10. An absorption spectrum is like the photographic negative of an emission spectrum.

11. German chemist, Robert Bunsen(1811-1899) was one of the first investigators to use line spectra to identify elements.

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Hydrogen Spectra

The source of radiation here is a hydrogen discharge tube. The discharge tube contains hydrogen gas at low pressure and high potential difference. The bright light emitted from the discharge tube is passed through a prism to cause dispersion. The emergent beam of light falls on a photographic plate and is recorded as the atomic spectrum of hydrogen. The hydrogen spectrum is the simplest of all the atomic spectra. It contains a number of groups of lines. They can be classified into various series. Only one such series is visible to the naked eye and is termed as the visible region of hydrogen spectrum. As it was discovered by Balmer, it is

called Balmer series.

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Balmer Series

The wavelength or wave number of various lines in the visible region can be expressed by an equation.

v¯ = 1λ = R[1n21−1n22]

where n1 = 2 which is constant for all the lines in Balmer series.

n2 = 3, 4, 5......

R is Rydberg constant and its value for hydrogen is 1,09,677 cm−1 (or) 1.09677 x 105 cm−1 Ryedberg constant value is not same for all the elements. The first line in Balmer series is called Hα line and its wavelength is 6563 A∘. The second line is called Hβ line and its wavelength is 4861 A∘ The spectral lines get closer when the n2 value is increased. If n2 is taken as infinity the wavelength of the limiting line in the series is obtained.

v¯∞ = 1λ∞ = R[122−1∞2] = R4 = 27,419 cm−1

The other series in the hydrogen spectrum are invisible. The wavelength or wave numbers of all the lines in all the series can be calculated by using Rydberg's equation

v¯ = 1λ = 1,09,678 (1n21−1n22 )

Maximum number of lines produced when an electron jumps from nth level to ground level = n(n−1)2

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Series of Hydrogen Spectrum

The value of R = 1,09,678 cm−1 is valid only for the lines in the hydrogen spectrum. For a spectral line of any one electron species like He+, Li2+ the value

of R = RH×Z2

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Postulates of Bohr's Model

Energy Levels: The electrons in an atom revolve around the nucleus in definite circular orbits or shells or energy levels. Stationary States: So far an electron revolves in a certain orbit, its energy remains constant and does not radiate energy. These orbits are called stationary orbits or stationary states. Angular Momentum of Electron in an Orbit: Electrons can revolve only in those stationary orbits in which their angular momentum is equal to integral multiple of h2π mvr = nh2π

where m = mass of electron v = velocity of electron, r = radius of orbit n = 1 , 2 , 3 ,4 ...... h = Planck's constant Just as linear momentum is the product of mass (m) and linear velocity (v), angular momentum is the product of moment of inertia (I) and angular velocity

(ω). For an electron of mass m, moving in a circular path of radius r

around the nucleus, angular momentum = I×ω

Since I = mer2, and ω = v/r where v is the linear velocity,

angular momentum = mr2×v/r = mvr

Difference in Energy Levels: When an electron drops from a higher orbit to a lower orbit, energy is released. When an electron jumps from a lower orbit to a higher orbit,energy is absorbed. The absorbed or evolved energy is equal to the difference in energies of two orbits, which is equal to quanta. ΔE = E2 - E1 = hv

The line spectrum is obtained due to the electronic transition from one orbit to another orbit. Force of Attraction: The force of attraction between the nucleus and the electron = −Ze2r2 The centrifugal force of the electron due to revolving around the nucleus

= −mV2r

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Radius of Bohr's Orbit

Expression for the radius of Bohr's orbit r = n2h24πmZe2

where r = radius of orbit n = 1, 2, 3, 4 ...... h = Planck's constant m = mass of electron Z = atomic number e = charge of electron Radius of Hydrogen atom, r = 0.529 x 10−8 x n2 cm

= 0.529 x n2 A∘ Radius of orbits in H atom like ions

r = 0.529×n2ZA∘

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FORMULA

Velocity of an Electron in Hydrogen Atom

Velocity of electron in hydrogen atom V = 2πZe2nh = 2.188×108n cm/sec

where V = velocity of electron e = charge of electron n = 1, 2, 3, 4 .......

h = Planck's constant For hydrogen atom, Vn = V1n

where Vn = Velocity of electron in nth orbit

V1 = Velocity of electron in first orbit

n = 1, 2, 3, 4 ........ For H atom like ions.

Vn = ZV1n

where Vn = Velocity of electron in nth orbit

V1 = Velocity of electron in first orbitof H-atom

n = 1, 2, 3, 4 ....... Z = Atomic number

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FORMULA

Energy in Bohr's orbit

Kinetic energy of electron = 12mV2

= Ze22r Potential energy of electron = −Ze2r Total energy of electron = KE+PE = Ze22r−Ze2r = −Ze22r

Expression for the energy of Bohr's orbit

E = −2π2mZ2e4n2h2

where E = energy of orbit m = mass of electron e = charge of electron n = 1, 2, 3, 4 ........ h = Planks constant

Effect of Kinetic,Potential and Total Energies as n Increases:

As we go to higher orbits, kinetic energy decreases, potential energy increases and the total energy increases. Energy of orbits in hydrogen atom ( Z = 1 ) Energy Expressions in Different Units:

E = −2.179×10−11n2 ergs

= −2.179×10−18n2 joules = −13.6n2 eV = −313.6n2 Kcal/mole = −1312.6n2 KJ/mole

1eV = 1.602×10−19 J

The energy of the electron in a hydrogen atom has a negative sign for all possible otbitals because the energy of the electrons in the atom is lower than

the energy of a free electron at rest. Energy of orbits in H atom like ions E = −2.179×10−11n2×Z2 ergs

En = −E1n2

where En = Energy of nth orbit in hydrogen atom

E1 = Energy of first orbit in hydrogen atom

n = 1, 2, 3, 4 ......... For Hydrogen atom like ions.

En = Z2n2×E1

where En = Energy of nth orbit in other ions like H - atom

Z = Atomic number n = 1, 2, 3, 4 ........

E1 = Energy of first orbit in hydrogen atom

Ionisation of Electron: Ionisation Potential is the energy required for removal of electron from the outermost orbit. For hydrogen atom, Ionization potential = −E1n2 For H atom, like ions, Ionisation potential = −E1×Z2n2

Ionisation potential of an atom or ion = 13.6 [Z2n2] eV

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Rydberg’s constant

R = 2π2mZ2e4h3C = 109680cm−1

Difference of energy between two Bohr orbits of hydrogen atom ΔE = Rhc[1n21−1n22]

where ΔE = Energy difference

R = Rydberg constant h = Planck's constant c = Velocity of light n1 = lower orbit,

n2 = higher orbit

As the value of n increases, the difference of energy becomes smaller. After a certain stage, the energy becomes nearly equal and this position of continuum is called critical energy. If energy is slightly greater than this given value, then the electron will be completely removed from the atom. Difference of energy between two orbits in H atom like ions.

ΔE = Z2Rhc[1n21−1n22] where Z = atomic number.

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Merits of Bohr’s model

It successfully explains the hydrogen spectrum and spectra of ions having one electron. The experimental values of the energies and radii of possible orbits in hydrogen atom are in good agreement with that calculated on the basis of Bohr's theory. The experimental value of Rydberg constant for hydrogen is in good agreement with that calculated from Bohr's theory. The calculated value of ionization energy of hydrogen using Bohr's theory is very close to the experimental value.

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Limitations of Bohr's Model

1. It failed to explain the spectra of atoms or ions having more than one electron. The fine structure of spectral lines cannot be explained by Bohr's theory.

2. It failed to explain Zeeman effect and Stark effect. 3. Bohr model of the hydrogen atom, not only ignores dual behavior of matter

but also contradicts Heisenberg uncertainty principle. Zeeman Effect: The splitting of spectral lines of an atom into a group of fine lines under the influence of a magnetic field is called Zeeman effect. Stark Effect: The splitting of spectral lines of an atom into group of fine lines under the

influence of an electric field is called Stark effect.

de-Broglie's hypothesis

1

de-Broglie Wave Theory:

The wave nature of electron was first proposed by de Broglie. According to de Broglie theory, all moving particles have wave properties. Wave properties are important only for particles of small mass and high velocity.

de Broglie equation is λ = hmv = hp

where λ = wave length

h = Planck's constant = 6.625 x 10−34 J.sec v= Velocity of the particle m = mass of the particle p = Momentum of the particle According to de Broglie's theory, electrons revolve around the nucleus in atomic orbits with stationary waves. Electrons revolve in those orbits, whose circumference must be equal to

integral multiple of wave length 2πr = nλ

where r = radius of the orbit n = 1, 2, 3, 4 .....

λ = wavelength

Number of waves in an orbit = n Number of revolutions of an electron per second in an orbit = Velocityofelectroncircumference

2

Bohr's Theory and de-Broglie's Concept

According to de Broglie, an electron behaves as a standing or stationary wave which extends around the nucleus in a circular orbit. If the two ends of the electron wave meet to give a series of crests and troughs, the electron wave is said to be in phase. In other words, there is constructive interference of electron waves and the electron motion has a character of standing wave or non-energy radiating motion. To be an electron wave in phase, the circumference of the Bohr's orbit should be an integral multiple of the wavelength of the electron wave.

nλ = 2πr Explanation of de Broglie's Concept:

In case, the circumference of the Bohr's orbit (2πr) is bigger or smaller

than nλ, the electron wave is said to be out of phase.

Then, destructive interference of waves occurs causing radiation of energy. Such an orbit cannot possibly exist. The wavelengths associated with ordinary objects are so short (because of

their large masses) that their wave properties cannot be detected.

Heisenberg's uncertainty principle

1

Heisenberg's Uncertainty Principle

It is impossible to determine accurately and simultaneously the position and momentum of a particle in an atom. It is called Heisenberg's uncertainty principle. The uncertainty principle equation is Δ x . Δ p ≥h4π

Δ x . mΔ v ≥h4π

where Δ x = uncertainty in position

Δ p = uncertainty in momentum

Δ v = uncertainty in velocity

m = mass of the particle h = Planck's constant The uncertainty principle is mainly applicable for microscopic particles. Explanation for Heisenberg's Uncertainty Principle: To observe an electron we can illuminate it with light or electromagnetic radiation. The light used must have a wavelength smaller than the dimesions of an electron. The high momentum photons of such light (p=hλ) would

change the energy of electrons by collisions. In this process we, no doubt, would be able to calculate the position of the electron, but we would know very little about the velocity of the electron after the collision. If one of tries to find the exact location of the electron, say to an uncertainty of

only 10−8 m,then the uncertainty Δv in velocity would be

10−4m2s−110−8m≈10+4ms−1 which is so large that the classical picture of electrons moving in Bohr's orbit

(fixed) cannot hold good.

Orbitals and Quantum numbers

1

Radial Probabilty Distribution

The probability of finding an electron at a certain distance from the nucleus is called radial probability.

The curves obtained by plotting probability function D = 4πr2drΨ2and radial

distance (r) are called radial probability distribution curves. Number of peaks obtained in a curve = n - l where n = principal quantum number

l = Azimuthal quantum number The nodal surface of 2s orbital exists at a distance of 2ao from the nucleus. Where ao is the Bohr radius 0.529 A∘ The curve for 2s orbital has two peaks, the curve passes through lower maximum at 0.53 A∘ and higher maximum at 2.6 A∘ radial distance.

2

Orbitals

The space around the nucleus of an atom in which there is a maximum probability of finding an electron is called an orbital. The maximum probability of finding an electron in an orbital is 95 %. The plane where the probability of finding the electrons is

zero (Ψ2=0 ) is called a nodal plane.

Number of nodal planes in an orbital = l.

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s orbital

The shape of s orbital ( l = 0 ) is spherical. s - orbital is a non directional orbital.

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p orbital

In a p - sub shell, the three orbitals are represented as px, py, and pz. These are degenerate orbitals. The shape of a p - orbital ( l = 1 ) is dumbbell. p - orbitals are oriented along the axes. So they are directional orbitals. Orbital : px py pz

m : ±1 ±1 0

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d orbital

In a d - sub shell, the five orbitals are represented as dxy, dyz,

dzx, d{x2−y2} and d2z.

These are degenerate orbitals. The shape of a d - orbital ( l = 2 ) is double dumbbell. dxy, dyz and dzx orbitals are oriented in between the axes dx2−y2 and d2z orbitals

are oriented along the axes. Orbital : dxy dyz dzx dx2−y2 d2z

m : ±2 ±1 ±1 ±2 0

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No. of nodal planes

When the number of nodal planes increases, the energy of the orbital increases. So the energy order of the orbitals is s < p < d < f

Number of radial nodes = n−l−1

where n =principal quantum number

l = Azimuthal quantum number

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Quantum Numbers

A set of numbers used to provide a complete description of an electron in an atom are called quantum numbers. There are four quantum numbers required for a complete explanation of electrons in an atom. The quantum numbers are n-Principal quantum number l-Azimuthal quantum number m- Magnetic quantum number

s-Spin quantum number

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Principle Quantum Number(n)

It was proposed by Niels Bohr. Possible Values of Principle Quantum Number: The values of n =1, 2, 3, 4 ..... or K, L,M, N ....... respectively. It indicates the size and energy of the orbit. The maximum number of electrons in an orbit = 2n2 Total number of orbitals = n2 where n = no.of the orbit Angular momentum of an electron in an orbit = nh2π

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Azimuthal Quantum Number(l)

It was proposed by Sommerfield. Possible Values of Azimuthal Quantum Number: The values of l = 0, 1, 2, .....( n -1 ).

The values of l represents various sub shells. When l = 0, 1, 2, 3 ...... etc are

called s, p, d, f ....... sub shells respectively. Energies are in the order of s < p < d < f . It indicates the shape of orbit or orbital and angular momentum of electron. Number of sub shells in an energy level = n where n = no.of the orbit Angular momentum of the electron in an orbital

= h2πl(l+1)−−−−−−√ = h¯l(l+1)−−−−−−√

(h¯=h2π )

where h = Planck's constant

l = Azimuthal quantum number

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Magnetic Quantum Number (m)

It was proposed by Lande. Possible Values of Magnetic Quantum Number:

The values of m = +l ..... 0 ..... -l.

The total m values = 2l + 1

The total number of m values indicates the total number of orbitals in the subshell. The number of orbitals in s, p,d and f sub shells are 1, 3, 5 and 7 respectively. It indicates the orientation of orbitals in space. The number of orbitals in an energy level n2

The number of orbitals in a sub shell = 2l + 1

Maximum number of electrons in a subshell 2(2l + l)

where l = Azimuthal quantum number.

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Spin Quantum Number (s)

It was proposed by Goudsmit and Uhlenbeck. Possible Value of Spin Quantum Number:

The values of s = +12 and −12

The clock wise direction spin is represented by +12 and anticlock wise

direction spin is represented by −12

For each value of m, there can be two values. It indicates the direction of the spin of the electron. Maximum number of electrons in an orbital = 2. The maximum number of electrons present in s, p, d and f shells are 2, 6, 10 and 14 respectively.

Important Principles

1

Pauli's Exclusion Principle

No two electrons in the same atom can have the same values for all the four quantum numbers.

Two electrons in a given orbital have same values of n, l and m. Electrons in the same orbital differ in their spin quantum number and they spin in opposite directions. Consequence of Pauli's Exclusion Principle: An orbital can not accommodate more than two electrons.

2

Aufbau Principle

Electron filling follows energy ranking. The orbitals are successively filled in the order of their increasing energy. Among the available orbitals, the orbitals of lowest energy are filled first.

Energy of the Orbital in terms of n+l: The energy value of an orbital increases as its (n+l) value increases.

If two orbitals have the same value for (n+l), the orbital having lower n value is

first filled.

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Hund's Rule

Orbitals of the same kind should be half filled before electron pairing takes place. Degenerate Orbitals: Orbitals having the same values for n and l are called degenerate orbitals. Unpaired electrons have parallel spin. Half filled and completely filled degenerate orbitals give greater stability to atoms. Example: Chromium (Z = 24) and copper (Z = 29) have anomalous electronic configuration due to this reason. Electronic configuration of chromium atom is 1s2 2s2 2p6 3s23p6 3d54s1 but

not 1s2 2s2 2p6 3s23p6 3d4 4s2.

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Stability of Completely Filled and Half Filled Sub-Shells

The valence electronic configurations of Cr and Cu are 3d54s1 and 3d104s1 respectively and not 3d4 4s2 and 3d94s2. Reasons: The completely filled and completely half filled sub-shells are stable due to the following reasons. Symmetrical distribution of electrons: It is well known that symmetry leads to stability. The completely filled or half filled subshells have symmetrical distribution of electrons in them and are therefore more stable. Exchange Energy: The stabilizing effect arises whenever two or more electrons with the same spin are present in the degenerate orbitals of a subshell. These electrons tend to exchange their positions and the energy released due to this exchange is called exchange energy. The number of exchanges that can take place is maximum when the subshell is either half filled or completely filled. As a result

the exchange energy is maximum and so is the stability.