NotesQuantum theory in terms of wavelength of radiationE =

: wavelength of radiationBy quantum theory in terms of λ wavelength of radiationE = c = Einstein's EquationE = E = Total energy of substancem = Total mass of substancePhotoelectric Effect

= = Frequency of incident radiation

V = Velocity of electronW = Work function of metalm = Mass of electronV = Stopping potentialBohr's TheorymVr = Quantization of angular momentumBohr's Theory for K.E.

K.E. =

r = radius of nth Bohr orbitBohr's Theory for R

=

Bohr's theory for V

=

Bohr's Theory for energy released

E =

Energy released in transition from orbit orbits is Rutherford's Nuclear Model of Atom=Rutherford performed an alpha particle (He ) scattering experiment on a thin gold foil and presented that :

(i) Most part of atom is empty.

(ii) Every atom possesses a highly dense, positively charged centre called "Nucleus".

(iii) Entire mass of atom is concentrated inside the nucleus.

(iv) Later Rutherford model was abandoned due to its failure to comply with classical theory of electromagnetic radiation. This theory also failedto explain the line spectrum of H-atom.Planck's Equation=

Bohr's Radius=

Radius of a stationary orbit r is

where, (For Bohr radius)Energy of Stationary Orbit (E )=

Potential Energy

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>Physical Chemistry>Atomic Structure

JEE

Chemistry

e

e

s

n

n

Å Ån

2+

n

Ån

=

Speed of Electron in nth Bohr Orbit=

Speed of Electron in nth Bohr OrbitEmission Spectrum of H-atom=

The frequency wavelengths or wave-number of a spectral line in any of the series in the emission spectrum of hydrogen atom can be calculatedusing the following Rydberg's equation :

(i) For Lyman series : n = 1, n = 2, 3 ..... (Occur in UV region)

(ii) For Balmer series : n = 2, n = 3, 4 ..... (Occur in visible region)

(iii) For Paschen series : n = 3, n = 4, 5 ....... (Occur in IR region)

(iv) For Brackett series : n = 4, n = 5, 6 ...... (Occur in IR region)

(v) For Pfund series : n = 5, n = 6, 7 .......... (Occur in IR region)Number of revolutions made by an electron in nth Bohr's orbit is =Revolution/second

Speed in nth orbit.Wave-Partical Duality (de-Broglie Relationship)=

Bohr's de-Broglie Relationship=

In a given stationary orbit, the number of de-Broglie wavelengths is equal to orbit number. If an electron at rest is accelerated by a potentialdifference of V volt then de-Broglie wavelength is :

Heisenberg's Uncertainty Principle=According to this principle, simultaneous and accurate measurement of both position and momentum of an electron in an atom is impossible.

Here is uncertainty in position and is uncertainty in momentum.The Schrodinger's Equation=

Wave-function, represents an orbital.

Total energy of an electron.

Potential energy associated with electron. Bohr's ModelBohr proposed an idea of stationary orbits in which electron revolves. According to Bohr's electron does not lose energy as long as it stays in anstationary orbit. When an electron jumps to lower stationary orbit, energy is lost in the form of electromagnetic radiation. Conversely whenenergy is supplied, electron jumps to a higher stationary orbit. According to Bohr's:(i) Angular momentum of an electron is quantized: Angular momentum ...(i) n = 1,2,3,..., (orbit number)

1 2

1 2

1 2

1 2

1 2

(ii) Centrifugal force of orbiting is exactly balanced by the electrostatic attraction between nucleus and electron ...(ii)

Atomic Weight (A)Atomic weight is the relative weight of one atom of an element with respect to a standard weight.

amu (atom mass unit) Weight1 amu th part by weight of an atom of isotope

Atomic Weight (A) x amu = Absolute atomic weight.

Note.Note Atomic weight is a relative weight that indicates the relative heaviness of one atom of an element with respect to amu weight. Atomicweight has no unit because it is the ratio of weights. One mole of an amu = 1.00 g.Change of Scale for Atomic Weight.If an amu is defined differently as (l/x)th part by weight of an atom of C-12 isotope rather (l/12)th part then the atomic weight (A') can be derivedas:

.where A = conventional atomic weight

Molecular Weight (MW).Like atomic weight, it is the relative weight of a molecule or a compound with respect to amu weight.

Molecular weight

Quantum NumbersQuantum Numbers To describe an electron completely inside the atom, four sets of quantum numbers are required. They are

Principal Quantum Number (n) This specify position and energy of an electron in the atom. Possible values of 'n' are 0,1,2,3,..

Angular Momentum (Azimuthal or Subsidiary) Quantum Number (l) This is used to specify subshell (orbital). Possible values of l are 0, 1,2, ... (n - 1). Orbitals with different values of l are denoted as:l = 0, s-orbital - sphericall= 1, p-orbital - dumb-bell shapel = 2, d-orbital — double dump-bell shapel = 3, f-orbital etc

The value of l also determine shape of orbital as mentioned above.

The value of l determines orbital angular momentum (L) as:

The value of 'l' also determine the magnitude of magnetic moment as:

Where,

Magnetic Quantum Number (m) It determines the preferred orientation of orbitals in three dimensional space. Its possible values are :

m= -l,...,0,...,+ l

eg, for l = 2, m = -2, -1, 0, +1, +2,

Total values of m for a given value of l=(2l+ 1) = number of orbitals in a given subshell.

Splitting of spectral lines occur when placed in a magnetic field (Zeeman effect) or in an electric field (Stark effect). Total lines from a single line inthe normal spectrum = (2l +1)

Total number of orbitals in nth orbit = n Total number of electrons in nth orbit = 2n

Spin Quantum Number (s) Electrons spin on its own axis like a top, in clockwise and anticlockwise directions- The two directions of spinning is denoted by spin quantumnumber as :

Principal Quantum Number (n)This specify position and energy of an electron in the atom. Possible values of 'n' are 0,1, 2, 3,...., ∞

Energy of electron in hydrogenic wave function Angular Momentum (Azimuthal or Subsidiary) Quantum Number This is used to specify subshell. Possible values of are . Subshells with different values of are denoted as :

, s-subshell- spherical

, p-subshell - dumb-bell shape

, d-subshell - double dumb-bell shape

, f-subshell etc.

The value of determines the orbital angular momentum vector

Magnetic Quantum Number (m)It determine the preferred orientation of orbitals in three dimensional space. Its possible values are :

for

Total values of m possible are i.e. no of orbitals in a subshell

s-subshell - 1 orbital

p-subshell - 3 orbitals

d-subshell - 5 orbitals

f-subshell - 7 orbitalsSpin Quantum Number (s) Electrons spin on its own axis like a top, in clockwise and anticlockwise directions. The two directions of spinning is denoted by spin quantumnumber as :

and . ,The spin quantum numbers are also denoted by up-half arrow and down-half arrow but neither the +1/2 and -1/2 or the 1 or arespecific for any direction, they just represent the two opposite directions of spinning of electrons.

Pauli's Exclusion Principle

No two electrons on any atom have all quantum Nos same, as orbital is characterised by quantum nos and it will have only two electronsone with clockwise other with anticlockwise spin.Electronic Configuration Electrons are filled in atomic orbitals in increasing order of their energy according to Aufbau principle:

Subshells receive electrons in the increasing order of their values, If they have same , the one with lower n value shall receiveelectron first.

Hund's Rule of maximum multiplicity i.e. feeding one electron to each degenerate orbitals of a subshell first with parallel spins, thereafterpairing of elctrons in orbitals of a subshell starts.

Pauli's Exclusion Principle i.e. only two electrons with antiparallel spins in each orbital.Quantum Mechanical Model (The Schrodinger's Equation)

ψ : Wave-function, represents an orbital.

E : Total energy of an electron.

V : Potential energy associated with electron.

Solution of second order differential equation give normalised wave functions . Normalised wave function, for some orbitals are ( in H-atom)

Total Radial probability, Probability density & Nodes

Wave function can be split into radial & angular parts

Probability density is

Total Radial Probability is

Radial nodes

Angular nodes

Radial Peaks

Rutherford Atomic ModelA central positively charged region called nucleus, comprising of neutrons & protons surrounded by empty space where in electrons revolving inclosed circular orbits.

Scattering of -particles by metal foil when 1 out of 10,000 particles retraced the path from distance of least approach

De Broglie Wave Equation

Each Bohr's orbit is occupied by n wave lengths of stationary electronic wave

or Where V in volts. will provide λ in

Heisenberg Uncertainty Principle

Bohr ModelTo explain the emission spectrum of H & H-like species (all one electron systems)

(i) Centripetal force for orbital motion of electron provided by electrostatic attraction between nucleus & electron.

(ii) Quantisation of angular momentum of electron moving in stationary Bohr's orbits

(n is principle quantum no. of orbit)

Radius of nth orbit

K.E of electron

P.E of electron

Total energy of electron

Speed of electron

fine structure constant

C is speed of light

Frequency of revolution in nth orbit

Energy Level Diagram For H-like Species

Å

Ionisation energy

Separation Energy

Wave length of Spectral line

Due to electronic transition between quantised energy levels, the energy difference is emitted out as spectral line, photon

Ritz Equation for spectral line wave length

R = Rydberg constant for H-atom

For Lymen series,

Balmer series,

Paschen series,

Brackett series,

Pfund Series series, ElectronThomson (1897) carried out the discharge through a vacuum tube having filled with a gas at very low pressure (10 to 10 mm) and noticed theemission. Fluorescence on the glass and influenced photographic plate. These rays are were called cathode rays.

The charge-mass ratio (e/m) determined (Thomson's experiment ) for cathode rays particle was found to be -1.76 x 10 C kg .

The specific charge (i.e., e/m ) was found to be independent of the nature of gas and electrode used.

Lorentz named "a subatomic particle, i.e., a fundamental constituent of all matter having a mass of 9.108 x 10 kg and charge equal to -1.602 x10 C" as electron. (usually represented as )

According to theory of relativity the mass of electron at high speed (m') is given by,

where v is the velocity of electron and c is velocity of light.ProtonGoldstein (1886) repeated discharge tube experiment using perforated cathode and noticed the emission of positive rays or canal rays.

The specific charge (e/m) of canal rays particles was found to vary with nature of gas and was maximum if H is used.

Å

H

-2 -3

11 -1

-31

-19

2

-27 -19

Thus , a subatomic particle, i.e., a fundamental constituent of all matter having a mass 1.673 x 10 kg and charge +1.602 x 10 C is called aproton.NeutronChadwick (1932) bombarded Be or B-atoms (sheet) with high speed alpha particles and noticed the emission of neutral particles, i.e.,neutrons of mass nearly equal to proton. A neutron is therefore, a subatomic particle, a fundamental constituent of matter having a mass 1.675 x10 kg and no net charge. (usually represented as )

Rutherford -Scattering ExperimentRutherford (1909) bombarded thin (10 m) Au foil with high speed -particles and noticed visible light scintillations on ZnS screen as shown infigure 1. He observed that :

Most of the -particles passed without any deflection or deflected through small angles of the order of 1 .

Some of the them were deflected away from their path by an angle as large as 90 or more.

Only a few (one in about 10,000) were returned back to their original direction of propagation.

The number of -particles scattered and their scattering angles are represented in Fig. 2(b).

The scattering i.e., number of scattered -particles (N) is inversely proportional to the square of the kinetic energy of the incident particle. Higherthe energy of the incident particle, the smaller will be number of scattered particle.

The scatttering of -particles involves perfectly elastic nuclear collisions and obey the law of conservation of energy, momentum and angularmomentum.

The scattering i.e., number of scattered -particles (N) is proportional to the square of the atomic number (Z) of both the incident particles ( i.e., ) and the target scatterer (Z ).

The scattering (N) is also proportional to the targets thickness for thin targets.

The number of -particles showing scattering is inversely proportional to the fourth power of , where is the scattering angle i.e.,

The number of -particles showing scattering is minimum when , i.e., .

These observations led Rutherford to propose the concept of nucleus in an atom. The size of the various nuclei can be calculated from therelation,

r = (1.33 x 10 )A )

where r is the radius of the nuclei (in cm) with mass number A.

-27 -19

-27

-4

o

o

t

-13 1/3

∴

(where A is mass number and r = 1.33 x 10 A )

= 1.68 x 10 g/cm

i.e., Density of nucleus is almost constant

Each atom consist of a small, heavy, positively charge portion located at its centre, known as nucleus.

All the positive charge of atom (i.e., protons) are present in nucleus.

The dimension of nucleus is of the order 10 nm = 10 m = 10 cm.

The dimension of atom is of the order 10 nm = 10 m = 10 cm.

The electrons are in continuous motion in extranuclear part round the nucleus like planets around sun. The centripetal force so developed keepsthem away from merging into the nucleus.

It does not obey the Maxwell law of thermodynamics : A small charged particle moving round an oppositely charged centre continuously loses itsenergy. ( Planets being uncharged do not show this feature. )

If an electron does so, it should also continuously lose its energy and should set up spiral motion ultimately falling into the nucleus. Calculationsshow that it should take an electron only 10 sec to spiral into the nucleus. This gave a vital blow to Rutherford model based on classical conceptand electromagnetic theory.

It could not explain the emission of various spectral lines (Lyman, Balmer, ..... series ) during emission spectrum of H.

It could not suggest for the discontinuous nature of spectrum.Planck Quantum TheoryRadiant energy is emitted or absorbed only in discrete units, i.e.. discontinuously (not continuous) in packet of energy called photon (quantum).The energy 'E' associated with a quantum is given by,

E = hv

where h is Planck's constant (6.626 x 10 J-sec or 6.626 x 10 erg-sec ) and v = frequency of radiation in sec .The dimension of Planck's constant is ML T like angular momentum.

where c is velocity of light and λ is wavelength of light. Thus, a body can radiate or absorb energy in whole number multiplies of a quantum. i.e.,hv, 2hv, 3hv, ... nhv ; where n is an integer.

The energy of photon decreases with increase in wavelength. That is why a photon of red light is less energitic than a photon of the blue light.

Note : Intensity of light (no. of photons falling per unit area per sec) depends upon the number of these photons and if intensity of light isreferred as amount of energy falling per unit area per sec; then it also depends upon the wavelength of photons used.

The rest of photon is zero.

Mass of moving photon m is given by;

or Electromagnetic radiationThus, wave motion represents propagation of a periodic disturbance carrying energy. A wave has five characteristics, i.e., wavelength, frequency,velocity, wave number and amplitude.

-13 1/3

14 3

-6 -15 -13

-1 -10 -8

-8

-34 -27 -1

2 -1

Å

Wavelength or

It is the distance between successive points of equal phase of a wave, i.e., the distance between two neighbouring crests or troughs. It is normallyexpresses in . (also in cm, m, nm, etc.)

Frequency or (v)

It is the number of cycles or oscillations or vibrations of a wave motion in unit time.

, expressed in Hz or sec

Velocity

It is the distance travelled by the wave in one second.

Velocity = wavelength x frequency; (expressed in cm sec-1 )

Wave number

It is defined as the number of waves in unit length. It is reciprocal of wavelength.

i.e.,

Amplitude

If any quantity is varying in an oscillatory manner about an equilibrium value, the maximum departure from that equilibrium is called amplitude.

Electromagnetic spectrum

The arrangement of the various types of electromagnetic radiation in order of their increasing (or decreasing) wavelengths frequencies is knownas electromagnetic spectrum.DeBroglie Wave EquationDe Broglie used Einstein special theory of relativity together with Planck's quantum theory to establish wave properties of particles.

For photon. E = mc = mc. c = p.c (where p is momentum of photon)

Also, E = hv

Thus,

or ...... (22)

This relation for photon was extended to all particles by de Broglie. Particle waves are matter waves and the wave-length expressed by Eq. (22) iscalled de Broglie wavelength of particle. Thus for electron if u is velocity for electron wave and λ is its wavelength, then

..... (23)

where, p is momentum of electron.

For a closed orbit, circumference should be integer multiple of wavelength.

i.e., or

i.e., angular momentum is quantised for closed orbit. de Broglie equation was verified experimentally by Davisson and Germer by their electrondiffraction experiment using Ni crystal. The electron microscope was constructed on the basis of de Broglie concept.

Å

-1

2

∵

∵

Note : The wavelength of an electron accelerated by a potential difference V can be given by :

For a gas molecule,

Average kinetic energy

or

where, k is Boltzman's constant.HeisenBerg Uncertainty PrincipleThese models were against the Heisenberg uncertainty principle. The principle sates that it is impossible to determine momentum and position ofa sub atomic particle precisely and simultaneously.

; or

or

and

is uncertainty in momentum;

is uncertainty in position along one axis

is uncertainty in velocity along same axis

As the mass of particle increases the uncertainty principle and de Broglie concept loses its significance in case of larger objects.Shapes of orbitalThe electron cloud represents the shape of orbital. Electron cloud is not uniform but it is dense where the probability for finding electron ismaximum.

∴

∴

∵

∴

∴

ℏ

ℏ

s-orbital do not vary with angles, i.e., they do not have directional dependence. Thus, all s-orbitals are called spherically symmetrical. Their sizeincreases with increase in the value of n. 1s-orbital has no nodal plane (the plane at which zero electron density is noticed.) 2s-orbital has onenodal plane; 3s-orbital has two nodal plane. It is thus, evident that number of nodal planes increases with increasing value of principle quantumnumber n.

All orbitals with have angular dependence. Therefore, p and d and other higher angular momentum orbitals are not sphericallysymmetrical. p-orbitals consist of two lobes to form dumbbell shaped structure. The three p-orbitals along x, y, z-axis named as p , p , p orbitalsare perpendicular to each other. All the three p-orbitals of a sub-shell have the same size and shape but differ from each other in orientation. Thesubscripts x, y and z indicate the axis along which orbitals are oriented and possess maximum electron density. Also the orbitals of a sub-shellhaving same energy are referred as degenerated orbitals.

The 'd' orbitals are resolved into five shapes. Each of the d-orbital possesses same energy but differ in their orientation in space. However, duringcomplex formation, these splits up into different energy levels in presence of ligands. Also four of the d-orbitals presence of ligands. Also four ofthe d-orbitlas (d , d , d and d ) contain four lobes while fifth, i.e., d consists of only two lobes along z-axis an a doughnut in the xy-plane.

The f-orbitals are resolved into 7 shapes, Their shape is complicated. However the seven orientations of f-orbitals are represented by the terms :

f , f , f , f , f , f . f

Note : 1. The plane and point at which zero electron density exists (i.e., ) is known as nodal plane or nodal point, e.g., the probability offinding electron between 1s and 2s-orbitals or between two lobes of p-orbitals is zero.

2. An orbital with quantum number n and possesses :

Angular or non-spherical nodes

Radial or spherical nodes

Thus, total nodes in an orbital

3. For one electron systems (an atom or an ion) the energy of orbital depends only on the number of nodes, i.e., on n and not on and . It istherefore, in H-atom or He energy levels of orbitals in a shell are same, i.e., energy level of 3s = 3p = 3d.

4. As r approaches zero, the wave function vanishes for all orbitals except the s-orbitals, thus, only an electron in 1s-orbital can penetrate thenucleus, i.e., have a finite probability of being found right at the nucleus.Photoelectric EffectAccording to Einstein-On exposing a metal surface to radiations, a part of photon's energy (say W) is used by the electrons to escape form themetal, the remaining imparts the kinetic energy (1/2 mu ) to the photo-electrons, If the incident radiation has frequency v, then its photons haveenergy hv, it follows form the conservation of energy principle that,

or

The equation shows that if KE is plotted against frequency of incident radiation, a straight line is obtained with a slope equal to Planck's constantThe equation expresses the fact that the metal provided the photon energy (i,e, hv) is greater than the binding energy or work function (W) of theelectron in the metal, Further the released electron will escape out with kinetic energy equal to (hv - W).

Instead of irradiating a metal, one can irradiate atoms with photons of known frequency, the above equation may be written as hv = IE + KE.

This suggests that the photon energy is partly used to knock out an electron form the atom (i.e., IE) and the remainder as the kinetic energy of thereleased photo-electron,

Note : An atom or a molecule can absorb only one photon.

x y z

xy yz xz x2 - y 2 z2

y(x - y )2 2 y(z - x )2 2 x(z - y )2 2 x(x - y )2 2 z(x - y )2 2 z(x - y )2 2 xyz

+

2

Characteristics of photo-electric emission

(i) The emission of photo-electrons takes place immediately after the light is incidented on metal surface, As soon as the photon falls on the metalsurface, one of the electrons in the metals absorbs it as such and it may get ejected. Thus, there is no time lag between incidence of light on metaland emission of photo-electrons,

(ii) The number of photo-electrons emitted per second is proportional to the intensity of light used.

(iii) The kinetic energy of photo-electrons varies from zero to maximum value depending upon the frequency of light used and not on intensity ofincident radiation of same wavelength. Here, the term intensity of light is used for number of photons falling per unit area per second. However,it intensity is referred in terms of amount of energy falling per unit area per second then kinetic energy of photo-electrons also varies with varying

of intensity of light, e.g., 100 photons of a wavelength, each having energy 'a' eV is falling over per unit area per second (a total of 100 eVenergy) and 50 photons of such wavelength that each having energy '2a' eV (a total of 100a eV energy) falling over per unit area per second willemit photo-electrons of different kinetic energy. Also, the number of electrons emitted per second will depend on the wavelength of radiationused for the same intensity of radiation.

(iv) Electrons are not emitted when the light has a frequency lower than a certain threshold frequency v . The value of v varies from metal tometal depending upon its ionisation energy. If KE = 0, then

hv = work function (W)

(v) The photo-electric effect is an evidence of particle nature of electron.

(vi) If V is the potential applied on surface so that velocity of photo-electrons becomes zero, the value (eV ) is known as stopping potentialwhich is numerically equal to kinetic energy, i.e.,

Note : Cs having lowest ionisation energy is commonly used in photo-electric cells.Special points for atomic spectra of H1. The first line of each series corresponds to n = 1 for Lyman, n = 2 for Balmer and n = n + 1 for the corresponding line.

2. Thus, mth and nth line of Lyman series can be given as :

Thus, ratio of mth and nth line of Lyman series can be written as

Lyman series :

Similarly For Balmer series

Balmer series :

3. The wavelength of line in each series is maximum for I line as is minimum.

4. The wavelength of line in each series is minimum for the line having is maximum, i.e., and for Lyman series; and for Balmer and so on.

5. For Balmer series, the first line is referred as line, the second line as line and so on.

6. The series limit of any series exists when electron jumps from infinity to n shell, i.e., last line represents series limit of each respective series.

7. The continuum in line spectra is noticed beyond a certain limit, i.e., after certain value of n.

8. Only Lyman series is observed in emission and absorption spectrum both. This is due to the fact that usually electrons in an atom lies in groundstate,

9. For n = 6 and n > 6, the Humphry series has also discovered,

10. The restriction of emission of line spectrum for H-atom or H-like species has been given by selection rule. The transition is allowed for H-likespecies or H-atom if it has.

0 0

0

0 0

1 1 2 1

1

1 2

It is thus evident that transition from 4p to 3p will not give line spectrum.Moseley's Work and Atomic NumberMoseley's introduced the concept of atomic number. He found that on bombarding the anti cathode with cathode rays, X-rays are emitted whosewavelength are characteristic of the element bombarded. Moseley showed that the frequency (v) of a given line in the spectrum of X-rays wasrelated to that atomic no. (Z) of the element by the expression :

where a is proportionality constant and b is a constant for all the lines of a given series. A straight line graph was obtained in values. This ledMoseley to modify the periodic law - The properties of elements are in periodic function to their atomic number and not atomic weight asproposed by Mandeleef.Photo electric effectWhen light of an appropriate frequency (or correspondingly of an appropriate wavelength) is incident on a metallic surface, electrons areliberated from the surface. This observation is known as photoelectric effect. Photoelectric effect was first observed in 1887 by Hertz. Forphotoemission to take place, energy of incident light photons should be greater than or equal to the work function of the metal.

or ......... (i)

or

Here, is the minimum frequency required for the emission of electrons. This is known as threshold frequency f .

Thus, (threshold frequency) ...... (ii)

Here, is the largest wavelength beyond which photoemission does not take place. This is called the threshold wavelength .

Thus, (threshold wavelength) ...... (iii)

Hence, for the photoemission to take place either of the following conditions must be satisfied.

...... (iv)Dipole MomentDipole moment = ionic charge × ionic distance

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