UPKAR PRAKASHAN, AGRA–2

ByDr. Hemant Kulshrestha

&Dr. Ajay Taneja

Department of ChemistrySt. Johns College,

AGRA

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ISBN : 978-81-7482-646-6

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Code No. 1542

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CONTENTS

Section-A

1. Atomic Structure………………………………………………………… 3–212. Chemical Periodicity…………………………………………………….. 22–343. Chemical Bonding…………………………………………………..…… 35–604. Oxidation States and Oxidation Number………………………………… 61–725. Acids and Bases…………………………………………………………. 73–826. Chemistry of Elements………………………………………………...… 83–1427. Extraction of Metals……………………………………………………… 143–1498. Nuclear Chemistry………………………………………………………. 150–1619. Coordination Compounds……………………………………………….. 162–189

10. Pollution and its Control………………………………………………… 190–196

Section-B1. Bonding and Shapes of Organic Molecules……………………………… 199–2112. Chemistry of Aliphatic Compounds…………………………………….. 212–2363. Stereochemistry of Carbon Compounds………………………………… 237–2594. Organometallic Compounds…………………………………………….. 260–2665. Active Methylene Compounds…………………………………………... 267–2786. Chemistry of Aromatic Compounds…………………………………….. 279–3117. Chemistry of Biomolecules……………………………..……………….. 312–3288. Basic Principles of Applications of UV, Visible, IR and NMR

Spectroscopy of Simple Organic Molecules………………..…………… 329–342

Section-C1. Kinetic Theory of Gases and Gas Laws………………………………… 345–3602. Thermodynamics………………………………………………………… 361–3773. Solutions and Colligative Properties……………………………………. 378–3884. Phase Rule and its Applications…………………………………………. 389–3995. Chemical Equilibria……………………………………………………… 400–4086. Electrochemistry………………………………………………………… 409–4347. Chemical Kinetics……………………………………………………….. 435–4448. Photochemistry…………………………………………………………... 445–4489. Catalysis…………………………………………………………………. 449–453

10. Colloids………………………………………………………………….. 454–464

Section–A

INORGANIC CHEMISTRY

1 ATOMIC STRUCTURE

Some Expressions According to Bohr’sModel● Angular momentum,

mvr = n h2π

● Number of waves in an orbit

=Circumference of that orbit

Wavelength

=2πrλ

= 2πrh/mv

= 2π(mvr)

h

=2π ( )nh/2π

h

= n

● Radius of orbit in hydrogen and hydrogen likespecies

rn =n2h2

4π2k mZe2

For H-atom Z = 1. For the lowest orbit n = 1.hence, the radius of the first orbit of H-atom(Bohr radius), on substituting the values inabove equation

r1 = 0·529 Å

= 0·529 × 10–10 m

Radius of nth orbit of H-atom,

rn = 0·529 × n2 Å

Radius of 1st orbit in He+ ion

rn =n2

Z × 0·529 Å

r1 (He+) =12

2 × 0·529 Å

rn (He+) =rn (H)

2

● Velocity of the electron in nth orbit of H-atomand hydrogen like species

vn =2πkZe2

nhSubstituting the values for various constants,we get

vn =2·18 × 106 × Z

n ms–1

v1(H) = 2·18 × 106 × 11 ms–1

= 2·18 × 106 ms–1 (∴

n = 1, Z = 1)

v1 (He+) = 2·18 × 21 ms–1

= 4·36 × 106 ms–1 (∴

n = 1, Z = 2)

vn (He+) = 2 × vn (H) and vn (Li2+

)

= 3 × vn (H)

● Number of revolution per second of anelectron in the nth orbit

=Velocity of the electron

Circumference of the orbit

=vn

2πrn

● Energy of the electron in nth orbit

En = – k2 2π2 me4 Z2

n2h2

= –1312n2 kJ mol–1

For H-like species,

En = Z2 En

In general, En ∝ – 1n2

and En =E1

n2

4C | Chem.

● Different spectral lines in the spectrum ofhydrogen.

Series Region n1 n2

Lyman Ultra-violet 1 2, 3, 4…Balmer Visible 2 3, 4, 5…Paschen Infra-red 3 4, 5, 6…Bracket Infra-red 4 5, 6, 7…Pfund Infra-red

(far)5 6, 7, 8…

Humphrey Infra-red (near)

6 7, 8, 9…

● Wave number of spectral lines in hydrogenatom

1λ

= –v (cm–1)

= R( )1n1

2 – 1

n22

R = Rydberg constant = 109678 cm–1

For hydrogen like species1λ

= –v(cm–1)

= R × Z2 ( )1n1

2 – 1

n22

● The number of spectral lines produced whenan electron jumps from n th level to theground level (n = 1)

=n(n – 1)

2● de Broglie equation

λ =h

mv =

hp (p = Momentum)

● Particle nature of radiation can account forphotoelectric effect, reflection and refractionwhile wave nature of radiation can accountfor interference and diffraction.

● Wave character of electron was experimen-tally verified by Davission and Germer bydiffraction experiment.

● Heisenberg uncertainty principle—It isimpossible to measure simultaneously boththe position and momentum (velocity) of amicroscopic particle (like electron) withabsolute accuracy.

Δx·Δp ≥h4π

Δx ∝1Δp

● Schrodinger’s wave equation—ErwinSchrodinger (1926) gave a wave equationwhich describes the wave motion of anelectron-wave propagating in three dimen-sions (X-, Y- and Z-axes) in space. In this, thediscrete energy levels or orbits proposed byBohr are replaced by mathematical functions,ψ (Psi), which are related with the probabilityof finding electrons at various places aroundthe nucleus. The equation is written asfollows :

∂2ψ∂x2 +

∂2ψ∂y2 +

∂2ψ∂z2 +

8π2mh2 (E – V)ψ = 0

( )∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 ψ + 8π2m

h2 (E – V)ψ = 0

x, y, z = Cartesian coordinatesm = Mass of the electronn = Planck’s constantE = Total energy of the systemV = Potential energy of the system

=–Ze2

r

or ∇2ψ + 8π2m

h2 (E – V) = 0

where ∇2 = ( )∂2ψ∂x2 +

∂2ψ∂y2 +

∂2ψ∂z2

∇2 is known as Laplacian mathematicaloperator. A mathematical operator such as ∫,ddx

‚ √ etc. is always accompanied by a

mathematical quantity or mathematical func-tion upon which it operates. In ∇2ψ, ∇2 is amathematical operator and ψ is amathematical function on which ∇2 operates.Thus ∇2ψ should not mean that ∇ 2 ismultiplied by ψ. The function ψ signifies theamplitude of the electron wave and is calledwave function.

● Other forms of Schrodinger’s waveequation

(i) [ ]– h2

8π2m∇2 + V ψ = Eψ

(ii) Hψ = Eψ

where H is called Hamiltonian operator andconsists of two parts, the kinetic energy part

( )–h2

8π2m∇2 and the potential energy part

(V).

Atomic Structure | 5C

● Eigen wave function and eigen value—Being a differential equation of second order,when we solve Schrodinger wave equation forψ, we get many values of ψ. Some of thesevalues are imaginary, while other are realvalues. The imaginary values of ψ can not beaccepted. Only those values of ψ which givedefinite and acceptable value of the totalenergy (E) of the electron are acceptable. Theacceptable values of the wave functions ψ arecalled eigen wave functions. The value of Egiven by eigen wave function is called eigenvalue. Only that value of ψ is an eigen wavefunction which has the following charac-teristics—(i) It should not have more than one value

at a point in space, i.e., it should besingle valued.

(ii) It should be continuous.(iii) The function must be finite.

(iv)∂2ψ∂x2 ‚

∂2ψ∂y2 and

∂2ψ∂z2 must be continuous

function of x, y and z respectively.(v) The function must be normalised, i.e., it

should obey the relation ∫

+ ∞

– ∞ ψ2dr = 1,

where dr is a small volume element andψ2dr represents the probability of findingthe electron in small volume element dr.

● Schrodinger’s wave equations for hydro-gen atomIn case of H-atom, a single electron havingcharge equal to –e moves round the nucleuswith change equal to +e.

∴ Potential energy of the electron, V = – e2

r.

Thus, Schrodinger’s wave equation in termsof Cartesian coordinates x, y, z takes the form

( )∂∂x2 +

∂∂y2 +

∂∂z2 ψ +

8π2mh2 ( )E +

e2

rψ = 0

or ∇2ψ + 8π2m

h2 ( )E – –e2

rψ = 0

The transformation from Cartesian coordi-nates x, y, z into polar coordinates facilitatessolution of wave equation. The positionvariables in polar coordinates are r, θ and φ,where r is the radial distance of a point fromthe origin, θ is the inclination of the radialline to the z-axis and φ is the angle made withthe x axis by the projection of the radial line

in the xy plane. After this transformation theSchrodinger equation can be factored into twoequations one depending only on r and otheronly on θ and φ. The conversion of Cartesiancoordinates into polar coordinates is as—

z = r cos θy = r sin θ sin φx = r sin θ cos φ

● Thus Schrodinger wave equation in terms ofpolar coordinates is as—1r2

∂∂r

( )r2 ∂ψ(r‚ θ‚ φ)∂r

+ 1

r2 sin θ ∂∂θ

( )sin θ ∂ψ(r‚ θ‚ φ)∂θ

+ 1

h2 sin θ ∂2ψ (r‚ θ, φ)

∂φ2

+ 8π2m

h2 ( )E + e2

rψ(r, θ, φ) = 0

The solution of this equation, evolution ofψ(r, θ, φ) is possible only if ψ function can bewritten in the form

ψ(r, θ, φ) = R(r) Θ(θ) Φ (φ)where R(r) is a function which depends on ronly (distance of the electron from nucleusbut independent of the other two polar coordi-nates θ and φ. Θ(θ) is a function dependent onθ and is independent of r and φ and Φ (φ) in afunction dependent on φ only but is indepen-dent of r and θ.

● The function R(r) is known as the radialwave function while the function Θ(θ) andΦ(φ) are called angular wave function.

● The radial wave function R(r) gives theenergy and size of the orbitals, i.e., principal(n) and angular quantum numbers (l). It iswritten as Rn, l (r) or Rn, l.

The angular wave function, which is theproduct of Θ(θ) and Φ(φ) gives the shape oforbitals, i.e., magnetic quantum number (m).It is written as Θl, m (θ) or Θl, m.

● Significance of ψψψψ and ψψψψ 2—ψ refers to theamplitude of electron wave. It has got nophysical significance. However, the square ofψ, i .e., ψ2 has a physical significance. Justlike light radiations where square ofamplitude gives the intensity of light,similarly, in electron wave, ψ2 gives theintensity of electron of any point, i.e., ψ2 ishelpful in assessing the probability of electronin a particular region. Thus ψ2 is calledprobability density and ψ is referred to

6C | Chem.

probability amplitude. In order to avoidimaginary values ψψ* is used for ψ2. ψ* is acomplex conjugate of ψ. We expect that if ψextends over a finite volume in space dx, dy,dz (= dτ), then the electron would be found inthat volume. So ψψ* dτ gives the probabilityof finding the electron in the volume dτ. Thewave function ψ may be positive, negative orimaginary but the probability density ψ ψ*

will always be positive and real.● Normalized wave function—ψψ* d τ is

proportional (not equal) to the probability offinding the electron in the given volumeelement dτ. However, if we have a value of ψsuch that the integration of ψψ* from –∞ to+∞ , that is over the entire space, is notproportional but equal to the total probabilityof finding the electron, then the probabilitythat the electron is present within the volumeelement dτ becomes unity (i.e., 100%),Mathematically

∫+ ∞

– ∞ ψψdτ = 1

This is the condition of normalization and ψsatisfying this expression is said to be nor-malised. Sometimes ψ may not be normalizedwave function but then it is possible tomultiply it by a constant N, so that Nψ is alsoa solution of the wave equation.Mathematically

N2 ∫ ψψ*dτ = 1

or ∫ ψψ*dτ = N2–

where N = Normalizing constant.

● Orthogonal wave function—This means thata particular wave function must be differentfrom all other wave functions over the entirespace. The condition for othogonality is thatthe product of two wave functions, integratedover the entire space must be equal to zero.Two wave functions ψ1 and ψ2 are said to beorthogonal if

∫ ψ1ψ*2 dτ = 0

and ∫ ψ*1

ψ*2 dτ = 0

Here ψ1* and ψ2

* are complex conjugate ofψ1 and ψ2 respectively.

● Wave functions which are both normalizedand orthogonal are called orthonormal wavefunctions.

● Radial probability—Radial wave functionRn, l or R2

n, l can not be directly related withprobability of finding an electron at a point,which is at a distant r from the nucleus. Theprobability of finding electron within smallradial shell of thickness dr around the nucleuscalled radial probability

Radial probability

= Probability density × Volume of radial shell

= ψ2·4πr2 dr

= l·4πr2dr

● The radial probability distribution curvesobtained by plotting radial probability versusdistance from the nucleus for 1s, 2s, 2p, 3s,3p and 3d orbitals are given below. In case of1s orbital, the probability (ψ2) of finding 1selectron is maximum near the nucleus.However, the volume element 4πr2dr is verysmall because r is very low near the nucleus.Hence, their product is quite small. As thedistance r from the nucleus increases, ψ2 goeson decreasing whereas volume 4πr2dr goeson increasing. Their product goes onincreasing till it reaches a maximum value ata distance of 0·53 Å and then begins todecrease. An important difference betweenBohrs model and wave mechanical model liesin the fact that whereas according to theformer the electron revolves at a constantdistance of 0·53 Å from the nucleus, accord-ing to the latter the electron is most likely tobe found at this distance but there is aprobability of finding the electron at distancesshorter as well as larger than 0·53 Å.

Atomic Structure | 7C

● Radial nodes—The number of minima appe-aring in a particular curve gives the numberof radial nodes or nodal points for theorbital. The number of radial nodes is equal to(n – l – 1). Thus for 1s, 2s, 2p, 3s, 3p and 3delectrons, the number of nodal points is equalto 0, 1, 0, 2, 1 and 0 respectively. Radial nodein that point at which the probability densityof electronic charge is zero. Due to greaternumber of radial nodes for 3s orbital (= 2)than that of 1s (= 0) and 2s (= 1), 3s orbital isbigger in size and move diffused.

● Angular probability—The angular part ofthe wave function determines the shape of theelectron cloud and varies depending upon thetype of orbital involved (s, p, d or f) and itsorientation in space. Thus, the use of ψθ

2 ψφ2

represent the probability in terms of θ and φ.The angular wave function is independent ofthe principal quantum number. Some typicalangular functions are—

l = 0, ml = 0, ΘΦ = ( )14π

1/2s-orbital

l = 1, ml = 0, ΘΦ = ( )34π

1/2 cos θ pz-orbital

l = 1, ml = 1, ΘΦ = ( )34π

1/2 sin θ cos θ

px-orbital

l = 2, ml = 0, ΘΦ = ( )516π

1/2 (3 cos2 θ – 1)

dz2 orbital

For an s-orbital angular wave function isindependent of angle and is of constant value.Hence, its graph is circular or in threedimensions spherical. For pz orbital we obtaintwo tangent spheres. The px and py orbitals areidentical in shape but are oriented along the x-and y -axes.

● The number of angular nodes in just equal tol.

● A complete representation of electrondistribution probability includes both radial

and angular probabilities and requires a threedimensional model.

● Total number of nodes in a shell = (n – 1)Angular nodes = l

Radial/spherical nodes = n – l – 1

● There are l nodal surfaces in the angulardistributional functions of all orbitals.

● Quantum numbers—The solution of theSchrodinger wave equation for hydrogenatom yields three quantum numbers viz.,principal, angular and magnetic. The fourthone viz., spin quantum number does notfollow from the wave mechanical treatment.

■ Principal quantum number (n)—Itarises from the solution of radial part ofwave function. It is required to explainthe main lines of a spectrum. It tellsabout the

(i) Main shell in which the electronresides.

(ii) Approximate distance of theelectron from the nucleus.

(iii) Energy of the shell, the energyincreases with increasing n.

(iv) Maximum number of electronspresent in the shell (2n2).

Values of n = 1, 2, 3, 4, 5 … ∞K, L, M, N, O …

As the value of n increases, the electrongets farther away from the nucleus andits energy increases.

■ Azimuthal or subsidiary quantumnumber (l)—It also arises from theradial part of wave function. It is requiredto explain the fine structure of the linespectrum. It tells about the

(i) number of sub-shells present in anymain shell.

(ii) relative energies of the sub-shells.

(iii) shapes of orbitals.

Its values for a particular value of n are, l= 0 to n – 1, i.e., when n = 1, l = 0 (onesub-shell), n = 2, l = 0, 1 (two sub-shells) n = 3, l = 0, 1, 2 (three sub-shell),n = 4, l = 0, 1, 2, 3 (four sub-shells)

Quantum No. l 0 1 2 3 4 5 …

Sub-shell notation s p d f g h …

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Publisher : Upkar Prakashan ISBN : 9788174826466 Author : Dr. HemanthKulseshtra, Dr.Ajay Taneja

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