Engineering Chemistry- Jain & Jain.pdf

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c t. po gs lo.b ty si er iv un al ic og ol hn ec at al r ke :// tp ht http://keralatechnologicaluniversity.blogspot.com  c t. po gs lo.b ty si er iv un http://keralatechnologicaluniversity.blogspot.com al ic og ol hn ec at please send ebooks questionpapers study materials...etc to [email protected] so that it will be helpful to your fellow students and al teachers r ke : // tp ht http://keralatechnologicaluniversity.blogspot.com  c t. po gs lo.b ty si er iv un al ic og ol hn ec at al r ke :// tp ht http://keralatechnologicaluniversity.blogspot.com  c t. po gs lo.b ty si Preface er iv The object of the present book is to serve the students with a very elementary knowledge of chemistry. The syllabi of chemistry taught in the name of engineering chemistry in different un engineering and technology degree colleges is of very diverse in nature. It is rather quite impos- sible to give a complete coverage of all the topics in a limited space, but the authors have dealt al with modern views of the topics of the syllabi and attempted to give a major coverage of the recent syllabi taught in various institutions. ic As teachers of some experience, the authors are well aware of the great value for attach- ing the short questions and answers as well as solutions of the numerical problems. They gave og due weightage regarding the matter in writing the book. In an effort to make the book as comprehensive as possible, a large number of topics have ol been dealt with and the authors hope that this will serve the purpose of making the book useful as a text book of chemistry for engineering degree colleges all over India. hn The authors wish to express deep sense of gratitude to their beloved student Sri Rajib Das who assisted throughout in writing the book. ec R Mukhopadhyay at Sriparna Datta r al ke :// tp ht http://keralatechnologicaluniversity.blogspot.com (v) C8—d:\N-engche\TITLE.pm5 ii  c t. po gs lo.b ty si er iv This page un intentionally left al ic blank og ol hn ec at al r ke :// tp ht http://keralatechnologicaluniversity.blogspot.com  c t. po gs lo.b ty si er iv This page un intentionally left al ic blank og ol hn ec at al r ke :// tp ht http://keralatechnologicaluniversity.blogspot.com  c t. po gs lo.b ty si Contents er iv 1 Atoms and Molecules.................................................................................... 1–11 Wave Mechanical Concept of Atom.................................................................................... 1 un Application of Schrödinger Equation................................................................................. 6 Probability Distribution...................................................................................................... 7 al Exercises............................................................................................................................. 11 ic 2 Valency and Chemical Bonding.............................................................. 12–42 Electronegativity............................................................................................................... 14 og Hydrogen Bond.................................................................................................................. 15 Dipole Moment.................................................................................................................. 16 ol Chemical Bonding (Wave-Mechanical Concept).............................................................. 18 VSEPR Theory and Molecular Model.............................................................................. 25 hn Aromatic Character........................................................................................................... 33 Short Questions and Answers.......................................................................................... 35 ec Exercises............................................................................................................................. 41 at 3 Nuclear Chemistry...................................................................................... 43–67 Radioactivity...................................................................................................................... 43 al Nuclear Fission.................................................................................................................. 47 Nuclear Reactors............................................................................................................... 50 r Uses of Radioisotopes........................................................................................................ 53 ke Short Questions and Answers.......................................................................................... 62 Exercises............................................................................................................................. 65 // 4 Thermodynamics....................................................................................... 68–103 : The First Law of Thermodynamics.................................................................................. 70 tp Thermochemistry.............................................................................................................. 80 Bond Energy...................................................................................................................... 83 ht Second Law of Thermodynamics...................................................................................... 84 Third Law of Thermodynamics........................................................................................ 87 Short Questions and Answers.......................................................................................... 98 Exercises........................................................................................................................... 100 5 Reaction Dynamics/Chemical Kinetics............................................. 104–136 Introduction..................................................................................................................... 104 http://keralatechnologicaluniversity.blogspot.com (vii) C8—d:\N-engche\TITLE.pm5 iii  c t. po (viii) gs Mathematical Formulation of First Order Reaction..................................................... 106 lo Mathematical Formulation of a Second Order Reaction.............................................. 108 Third Order Reaction...................................................................................................... 111.b Disturbing Factors in the Determination of an Order.................................................. 116 Collision Theory............................................................................................................... 120 ty Solved Examples.............................................................................................................. 121 Short Questions............................................................................................................... 131 si Short Questions and Answers........................................................................................ 131 Exercises........................................................................................................................... 135 er 6 Catalyst......................................................................................................137–149 iv Catalyst............................................................................................................................ 137 Definition......................................................................................................................... 137 un Types of Catalyst............................................................................................................. 138 Short Questions and Answers........................................................................................ 143 al Catalytic Applications of Organometallic Complexes................................................... 145 Exercises........................................................................................................................... 148 ic 7 Mechanism of Organic Reactions....................................................... 150–183 og Reaction Types................................................................................................................. 150 Energy Changes During the Progress of a Reaction..................................................... 153 ol Resonance........................................................................................................................ 157 Steric Hindrance.............................................................................................................. 158 hn Isomerism......................................................................................................................... 160 R-S System of Nomenclature.......................................................................................... 166 ec E and Z Nomenclature.................................................................................................... 170 Short Questions and Answers........................................................................................ 171 at Exercises........................................................................................................................... 182 al 8 Ionic Equilibrium.................................................................................... 184–203 Law of Mass Action and Ionisation................................................................................ 184 r Acids and Bases............................................................................................................... 185 ke pH Scale........................................................................................................................... 187 Buffer Solutions............................................................................................................... 189 // Solubility Product............................................................................................................ 192 Solved Examples.............................................................................................................. 195 : Short Questions and Answers........................................................................................ 197 tp Exercises........................................................................................................................... 202 ht 9 Electrochemistry..................................................................................... 204–229 Introduction..................................................................................................................... 204 Electrolysis....................................................................................................................... 204 Faraday’s Law of Electrolysis......................................................................................... 205 Relative Speeds of Ions During Electrolysis (Transport Number)............................... 208 Determination of Transport Number (Hittorf’s Method).............................................. 210 Specific Conductance....................................................................................................... 211 Conductometric Titration............................................................................................... 217 http://keralatechnologicaluniversity.blogspot.com C8—d:\N-engche\TITLE.pm5 iv  c t. po (ix) gs Solved Examples.............................................................................................................. 219 lo Short Questions and Answers........................................................................................ 223 Exercises........................................................................................................................... 228.b 10 Electrochemical Cells............................................................................. 230–265 ty Electrode Potential.......................................................................................................... 230 Interpretation of the Electrochemical Series................................................................ 234 si Latimer Diagram............................................................................................................. 235 Frost Diagram.................................................................................................................. 237 er Concentration Cell........................................................................................................... 239 Indicator Electrodes........................................................................................................ 242 iv Battery............................................................................................................................. 248 Solved Examples.............................................................................................................. 252 un Short Questions and Answers........................................................................................ 256 Exercises........................................................................................................................... 263 11 al Phase Rule................................................................................................. 266–277 ic Introduction..................................................................................................................... 266 The Phase Rule................................................................................................................ 267 og The Water System........................................................................................................... 268 Sulphur System............................................................................................................... 269 ol Eutectic Systems............................................................................................................. 271 Tin-Magnesium System.................................................................................................. 273 hn Iron-Carbon Alloy System............................................................................................... 273 Solved Problem................................................................................................................ 275 ec Short Questions and Answers........................................................................................ 275 Exercises........................................................................................................................... 277 at 12 Colloids....................................................................................................... 278–291 al Introduction..................................................................................................................... 278 Classification of Colloids................................................................................................. 278 r Preparation of Colloidal Solutions................................................................................. 280 ke Purification of Colloidal Solutions.................................................................................. 282 Properties of Colloidal Solutions.................................................................................... 283 // Coagulation of Colloids................................................................................................... 285 Protection of Colloid........................................................................................................ 286 : Application of Colloids..................................................................................................... 288 tp Short Questions and Answers........................................................................................ 289 Exercises........................................................................................................................... 290 ht 13 Transition Metal Chemistry................................................................. 292–307 Transition Metals............................................................................................................ 292 Crystal Field Theory (CFT)............................................................................................. 303 Short Questions and Answers........................................................................................ 306 Exercises........................................................................................................................... 307 http://keralatechnologicaluniversity.blogspot.com C8—d:\N-engche\TITLE.pm5 v  c t. po (x) gs 14 Metallurgy................................................................................................. 308–328 lo Introduction to the Study of Metals............................................................................... 308 Common Minerals........................................................................................................... 309.b Ores.................................................................................................................................. 311 Fluxes............................................................................................................................... 313 ty Furnaces........................................................................................................................... 314 si Powder Metallurgy.......................................................................................................... 320 Some Specific Examples of Extraction of Metals.......................................................... 322 er Exercises........................................................................................................................... 326 15 Adhesives................................................................................................... 329–337 iv Adhesives......................................................................................................................... 329 Adhesive Strength Development.................................................................................... 330 un Technique of Bonding...................................................................................................... 331 Classification of Adhesives............................................................................................. 333 al Short Questions and Answers........................................................................................ 336 Exercises........................................................................................................................... 336 ic 16 Explosives and Propellants..................................................................338–349 og Explosives........................................................................................................................ 338 Classification of Explosives............................................................................................. 339 ol Manufacture of Important Explosives........................................................................... 343 Propellants....................................................................................................................... 345 hn Short Questions and Answers........................................................................................ 348 Exercises........................................................................................................................... 348 ec 17 Water Treatment..................................................................................... 350–378 at Sources of Water.............................................................................................................. 350 Hardness of Water........................................................................................................... 351 al Sludge and Scale Formation in Boilers.......................................................................... 352 Softening of Water........................................................................................................... 356 r Cold lime-Soda Process................................................................................................... 357 ke Hot lime-Soda Process..................................................................................................... 358 Permutit or Zeolite Process............................................................................................. 359 // Ion Exchange or Demineralization................................................................................. 360 Treatment of Water for Domestic Use........................................................................... 362 : Chemical Analysis of Water............................................................................................ 366 tp Short Questions and Answers........................................................................................ 372 Exercises........................................................................................................................... 375 ht Problems........................................................................................................................... 377 18 Fuels and Combustion........................................................................... 379–414 Introduction..................................................................................................................... 379 Calorific Value................................................................................................................. 379 Solid Fuels........................................................................................................................ 383 Liquid Fuels..................................................................................................................... 390 http://keralatechnologicaluniversity.blogspot.com C8—d:\N-engche\TITLE.pm5 vi  c t. po (xi) gs Gaseous Fuels.................................................................................................................. 399 lo Solved Examples.............................................................................................................. 404 Short Questions and Answers........................................................................................ 407.b Exercises........................................................................................................................... 412 ty 19 Silicate Technology................................................................................. 415–438 Introduction..................................................................................................................... 415 si Cement............................................................................................................................. 418 Glass................................................................................................................................. 422 er Pottery and Porcelain...................................................................................................... 427 Refractories...................................................................................................................... 430 iv Short Questions and Answers........................................................................................ 435 Exercises........................................................................................................................... 437 un 20 Polymers.................................................................................................... 439–470 al Polymerization................................................................................................................. 439 Plastics (Resins)............................................................................................................... 445 ic Important Thermoplastics.............................................................................................. 449 Important Thermosetting Resins................................................................................... 453 og Rubber.............................................................................................................................. 460 Miscellaneous Polymers.................................................................................................. 464 ol Short Questions and Answers........................................................................................ 465 Exercises........................................................................................................................... 468 hn 21 Paints.......................................................................................................... 471–480 ec Paints............................................................................................................................... 471 Varnishes......................................................................................................................... 475 at Lacquers........................................................................................................................... 477 Enamels and Japans....................................................................................................... 477 al Short Questions and Answers........................................................................................ 478 Exercises........................................................................................................................... 480 r 22 Solid State Chemistry............................................................................ 481–503 ke Crystal.............................................................................................................................. 481 // Fundamental Law of Crystallography........................................................................... 482 Crystal Lattice................................................................................................................. 485 : Cubic Crystals................................................................................................................. 487 tp Transistors (Semiconductor Triodes)............................................................................. 492 Elements of Band Theory............................................................................................... 493 ht Conductors, Semiconductors and Insulators................................................................. 496 Problems........................................................................................................................... 501 Exercises........................................................................................................................... 503 23 Chromatography..................................................................................... 504–511 Introduction..................................................................................................................... 504 Types of Chromatography............................................................................................... 504 Exercises........................................................................................................................... 511 http://keralatechnologicaluniversity.blogspot.com C8—d:\N-engche\TITLE.pm5 vii  c t. po (xii) gs 24 Instrumental Methods of Analysis...................................................... 512–547 lo Introduction..................................................................................................................... 512 Some Terms Concerning UV.......................................................................................... 513.b Beer-Lambert’s Law........................................................................................................ 519 Infrared Spectroscopy..................................................................................................... 522 ty Some IR Spectra.............................................................................................................. 525 si Shielding, Deshielding and Chemical Shift................................................................... 529 Mass Spectrometry.......................................................................................................... 537 er Short Questions and Answers........................................................................................ 540 Exercises........................................................................................................................... 547 iv 25 Photochemistry........................................................................................548–560 Singlet and Triplet States............................................................................................... 549 un Properties of the Excited States..................................................................................... 549 Photolysis......................................................................................................................... 550 al Types of Photophysical Pathways.................................................................................. 552 Photochemical Processes for Excited Molecules............................................................ 554 ic Photosynthesis................................................................................................................. 556 Exercises........................................................................................................................... 560 og 26 Role of Metals in Biology....................................................................... 561–570 ol Iron................................................................................................................................... 562 Copper.............................................................................................................................. 563 hn Zinc................................................................................................................................... 563 Manganese....................................................................................................................... 564 ec Cobalt............................................................................................................................... 564 Nickel............................................................................................................................... 564 at Calcium and Magnesium................................................................................................ 565 Electron Transfer........................................................................................................... 565 al Electron Transport and Oxidative Phosphorylation..................................................... 567 Short Questions and Answers........................................................................................ 569 r Exercises........................................................................................................................... 570 ke 27 Pollution Prevention and Waste Minimisation............................... 571–589 // Air Pollution..................................................................................................................... 571 Water Pollution................................................................................................................ 579 : Soil Pollution.................................................................................................................... 583 tp Radioactive Pollution...................................................................................................... 584 Noise Pollution................................................................................................................. 585 ht Thermal Pollution............................................................................................................ 586 Short Questions and Answers........................................................................................ 587 Exercises........................................................................................................................... 589 http://keralatechnologicaluniversity.blogspot.com C8—d:\N-engche\TITLE.pm5 viii  c t. po gs lo 1.b ty si Atoms and Molecules er iv WAVE MECHANICAL CONCEPT OF ATOM un Wave Nature of the Electron We have seen upto Bohr’s concept of atom that the electron is treated as a particle. The concept al of wave nature of matter came from the dual character of radiation, which sometimes behaves as a wave and sometimes as a particle. de-Broglie pointed out in 1924 that radiation including ic visible light, infrared, ultraviolet and X-rays behave as waves in propagation experiments based on interference and diffraction. These experiments conclusively proved the wave nature og of these radiations. However, radiation behaves as a particle in interaction experiments which include black body radiation, photoelectric effect and Compton effect. Here radiation interacts ol with matter in the form of photons or quanta. Of course radiation cannot exhibit its particle and wave properties simultaneously. hn • A wave is specified by its (i) frequency, (ii) wavelength (λ), (iii) phase, (iv) amplitude, (v) intensity. ec • A particle is specified by its (i) mass (m), (ii) velocity (v), (iii) momentum (p), (iv) energy (E). Moreover, a particle occupies a definite position in space. at In view of the above facts, it is rather difficult to accept two conflicting ideas, that radiation is a wave which is spread out over space and also a particle which is localised at a point in al space. r de-Broglie’s Equation (de-Broglie’s matter waves) ke His suggestion was based on: as radiation like light can act sometimes as a wave and sometimes like a particle, small particles like electron which are considered as minute particles // should also act as waves for sometimes. According to his hypothesis, all matter particles like electrons, protons, neutrons, atoms or molecules have an associated wave with them which is : called matter wave or pilot wave or de-Broglie’s wave. tp The wavelength of the matter wave is given by h h ht λ= = p mv where ‘m’ is the mass of the material particle, ‘v’ is the velocity and ‘p’ is the momentum. The above equation is known as de-Broglie’s wave equation. According to the wave mechanical model of the atom, an electron behaves as a standing wave which goes round the nucleus in a circular orbit. The only necessary condition for the establishment of such a stationary wave is that the length of the orbit should be a whole number multiple of the wavelength of the electron as shown in the following Fig. (1.1). http://keralatechnologicaluniversity.blogspot.com 1  c t. po 2 ENGINEERING CHEMISTRY gs If r is the radius of the circular orbit, then lo 2πr = nλ. Now, λ = h/mv..b nh r ty ∴ 2πr = mv or mvr = nh/2π si where n = 1, 2, 3, etc. Since ‘mvr’ is the angular momentum of the electron as a er particle, we see that wave mechanical picture leads naturally to Fig. 1.1 Bohr’s postulate that the angular momentum is an integral multiple of h/2π. Other quantum iv conditions can also be derived similarly, i.e., the angular momentum is quantised. Moreover, the concept of an electron as a standing wave rather than a particle revolving un in an orbit also removes the difficulty met in Bohr’s theory regarding non-radiation of energy by the revolving electron. The New Atomic Picture al ic In Bohr’s atomic model, nucleus of an atom is surrounded by particles known as electrons, which revolve in defined shells or orbits. This model has been replaced by wave mechanical og model, i.e., replaced by de-Broglie’s electron wave. These waves form stationary waves with their nodes and antinodes. Hence, instead of being localised at a point, the whole of electron ol mass and charge is spread out uniformly throughout the space immediately surrounding the nucleus of the atom (Fig. 1.2). hn At nodes where the motion is practically zero, there is assumed to be a little or no charge while at the antinodes the amount of charge is maximum. However, it is still customary to talk ec of orbits and shells for the simple reason that even according to the new picture of the atom, electrons are found to distribute their charges in such a way that something analogous to at shells is formed. New wave model accepts the electron distribution to be three dimensional. al Charge Density r ke // Distance : tp ht K L M N O Fig. 1.2 Old and new models of an atom. Old Bohr’s model is represented by orbits of an atom (Z = 37) (Fig. 1.2) and the new wave model is represented by a graph of spherically symmetrical electronic charge with several maxima corresponding to the discrete K, L, M, N shells. http://keralatechnologicaluniversity.blogspot.com C-8\N-ENGCHE\ECH1-1.PM5 2  c t. po ATOMS AND MOLECULES 3 gs Heisenberg’s Uncertainty Principle lo The dual nature of the electron implies that any precise measurement of its position would create uncertainty in measurement of its momentum and position. The Heisenberg.b uncertainty principle states that l It is impossible to determine simultaneously both the position and the ty momentum of a particle with accuracy. si ∆ x.∆p ≥ h/2π. The above expression is known as uncertainty relation where ∆ x = change in position, er ∆p = change in momentum and h = Planck’s constant. The relation implies that a simultaneous and precise measurement of both position and iv momentum (velocity) of a dynamic particle like electron is impossible and the extent of inherent uncertainty in any such measurement is of the order of h (Planck’s constant). un Uncertainty Principle and Bohr’s Theory—Concept of Probability Bohr had postulated that electrons revolve in well defined orbits with fixed velocities al (energy). But according to uncertainty principle since an electron possesses wave nature, it is impossible to determine its position and momentum simultaneously. On the basis of this ic principle therefore Bohr’s model of atom no longer stands. The best way is to predict the probability of finding an electron with probable velocity with definite energy in a given region og of space in given time. Thus the uncertainty principle which gives the wave nature of the electron only provides probability of finding an electron in a given space. It is for this reason ol the classical concept of Bohr’s model of atom has been replaced by probability approach. Schrödinger Wave Equation hn It is a differential equation capable of describing the motion of an electron. In an ordinary material wave the displacement of whatever is vibrating about its mean position is given by ec x FG IJ y = a sin 2π ft − H K...(1) at λ where, y = displacement at time t and at distance x from origin. al a = maximum displacement from mean position. λ = wavelength. r f = frequency of vibration. ke When differentiated twice with respect to x, it becomes d2y 4π 2 // + y=0...(2) dx 2 λ2 : d2 y or + k2 y = 0...(3) tp dx 2 The above equation involves only distance as the independent variable. Its solution is ht y = a sin 2π x/λ which defines a standing wave. At each point along the wave in space, y varies periodically with frequency f. Let us now see how this equation can be applied to specify an electron in motion. As we know, the total energy E of an electron is partly kinetic and partly potential. 1 E= mv2 + V 2 ∴ mv = 2m(E − V). http://keralatechnologicaluniversity.blogspot.com C-8\N-ENGCHE\ECH1-1.PM5 3  c t. po 4 ENGINEERING CHEMISTRY gs Now, de-Broglie wavelength is given by lo h h λ= =..b mv 2m(E − V) Substituting this value in equation (1) and replacing y by ψ as is customary, we get ty d 2ψ 8 π 2m + (E − V) ψ = 0...(4) si dx 2 h2 The equation (4) is known as the time-independent Schrödinger equation in one er dimension. For three-dimensional motion of an electron, this equation becomes, iv d 2 ψ d 2 ψ d 2 ψ 8π 2 m + + + (E − V)ψ = 0 dx 2 dy 2 dz 2 h2 un 8π 2m or ∇2ψ + (E − V)ψ = 0 al...(5) h2 ic This Schrödinger equation is a basic equation of wave mechanics. As, p = mf is a starting point of classical mechanics. It is seen from equations (3) and (4) that og 8 π 2m k2 = × (E − V)...(6) h2 ol So, Schrödinger equation can be employed for determining the total energy of an electron. The potential energy V of an electron is specified in terms of space co-ordinate not on time, i.e., hn Schrödinger wave equation is time independent. Like other differential equation, the Schrödinger wave equation is also governed by boundary conditions. Generally its solutions are only attained ec for certain energy values called characteristic or eigen values. The corresponding wave function ψ which is generally complex is called characteristic or eigen function. So, we can write, at ψ = a + ib and its conjugate ψ* = a – ib and their product, ψ.ψ* = (a – ib) (a + ib) = a2 + b2, which is real. al For a one dimension system, |ψ2|dx represents the probability of finding an electron within a range of x and x + dx. But in three dimension system |ψ2|dV represents the probability r of an electron within the volume range of V to (V + dV). ke Significance of ψ and ψ2 // l ψ denotes the amplitude of a three dimensional stationary electron wave. l According to the Heisenberg’s uncertainty principle, it is impossible to locate an electron : in an atom with precision but the nature of the wave function ψ is such that |ψ|2 tp expresses the probability of finding an electron in a definite volume of space around the nucleus. This mathematical expression displays how the probability of finding an ht electron varies in space. The total probability of finding an electron in space extending to infinity is expressed as follows: z +∞ ψ2d V = 1 −∞ where dV = dx.dy.dz. This is known as the condition of normalisation and the corresponding wave function ψ is said to be normalised. Because of the spherical symmetry of an atom the wave functions are most satisfactorily expressed in terms of http://keralatechnologicaluniversity.blogspot.com C-8\N-ENGCHE\ECH1-1.PM5 4  c t. po ATOMS AND MOLECULES 5 gs spherical polar co-ordinate system with nucleus at the origin, changing from the Cartesian lo co-ordinates to polar co-ordinates, the Schrödinger wave equation assumes the following form..b FG 1 ∂ 2 ∂ψ IJ 1 ∂2 ψ 1 ∂ ∂ψ FG IJ r H + K + 2. sin θ. H + 8π 2 µ(E − V) = 0 K...(A) ty 2 2 2 2 r ∂r ∂r r sin θ ∂ϕ r sin θ ∂θ ∂θ where r, θ and ϕ are the polar co-ordinates of the electron with respect to the nucleus si (N) as origin [see Fig. 1.3 (a) and (b)]. er iv m e m e un r N E r al E N  ic og (a) (b) Fig. 1.3 ol It can be shown mathematically that each permitted solution of the wave equation (A) i.e., wave function ψ(r, θ, φ) can be expressed as hn ψ (r, θ, ϕ) = R(r). Θ(θ). Φ(φ) ec where, R(r) is a function that depends on the distance from nucleus, which depends on the quantum number n, l (the principal and azimuthal quantum numbers, respectively). The at function Θ(θ) is function of θ depending on azimuthal quantum number (l) and magnetic quantum number (ml). Φ(φ) is a function of ϕ which depends on magnetic quantum number (ml). The total wave function ψ(r, θ, φ), which gives the total probability of finding an electron is called al the atomic orbital. The wave function ψ(r, θ, φ) is denoted as a product of two functions, (i) radial wave function and (ii) angular wave function. r ke The radial wave function R(r) shows the variation of wave function with r keeping θ and ψ constants i.e., it represents the variation of ψ in the same direction. The angular wave function is a joint function θ and φ which determines the variation of ψ in different directions at a fixed // radial distance r. : For any s orbital (1s, 2s, 3s, etc.), the angular part of the wave function is always the tp same whatever be the principal quantum number n. The angular dependence of p-orbital is also not influenced by the principal quantum number. This is also same for d-orbital. ht Highlight: For a small particle like an electron moving in three dimension in the field of nucleus, the Schrödinger wave equation is ∂ 2 ψ ∂ 2 ψ ∂ 2 ψ 8π 2 m + 2 + 2 + (E − V)ψ = 0 ∂x 2 ∂y ∂z h2 where m = mass of the electron, (contd...) http://keralatechnologicaluniversity.blogspot.com C-8\N-ENGCHE\ECH1-1.PM5 5  c t. po 6 ENGINEERING CHEMISTRY gs V = potential energy of the electron, lo E = total energy of the electron,.b h = Planck’s constant, ψ = wave function or the eigen function of the electron representing the ty amplitude of the wave associated with the electron, x, y, z = co-ordinates of the electron. si The solution of the above second order differential equation furnishes the values of E, the quantised or allowed or permitted energies of electrons and ψ is the wave function. er Alternatively, the solution of the above equation produces the electron distribution in space as well as the allowed energy state of the electron. When this equation is applied to iv hydrogen atom, it is found that the equation can be solved when E assumes certain values which are related by integers. Hence, the concept of quantised energy levels and quantum un numbers are the consequences of the wave theory. Application of Schrödinger Equation al ic Particle in a box illustrating energy quantization (Like electrons in metals) Let us consider a particle of mass m which is free to move in a one dimensional box of og length l as shown in Fig. 1.4. ol   n=3 n=2 n=1 2 [x] || hn n=1 V V ec 0 L 0 L at V=0 al n=3 n=2 x=0 x=L (a) (b) (c) r ke Fig. 1.4 The potential energy V of the electron at the bottom of the box is constant, may be taken // as zero. Hence, inside the box V = 0. Let the potential energy be infinite for x < 0 and x > l. So, ψ function has to be zero at : x = 0 and for all negative values of x, since the particle is not allowed over the walls of the box. tp Similarly, ψ function must be zero for all values x > l. Alternatively, it can be stated that the particle is confined to the box and cannot exist outside. ht A general solution of Schrödinger equation is: ψ(x) = a sin kx + b cos kx For boundary conditions, we have, ψ(x) = 0 at x = 0 or ψ(0) = 0 0 = a sin 0 + b cos 0 or b = 0 Also ψ(x) = 0 at x = l http://keralatechnologicaluniversity.blogspot.com C-8\N-ENGCHE\ECH1-1.PM5 6  c t. po ATOMS AND MOLECULES 7 gs ∴ a sin kl + b cos kl = 0 lo It can only be possible when,.b nπ kl = nπ or k= l ty where x is called quantum number and is equal to 1, 2, 3... ∞. Substituting this value in eqn. 4 and also putting V = 0; we get, si n2π2 8π 2 m = E er l2 h So, (E, the total energy of the electron) = K.E. of the electron. iv n 2h 2 ∴ En = where n = 1, 2, 3... ∞. un 8ml 2 The above equation means that the particle in a box does not possess any arbitrary al amount of energy, rather it possesses discrete set of energy values i.e., its energy is quantised. A few energy levels are given below: ic n=3 h2 E3 E1 = 8ml 2 og Energy 4h 2 ol E2 = n=2 E2 8ml 2 hn n=1 E1 9h 2 E3 = 0 8ml 2 ec x=0 x=L The reason why a particle in a box i.e., bound particle possesses Fig. 1.5 a quantised energy whereas a free particle does not, can easily be at deduced from the above equation. al Highlight: Calculation of minimum energy of a particle in a box from Heisenberg’s uncertainty principle. r ke Here ∆ x = l, and the particle bounces back in the box. So, ∆p = 2p because the momentum changes from + p to – p continuously. So we can write, // h ∆ x × ∆p = l. (2p) = h or, p= 2l : tp 1 ( mv ) 2 p2 Now, E= mv2 = = 2 2m 2m ht h2 ∴ E=. 8ml 2 PROBABILITY DISTRIBUTION The directional properties of an election in an orbital of the hydrogen atom cannot be represented in one diagram. Two separate diagrams are required to meet the demand. http://keralatechnologicaluniversity.blogspot.com C-8\N-ENGCHE\ECH1-1.PM5 7  c t. po 8 ENGINEERING CHEMISTRY gs These are namely (i) an angular probability distribution i.e., how the angular position θ lo varies from nucleus with respect to a fixed axis and (ii) a radial probability distribution i.e., how r, the distance from the nucleus does vary..b (i) Angular probability distributions of orbitals i.e., shapes or boundary surfaces of orbitals gives the probability of finding an electron in a particular direction. ty An s-electron has no preferred direction in space i.e., there is an equal chance of finding si it in any direction around the nucleus. This is graphically shown in Fig. 1.6 below. y er z iv un + x al ic Fig. 1.6 og In this diagram, the nucleus of the atom is at the origin and the surface of the sphere represents the probability of finding the s-electron which is therefore same in all directions. ol A p-level has an accommodation for six electrons distributed over three p-orbitals to hn each value of the principal quantum number. These three orbitals are at right angles to each other and the three angular probability distributions are shaped like dumb-bells along the ec three co-ordinate axes and are named as px, py and pz orbitals. In the absence of magnetic field these orbitals are equivalent in energy content and are said to be triply degenerate. In the at presence of external magnetic field these vary in their energy content depending on the magnetic quantum number. The different shapes are shown in Fig. 1.7 below. al y y y r z z z ke + + // – + x x x – : – tp px py pz ht (a) (b) (c) Fig. 1.7 It is most likely that in the px orbital the electron will be found in the direction of x-axis and there is no chance of it to be found in any of the directions perpendicular to x-axis i.e., y-z plane in this case is nodal plane with zero electron density. Similarly, in the case of py and pz orbitals the electron will be found along y and z axes. http://keralatechnologicaluniversity.blogspot.com C-8\N-ENGCHE\ECH1-1.PM5 8  c t. po ATOMS AND MOLECULES 9 gs The angular probability distribution of five d-orbitals that can accommodate ten electrons lo is quite complicated. These five orbitals are named as dxy, dyz, dxz, dx2 − y 2 , dz2. These are shown.b in Fig. 1.8 below. ty y y y z z si z + – – + + – er x x x + – – + – iv + un dxy dyz dxz (a) y (b) al y (c) ic z z Negative og dough nut in – + x + + x ol x-y plane + x-y plane – hn ec dz2 px2–y2 (d) (e) at Fig. 1.8 al All the d-orbitals are equivalent in energy in the absence of magnetic field and are said to be five fold degenerate. r The set of the three orbitals namely dxy, dyz, dzx have their lobes lying symmetrically in ke between the co-ordinated axes indicated by the subscripts to d, e.g. the lobes of dxy orbital are pointing or lying in between the x and y axes. This set is referred to as t2g set. // The set of two orbitals i.e., dx 2 − y2 and dz 2 orbitals form eg set having their lobes along : the axes, e.g. The lobes of dx 2 − y2 orbital lie along the x and y axes while those of dz2 orbital lie tp along z-axis. ht (ii) The Radial Probability distribution—The angular probability graphs indicate the most probable directions of the electrons but they do not give any indication of the probable distance of the electron from the nucleus. The pr