CH111 Physical Chemistry Lectures 1-3 PDF

CH111 : Physical Chemistry G. Naresh Patwari Room No. 206; Department of Chemistry [email protected] [email protected] Recommended Texts (Physical Chemistry) Physical Chemistry –I.N. Levine Physical Chemistry – P.W. Atkins Physical Chemistry: A Molecular App...

CH111 : Physical Chemistry G. Naresh Patwari Room No. 206; Department of Chemistry [email protected] [email protected] Recommended Texts (Physical Chemistry) Physical Chemistry –I.N. Levine Physical Chemistry – P.W. Atkins Physical Chemistry: A Molecular Approach – McQuarrie and Simon Websites: http://www.chem.iitb.ac.in/~naresh/courses/CH103-L1.pdf www.chem.iitb.ac.in/academics/menu.php IITB-Moodle http://moodle.iitb.ac.in http://ocw.mit.edu/OcwWeb/web/courses/courses/index.htm#Chemistry http://education.jimmyr.com/Berkeley_Chemistry_Courses_23_2008.php What Do You Get to LEARN?  Why Chemistry?  Classical Mechanics Doesn't Work all the time!  Is there an alternative? QUANTUM MECHANICS  Origin of Quantization & Schrodinger Equation  Applications of Quantum Mechanics to Chemistry  Atomic Structure; Chemical Bonding; Molecular Structure  Why should Chemistry interest you? Chemistry plays major role in 1. Daily use materials: Plastics, LCD displays 2. Medicine: Aspirin, Vitamin supplements 3. Energy: Li-ion Batteries, Photovoltaics 4. Atmospheric Science Green-house gasses, Ozone depletion Haber Process 5. Biotechnology Insulin, Botox 6. Molecular electronics Transport junctions, DNA wires LCD Display Transport Junctions Haber Process The Haber process remains largest chemical and economic venture. Sustains third of worlds population Quantum theory is necessary for the understanding and the development of chemical processes and molecular devices Classical Mechanics Newton's Laws of Motion Classical Mechanics Newton's Laws of Motion 1. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. 2. The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma. The direction of the force vector is the same as the direction of the acceleration vector 3. For every action there is an equal and opposite reaction. Black-Body Radiation; Beginnings of Quantum Theory Hot objects glow Rayleigh-Jeans law was based on equipartitioning of energy Planck’s hypothesis The permitted values of energies are integral Towards multiples of frequencies Ultraviolet Catastrophe E = nh nhc/n = 0,1,2,… b max  Value of ‘h’ (6.626 x 10-34 J s) T 8 kT  was determined by fitting the 4 experimental curve to the Planck’s radiation law Planck’s radiation law 8 hc Planck did not believe in the   e 5  hc  kT 1  quantum theory and struggled to avoid quantum theory and make its influence as small as possible  Heat Capacities of Solids Element Gram heat Atomic Molar heat capacity weight capacity J deg-1 g-1 J deg-1 mol- 1 Bi 0.120 212.8 25.64 Au 0.125 198.9 24.79 Pt 0.133 188.6 25.04 Sn 0.215 117.6 25.30 Zn 0.388 64.5 25.01 Ga 0.382 64.5 24.60 Cu 0.397 63.31 25.14 Ni 0.433 59.0 25.56 Fe 0.460 54.27 24.98 Ca 0.627 39.36 24.67 S 0.787 32.19 25.30 Um  3 N A kT  3RT  U  CV , m   m   3R  25kJmol 1  T V Dulong – Petit Law The molar heat capacity of all solids have nearly same value of ~25 kJ  Heat Capacities of Solids Einstein formula Einstein considered the oscillations of atoms in the crystal about its 2 E equilibrium position with a single  2    E   e 2T  h frequency ‘’ and invoked the Planck’s CV , m  3R    ;    T   e E T 1  E k hypothesis that these vibrations have   quantized energies nh 3N A h Um  h e kT 1  Heat Capacities of Solids 3  D  D x 4e x h D CV , m  3R    dx; D  T T  o (e x  1) 2 k Debye formula Averaging of all the frequencies D  Rutherford Model of Atom Alpha particles were (He2+) bombarded on a 0.00004 cm (few hundreds of atoms) thick gold foil and most of the alpha particles were not deflected  Rutherford Model of Atom Positive Charge Thompson’s model of atom is incorrect. Cannot explain Rutherford’s experimental results Negatively Charged Particles Planetary model of atoms with Classical electrodynamics predicts that central positively charged nucleus such an arrangement emits radiation and electrons going around continuously and is unstable  Atomic Spectra Balmer Series The Rydberg-Ritz Combination Principle states that the spectral lines 410.1 nm of any element include frequencies   434.0 nm 1  R  12  12  that are either the sum or the 486.1 nm   n1 n2  difference of the frequencies of two 656.2 nm R  1.09678 x 102 nm 1 other lines. “RH is the most accurately measured fundamental physical constant”  Bohr Phenomenological Model of Atom Electrons rotate in circular orbits around a central (massive) nucleus, and obeys the laws of classical mechanics. Allowed orbits are those for which the electron’s angular momentum equals an integral multiple of h/2π i.e. mevr = nh/2π Energy of H-atom can only take certain discrete values: “Stationary States” The Atom in a stationary state does not emit electromagnetic radiation When an atom makes a transition from one stationary state of energy Ea to another of energy Eb, it emits or absorbs a photon of light: Ea – Eb = hv Bohr Model of Atom Angular momentum quantized nh mvr  n=1,2,3,... 2 (2 r  n ) Energy expression me e 4 1 En   2 2. 2 8 0 h n Spectral lines m ee 4  1 1  E   2  2   h n i , n f  1, 2 , 3,... 8 2 h 2  ni nf  Explains Rydberg formula me e4 R  2 2  1.09678 x 102 nm 1 8 h Ionization potential of H atom 13.6 eV  Bohr Model of Atom The Bohr model is a primitive model of the hydrogen atom. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics  Photoelectric Effect: Wave –Particle Duality Electromagnetic Radiation E  E0 Sin(kx  t ) Wave energy is related to Intensity, I  E20 and is independent of  Experimental Observations Increasing the intensity of the light increased the number of photoelectrons, but not their maximum kinetic energy! Red light will not cause the ejection of electrons, no matter what the intensity! Weak violet light will eject only a few electrons! But their maximum kinetic energies are greater than those for very intense light of longer (red) wavelengths  Photoelectric Effect: Wave –Particle Duality Einstein borrowed Planck’s idea that ΔE=hν and proposed that radiation itself existed as small packets of energy (Quanta)now known as PHOTONS EP  h Energy is frequency dependent 1 2 EP  hv  KEM    mv   2  = Energy required to remove electron from surface Diffraction of Electrons : Wave –Particle Duality Davisson-Germer Experiment A beam of electrons is directed onto the surface of a nickel crystal. Electrons are scattered, and are detected by means of a detector that can be rotated through an angle θ. When the Bragg condition mλ = 2dsinθ was satisfied (d is the distance between the nickel atom, and m an integer) constructive interference produced peaks of high intensity  Diffraction of Electrons : Wave –Particle Duality G. P. Thomson Experiment Electrons from an electron source were accelerated towards a positive electrode into which was drilled a small hole. The resulting narrow beam of electrons was directed towards a thin film of nickel. The lattice of nickel atoms acted as a diffraction grating, producing a typical diffraction pattern on a screen de Broglie Hypothesis: Mater waves Since Nature likes symmetry, Particles also should have wave-like nature De Broglie wavelength h h   p mv Electron moving @ 106 m/s h 6.6x10-34 J s    7  10 10 m mv 9.1x10 Kg  1x10 m/s -31 6 He-atom scattering Diffraction pattern of He atoms at the speed 2347 m s-1 on a silicon nitride transmission grating with 1000 lines per millimeter. Calculated de Broglie wavelength 42.5x10-12 m de Broglie wavelength too small for macroscopic objects Diffraction of Electrons : Wave –Particle Duality The wavelength of the electrons was calculated, and found to be in close agreement with that expected from the De Broglie equation  Wave –Particle Duality Light can be Waves or Particles. NEWTON was RIGHT! Electron (matter) can be Particles or Waves Electrons and Photons show both wave and particle nature “WAVICLE” Best suited to be called a form of “Energy” Wave –Particle Duality Bohr – de Broglie Atom Electrons in atoms behave as standing waves Constructive Interference of the electron-waves can result in stationary states (Bohr orbits) If wavelength don’t match, there can not be any energy level (state) Bohr condition & De Broglie wavelength 2 r  n n=1,2,3,... h  mv nh mvr  n=1,2,3,... 2 Uncertainty Principle Uncertainty principle h x.px  4  Schrodinger’s philosophy PARTICLES can be WAVES and WAVES can be PARTICLES New theory is required to explain the behavior of electrons, atoms and molecules Should be Probabilistic, not deterministic (non-Newtonian) in nature Wavelike equation for describing sub/atomic systems  Schrodinger’s philosophy PARTICLES can be WAVES and WAVES can be PARTICLES let me start with classical wave equation A concoction of 1 2 p2 E  T  V  mv  V  V 2 2m E  h   Wave is Particle h 2   Particle is Wave p k Do I need to know any Math? Algebra A  c1 f1 ( x)  c2 f 2 ( x)  c1 Af1 ( x)  c2 Af 2 ( x) Trigonometry Sin(kx) Cos (kx) eikx Differentiation d d2  2 dx dx 2 x x 2 Integration b  dx  ikx e f ( x)dx a Differential equations  2 f ( x)  2 f ( y) f ( x) f ( y )     mf ( x)  nf ( y )  k x 2 y 2 x y  Schrodinger’s philosophy  2 ( x , t ) 1  2 ( x , t )  2 Classical Wave Equation x 2 c t 2 ( x ,t )  Amplitude i x  ( x ,t )  Ce ; Where   2   t  is the phase   Remember! E  h   h 2   p k x  x  p  E t   2   t      Schrodinger’s philosophy i x  p  E t ( x ,t )  Ce and   ( x ,t ) i    E   iCe   i  ( x , t )   i  ( x , t )    t t t     ( x ,t )  E  ( x , t ) i t Schrodinger’s philosophy i x  p  E t ( x ,t )  Ce and   ( x ,t ) i    px   iCe   i  ( x , t )   i  ( x ,t )    x x x     ( x , t )  px   ( x , t ) i x Schrodinger’s philosophy i x  p  E t ( x ,t )  Ce and   ( x ,t ) i    E   iCe   i  ( x , t )   i  ( x , t )    t t t     ( x ,t )  E  ( x , t ) i t ( x , t ) i    px   iCe   i  ( x , t )   i  ( x , t )    x x x     ( x ,t )  px   ( x , t ) i x  Operators  ( x ,t )  ( x , t )  E  ( x ,t )  px   ( x , t ) i t i x     ( x , t )  E  ( x , t )  ( x , t )  px   ( x , t ) i t i x          i  E  i   px Operators i t t i x x Operator A symbol that tells you to do something to whatever follows it Operators can be real or complex, Operators can also be represented as matrices Operators and Eigenvalues Operator operating on a function results in re-generating the same function multiplied by a number  A  f ( x)  a  f ( x) Eigen Value Equation The function f(x) is eigenfunction of operator  and a its eigenvalue Sin  x  is an eigenfunction of f ( x)  Sin  x  d2 operator 2 and  2 is its d dx f ( x)    Cos  x  eigenvalue dx d2 d 2 f ( x )     Cos    x    2  Sin    x   2  f ( x) dx dx Laws of Quantum Mechanics The mathematical description of quantum mechanics is built upon the concept of an operator The values which come up as result of an experiment are the eigenvalues of the self-adjoint linear operator. The average value of the observable corresponding to operator  is a    * Aˆ d The state of a system is completely specified by the wavefunction Ψ(x,y,z,t) which evolves according to time-dependent Schrodinger equation Probability Distribution and Expectation Values Classical mechanics uses probability theory to obtain relationships for systems composed of larger number of particles For a probability distribution function P(x) the average value is given by n n Mean : x   x j Pj ( x j ) and x 2   x 2j Pj ( x j ) j 1 j 1 Probability Distribution and Expectation Values Let us consider Maxwell distribution of speeds 3  M  2  Mv2 f (v)  4   v2 e 2 RT  2 RT  The mean speed is calculated by taking the product of each speed with the fraction of molecules with that particular speed and summing up all the products. However, when the distribution of speeds is continuous, summation is replaced with an integral 1   8 RT  2 v vf (v)dv    0   M   Born Interpretation In the classical wave equation Ψ(x,t) is the Amplitude and |Ψ(x,t)|2 is the Intensity The state of a quantum mechanical system is completely specified by a wavefunction Ψ(x,t) ,which can be complex All possible information can be derived from Ψ(x,t) From the analogy of classical wave equation, Intensity is replaced by Probability. The probability is proportional to the square of the of the wavefunction |Ψ(x,t)|2 , known as probability density P(x) Born Interpretation Probability density 2 P ( x )  ( x , t )    ( x , t )   ( x , t ) Probability 2 P ( xa  x  xa  dx )  ( x ,t ) dx    ( xa ,t )  ( xa ,t )dx Probability in 3-dimensions P( xa  x  xa  dx, ya  y  ya  dy, za  z  za  dz )  * ( xa , ya , za , t '). ( xa , ya , za , t ')dxdydz 2   ( xa , ya , za , t ') d  Normalization of Wavefunction Since Ψ*Ψdτ is the probability, the total probability of finding the particle Ψ somewhere in space has to be unity ∞ ∞  all space  * ( x, y, z ). ( x, y, z )dxdydz x   all space  *d     1 ∞ ∞ If integration diverges, i.e.  ∞: Ψ can not be normalized, and therefore is NOT an Ψ acceptable wave function. However, a x constant value C ≠ 1 is perfectly Unacceptable wavefunction acceptable. Ψ must vanish at ±∞, or more appropriately at the boundaries and Ψ must be finite Laws of Quantum Mechanics The mathematical description of QM mechanics is built upon the concept of an operator Classical Variable QM Operator Position, x x   d d Momentum, px  mv px   i  i dx dx px2   2 d 2 Kinetic Energy, Tx  Tx  2m 2m dx 2  2   2 2 px2 py pz2  2 2  Kinetic Energy, T  +  T  2 2 2 2m 2m 2m 2m  x y z  Potential Energy, V ( x )  (x ) V Laws of Quantum Mechanics The values which come up as result of an experiment are the eigenvalues of the self-adjoint linear operator In any measurement of observable associated with operator Â, the only values that will be ever observed are the eigenvalues an, which satisfy the eigenvalue equation:  A   n  an   n Ψn are the eigenfunctions of the system and an are corresponding eigenvalues If the system is in state Ψk , a measurement on the system will yield an eigenvalue ak Laws of Quantum Mechanics Only real eigenvalues will be observed, which will specify a number corresponding to the classical variable If  ( x )  Sin(cx) d  ( x)  c  Cos (cx) dx d2 2  ( x )   c 2  Sin ( cx )   c 2   ( x) dx There may be, and typically are, If  ( x )  e x many eigenfunctions for the same d QM operator!  ( x)    e x dx d2 2  ( x )   2  ex   2   ( x) dx