QUANTUM MECHANICS AND MOLECULAR SPECTROSCOPY

Transcription

1 QUANTUM MECHANICS AND MOLECULAR SPECTROSCOPY CHEM 330 B. O. Owaga Classical physics Classical physics is based on three assumptions i. Predicts precise trajectory for particles with precisely specified locations and momenta at each instant, and ii. Allows any type of motion to be excited to any energy simply by controlling the forces that are applied. iii. Waves and particles are distinct concepts. Conclusions agree with everyday experience but not when one looks at individual atoms. APRIL Classical physics Classical description of an atom: Coulomb force force of attraction. Think of the simplest atom H 2. e + nucleus r Coulombs force law F(r) = ( e)(e) 4πε 0 r 2 e = absolute value of an electron charge r = distance between two charges ε = permittivity constant of a vacuum (8.854 x C 2 J -1 m -1 ) e - electron 3 So what happens when r goes to infinity? F(r) = ( e)(e) 4πε 0 r 2 Classical physics As r, F(r) = 0 [no force] As r 0, F(r) = - (negative is convention) Coulombs law is effective in showing F as a function of distance (r) but not how r changes with time, t i.e. r(t). The change in r with time is shown by Newtons 2 nd Law of motion. 4 1

2 Failures of classical physics It gives force as a function of acceleration F = ma F = m dv F = m d2 r dt dt 2 Plug in coulombs force by combining the two equations F = m dv = m d2 r dt dt 2 and take initial distance, r as 10 Å. Solution:- r = 0 in as little as sec. - (0.1 ns). 5 Classical description of an atom Implying that either the Coulombs Law or the Newtonian mechanics have a problem. Newtonian mechanics cannot work at the level of an atom And that s where QUANTUM MECHANICS comes into play!!! It accurately describes the behaviour of particles at small scale. 6 Has two important properties: Quantum mechanics 1. Matter and radiation have both particle and wavelike properties. 2. Light consists of tiny packets of energy called Quanta. Packets that behave like particles. Careful experiments have shown that laws of classical physics cannot describe the behaviour of VERY SMALL PARTICLES Classical mechanics - only a approximation of motion of particles 7 Quantum mechanics Quantum theory originated in 1900 with Max Planks. One failure of classical physics is the explanation of observed frequency distribution of radiant energy emitted by a hot body. Take three examples of materials we see in everyday life: Glass, Aluminium metal Charcoal or coal 8 2

3 Structure of body and radiation Glass Shinny metal Coal Has charges electrons which will move/oscillate when in contact with photons. Also has charges electrons which move freely through the entire body. Also have free electrons (unattached) that can move through the entire body. Coal radiates when heated since; As the temperature is raised, the lattice energy of atoms vibrates more and more (ceaselessly), these vibrations scatter and accelerate the electrons. Even glass glows at high enough temperatures, as the electrons are loosened and vibrate. Structure of body and radiation Electrons in glass are so tightly bound to atoms that they will only oscillate at certain frequencies. None of the frequencies correspond to VISIBLE LIGHT ( nm). At frequencies outside the visible region - glass is opaque. The electrons when hit by photons are driven by large oscillations which are radiated (reflected light). So they hardly absorb radiant energy. Electrons are free they bump into other particles all the time that they have no mean path (hence poor conductance). When they bump they give K.E. in form of heat. Can pick up energy from electric field hence good materials for transforming energy from light into heat 9 A hot object emits electromagnetic radiation because its atoms and electrons are ceaselessly being accelerated: the atoms vibrate around their mean positions and the electrons are moved from location to location. At high temperatures, an appreciable proportion of the radiation is in the visible region of the spectrum. A higher proportion of short-wavelength blue light is generated as the temperature is raised (e.g. a red-hot iron bar). 10 Electromagnetic spectrum Black-body radiation Black-body - an idealized body which is a perfect absorber, and therefore also (from the above argument) a perfect emitter. Electromagnetic wave travelling at the speed of light but all with different wavelengths 11 An example - a pinhole in an empty container maintained at constant temperature. radiation that goes through the hole bounces around inside, a lot getting absorbed on each bounce, and has little chance of ever getting out again. 12 3

4 Consider the energy density of the radiation in a cavity, te total energy in cavity i.e., volume of cavity and in particular the contribution to the total energy density from radiation of different wavelengths. Shorter λ - contribute more to the energy density as T is raised (shift to the blue). This shift is summarized by Wien s displacement law: Tλ max = constant = 2.9 mm K max = λ of the radiation that makes the greatest contribution to the energy density when the absolute temperature is T. Implying that As T increases max decreases enough to preserve the value of T max. Black-body radiation There is a sharp rise in the emittance M (the total power emitted by a black body divided by the surface area of the body) as the temperature is raised and this is called the Stefan-Boltzmann law: M = at 4 with a = 56.7 nw m -2 K -4 Implying - each cm 2 of a surface of a black body at 1000 K radiates about 5.7 W Rayleigh-Jeans law: predicted that the density of energy in a region of the electromagnetic field due to radiation of wavelength is proportional to 1/ 4. This is a problem!! Power emitted at a particular λ is proportional to the density of energy at that λ. So as decreases, the power increases without limit, meaning that oscillations of very short wavelength are excited even at room temperature. Ultraviolet catastrophe. Black-body radiation Max Planck - Energy of each electromagnetic oscillator is limited to discrete values and cannot be varied arbitrarily. The limitation of energy to discrete values is called the quantization of energy. Energy of an oscillator of frequency v, is restricted to an integral multiple of the quantity hv (h = Planks constant) ΔE = nhv (n = 0, 1, 2,.. ) Black-body radiation Heat capacities of solids Heat capacity, C, is the constant proportionality between the rise in temperature, ΔT and the heat, q, supplied. q = C T Dulong and Petit - heat capacity of monatomic solids = 25 J K -1 mol -1. This is as accounted for through classical mechanics The Planck distribution resembles the Rayleigh Jeans law apart from the allimportant exponential factor in the denominator. With technological measurements of heat capacities at low temperatures of all substances were found to have values significantly lower than 25 J K -1 mol -1. Diamond for instance = 6.1 j K -1 mol -1 For short wavelengths, hc/λkt >> 1 and e hc/λkt faster than λ 5 0; therefore ρ 0 as λ 0 or ν. Hence, the energy density approaches zero at high frequencies, in agreement with observation. 8πhc ρ λ, T = λ 5 e c λkt 1 15 In fact, at very low temperatures, the heat capacity was found to approach zero. i.e. C v 0 as T

5 Einstein deduced that instead of C m = 3R, a better formula is C m = 3Rf(T) where f(t) 0 as T 0 and f(t) 1 as T becomes large (such as room temperature). The physical reason for the success of Einstein s model is that an atom can start to oscillate only if it can acquire a certain minimum energy (h ). Heat capacities of solids At low temperatures there is only enough energy available for a few atoms to be able to oscillate. 17 Because so few atoms can be involved in taking up energy, the solid cannot absorb heat readily and consequently its heat capacity is low. At higher temperatures there is enough energy available for all the oscillators to become active: all 3N oscillators contribute, and the molar heat capacity reaches its classical value of 3R. Debye - allowed for the atoms to oscillate with a range of frequencies (as opposed to the single frequency by Einstein). At low temperatures, the heat capacity of a solid is expected to be proportional to T 3. This dependence, which is called the Deybe T 3 law, is used to extrapolate measurements of heat capacities to T = 0 in the experimental determination of entropies. C m = 3Rf D (T) Heat capacities of solids 18 The most compelling evidence for the quantization of energy comes from the frequencies of radiation absorbed or emitted by atoms and molecules. Atomic and molecular spectra This can only be understood if the energy of atoms and molecules is confined to discrete values, as then energy can be discarded only in packets (hν). Atomic and molecular spectra Radiation is emitted (or absorbed) at a series of discrete frequencies. Atomic emission spectrum Molecular absorption spectrum

6 Particle nature of light - Photoelectric effect The easiest way of showing particle nature of light is through the Photoelectric effect. In 1905, Einstein showed that one can eject an electron from metal surfaces by light. UV light (ν) e - Threshold frequency!! If ν < ν 0 Metal surface K. E. = 1 2 mv2 Metal surface No electron is ejected Threshold frequency!!! If ν > ν 0 UV light (ν) e - This suggests that the electron is ejected in a collision with a particlelike projectile, provided that the projectile carries enough energy to expel the electron from the metal. Metal surface Classical physics cannot make any connections between the frequency of (ν) of light and the energy (E) of an electron given out. 23 If we suppose the projectile is a photon of energy hν, then the conservation of energy requires that the kinetic energy of the electron (½m e v 2 ) should be equal to the energy supplied by the photon less the energy Φ required to remove the electron from the metal: E K = hν - Φ 24 6

7 Φ is called the work function of the metal, the analogue of the ionization energy of an atom. When hν < Φ, photoejection cannot occur. The above equation predicts that the kinetic energy of an ejected electron should increase linearly with the frequency. When a photon collides with an electron, it gives up all its energy, so we should expect electrons to appear as soon as the collisions begin, provided the photons carry sufficient energy. Thus the photoelectric effect is strong evidence for the existence of photons. E K = hν - Φ The work function for Caesium is 3.43 x J. Class Tutorial i. Calculate the energy of an electron liberated by radiation of 550 nm (energy of photon). ii. The stopping voltage. iii. The number of electrons generated if the total number absorbed at 5.5 x 10-7 m is 1.00 x 10-3 J. 2. If a beam of light with energy equal to 4 ev hits a surface of gold metal, what is the maximum K.E. of ejected electrons? [ϕ for gold is = 5.1 ev]. Would this be enough to eject an electron off the gold surface? 26 Solution 1: i. E photon = hv = c λ 6.63 x 10 = 34 (J m s _1 )(2.998 x 108)(m s _1 ) 5.5 x 10 7 (m) = 3.62 x J ii. E electron = E photon ϕ = = 0.12 V. Solid + photon Solid + + e - Below threshold? K.E. max = hv - ϕ 1.00 x 10 3 iii. E = Nhv and N = = 2.76 x 3.62 x electrons 27 The diffraction of electrons wave character of particles Diffraction is the interference between waves caused by an object in their path, and results in a series of bright and dark fringes where the waves are detected. It is a typical characteristic of waves. Davisson and Germer observed the diffraction of electrons by a crystal. Thus particles have wave-like properties and waves have particle-like properties. The concepts of particle and wave melt together. This joint waveparticle character of matter and radiation is called wave-particle duality. 28 7

8 The diffraction of electrons de Broglie suggested that any particle traveling with a linear momentum, p, should have a wavelength λ given by the de Broglie relation Tutorial Is the wave behavior of particles observed in day-to-day life? Tshoshobe bowls a 142 g ball at a velocity of 42 m/s. What s the wavelength of the ball? If light is a stream of particles, each of these particles must have a momentum, p. λ = mv = kg m 2 s kg 42 m s = m p = v and since c = vλ then p = c λ ν > ν Compton effect. ν e - ν e - 29 λ = m is so small compared to the size of the ball. What is the wavelength of a gaseous electron travelling at 1 x 10 5 m/s? e- diameter = Å 30 The Schrödinger equation A particle is spread through space like a wave. Describing electrons by their wavelike property There is only a probability that it may be found at a specific location at any instant given by a wavefunction, ψ. Schrödinger proposed an equation for calculating the wavefunction of a system. The Schrödinger equation - for a single particle of mass m moving with energy E in one dimension is d 2 Ψ + V(x)Ψ = EΨ 2m dx2 ћ2 31 ћ = 2π = x 10_34 J 32 8

9 The Schrödinger equation We shall look at only simple functions where we need to see the explicit forms of the solution of Schrödinger's equation. For example: The wave function of a free moving particle is just sin x (de Broglie s wave) For an oscillating particle - e x2 Only certain ψ functions are allowed and each correspond to a certain energy implying only certain energies values (eigenvalues) are acceptable. ψ must be continuous. Discontinuities are not possible in nature. The Schrödinger equation Ψ for an e - For an electron in a hydrogen atom - e r Common feature of any solution: If x is a solution, then a sin bx, is also a solution. a and b are arbitrary constants Each value corresponds to some value of Energy 33 ψ must be single-valued at any point in space. ψ must always be finite at any point 34 9