CHE1043 Lecture 3: Physical Processes in Chemistry PDF

Physical processes in Chemistry (CHE1043) Lecture 3 Dr Daniela Carta [email protected] Learning outcomes By the end of this lecture, you should be able to: Understand the principles of quantum mechanics: the De Broglie equation the Heisenberg uncertainty principle the Schrödinger equation Underst...

Physical processes in Chemistry (CHE1043) Lecture 3 Dr Daniela Carta [email protected] Learning outcomes By the end of this lecture, you should be able to: Understand the principles of quantum mechanics: the De Broglie equation the Heisenberg uncertainty principle the Schrödinger equation Understand the limitations of Bohr’s atomic model Be able to explain the wave nature of small particles such as electrons 3 Structure of the atom: timeline Modern atomic theory Quantum mechanics Limitations of Bohr’s model Bohr’s model could explain the emission spectrum of the H atom But it could not explain spectra of atoms with more than one electron E2−E1 = Δ𝐸 = ℎ𝜈 = ℎ𝑅𝐻 Electrons revolve round the nucleus in stable ‘stationary states’ or orbits, reflecting a particular energy, and do not radiate energy during this revolution. Electrons can only change energy by undergoing a transition from one orbit to another During transition, electrons absorb or emit an amount of energy exactly equal to the energy difference between the two states, E1 and E2 1 𝑛12 1 − 2 𝑛2 J Wave-particle theory The simplistic theory of electrons circulating a nucleus is insufficient to explain experimental results. The currently accepted theory for explanation of atomic structure involves looking at an electron in a different way, as a wave instead as a particle. Rules of classical mechanics are no longer valid for very small objects Quantum mechanics rules have to be considered instead 6 Key findings… In the 1920s, DRAMATIC CHANGE IN THINKING OCCURRED, BROUGHT ABOUT BY........... Louis De Broglie (1924): wave-particle duality Moving electrons, besides exhibiting particulate ("corpuscular") properties, should also possess wave properties characteristic of light. Werner Heisenberg (1927): uncertainty principle It is impossible to define simultaneously and exactly the momentum and the position of a particle. 7 Wave properties of matter De Broglie equation (1924): p = h p = mv Objects with high mass (e.g. tennis ball) mv= h p = momentum m = mass v= velocity  = wavelength WE CAN APPRECIATE WAVE PROPERTIES ONLY IF THE PARTICLE IS VERY SMALL SUCH AS AN ELECTRON ( m = 9.11 10-31 Kg) Objects with low mass (e.g. electrons) SIZE MATTERS!! 8 De Broglie equation: derivation According to Einstein: E = mc2 (energy proportional to mass, a particle property); According to Planck: E = h (energy proportional to frequency, a wave property): Since de Broglie believed that particles and waves have similar properties, he combined the two: mc2 = h (h x frequency) As real particles do not travel at speed of light, v was substituted for c: mv2 = h = hv/ thus =hv/mv2 = h/mv The product of mass (m) and velocity (v) is known as momentum, p. p = m v thus =h/p or p  = h The De Broglie equation is obtained by considering a particle having both particle and wave properties 9 Diffraction proves the wave nature of the electromagnetic radiation (T Young, 1801) Electromagnetic radiation is usually considered as a WAVE. When electromagnetic radiation passes through two closely spaced slits each slit gives rise to a circular wave and these waves interfere with each other to give a diffraction pattern consisting of a series of bright and dark lines (double slit experiment, T. Young, beginning of 1800). DOUBLE SLITS EXPERIMENT DIFFRACTION PATTERN The series of lines result from constructive and destructive interference of two circular waves generated by the slits When the peak of the 1st wave coincides with the peak of the 2nd wave, constructive interference occurs and the amplitude of the 2 waves add together (BRIGHT LINES). When the peak of the 1st wave coincides with the trough of the 2nd wave, destructive interference occurs and the 2 waves cancel each other out (DARK LINES). 10 Wave properties of the matter: electron diffraction(1925) The first experimental evidence of the wave properties of matter was obtained in 1925 in independent experiments carried out by Davisson & Germer in the USA and by GP Thomson in the UK. The scattering of an electron beam from a Ni crystal shows a variation in intensity characteristics of a diffraction experiment in which waves interfere constructively and destructively in different direction. Electron diffraction pattern Transmission Electron Microscopy TEM https://analyticalscience.wiley.com/do/10.1002/was.000204662 Thomson JJ (father) Nobel prize in 1906 for showing that electrons are particles Thomson GP (son) Nobel prize in 1937 for showing that electrons are waves Both were right 11 The de Broglie relationship indicated that electrons cannot be regarded as classical particles but as waves so… where are the electrons in an atom? 12 Heisenberg uncertainty principle (1927) It is impossible to specify simultaneously, with arbitrary precision, both the momentum (p) and the position (x) of a particle. p = mv (momentum) x = position Δ = uncertainty The more accurately you know the particle’s location, the less accurately you know how fast is moving, and viceversa. This principle is important for very small objects such as atoms, electrons and sub-atomic particles. 13 Heisenberg uncertainty principle (1927) It is impossible to specify simultaneously, with arbitrary precision, both the momentum (p) and the position (x) of a particle. Classical mechanics (Newtonian) p = mv (momentum) particle may have a well-defined trajectory, precisely specified position and momentum simultaneously at each instant Quantum mechanics a particle cannot have a precise trajectory; there is only a probability that it may be found at a specific location at any instant. Classical mechanics Quantum mechanics Once the wave-duality of electrons had been established, this could be incorporated into mathematical models of the atom. 14 Quantum mechanics approach Quantum mechanical view: A particle is spread through space like a wave. To describe this distribution, the wavefunction  (psi) is used to replace the idea of ‘trajectory’ of a particle, which is a mathematical function detailing the behaviour of an electron-wave. The probability of finding an electron at a given point in space is reflected by the function 2 (Born interpretation). the wavefunction is represented by areas of shading. It is a mathematical function that describes the behaviour of the electron spread through space like a wave. To get information on  and, more importantly, 2, it is necessary to use an equation derived by Erwin Schrödinger, universally known as Schrödinger’s Wave Equation (1926). A wavefunction is a mathematical function that contains all the dynamical information about the state of a system. 15 Schrödinger equation (1926) In 1926, the Austrian physicist Erwin Schrödinger proposed an equation for calculating the wavefunction (, psi) of a particle. The wavefunction is a mathematical function of the position coordinates x, y and z which describes the behaviour of a particle (e.g. an electron). The Schrödinger wave equation is a fairly complicated second order differential equation. A partial differential equation shows how a function depends on one variable while several are changing. In quantum mechanics, the time-independent Schrödinger equation can be represented in compact form as: H = Hamiltonian operator E = total energy (or eigenvalue)  = wavefunction To apply the equation, we need to write the Hamiltonian operator H for the system, considering kinetic and potential energy for the particles that form the system. Then we need to insert H into the Schrodinger equation Partial differentials Hamiltonian operator: H= 2 2 2 ( 2 + + ) + V ( x, y , z ) 2 2 2 8 m x y z h2 Kinetic Energy Term It governs the wavefunction of a quantum-mechanical system Potential Energy Term 16 Schrödinger equation By replacing into the Schrodinger equation we get: Kinetic Energy Term Potential Energy Term  +  Total Energy Term = E A ‘simple’ form of the time-independent Schrödinger equation for a single particle of mass m moving with energy E in 1dimension (x) is  2 − 2 + E PotentialEnergy  = ETotalEnergy  2 8 m x h2 Kinetic Energy Term What Newton’s 2nd Law (F = ma) is to classical mechanics, Schrödinger’s equation is to quantum mechanics. Newton’s 2nd Law: mathematical prediction of the behaviour of a physical system. Schrödinger’s equation: mathematical prediction of the behaviour of a wavefunction (and probability). 17 St. Oswald's church, Alpbach village, Tyrol, Austria 18 Are all the solutions of the Schrödinger equation allowed? NO ! Acceptable solutions and boundary conditions Although an infinite number of solutions of the Schrödinger equation exist, not all of them are physically acceptable. The Schrödinger equation only has a finite number of physically acceptable solutions called wavefunctions , each corresponding to a certain energy. The wavefunctions MUST satisfy certain BOUNDARY conditions in order to be acceptable. Because each wavefunction corresponds to a characteristic energy, and the boundary conditions rule out many solutions, only a finite number of allowed energy values are allowed also known as energy levels (quantisation). 20 Boundary conditions UNACCEPTABLE WAVEFUNCTIONS The wavefunctions MUST satisfy certain BOUNDARY conditions in order to be acceptable: 1) It must be single valued (that is, have only a single value at each point): there cannot be more than one probability density at each point. 2) It cannot become infinite over a finite region of space: the total probability of finding a particle in a region cannot exceed 1. 3) The wavefunction is continuous everywhere. 4) It has a continuous slope everywhere. These wavefunctions are unacceptable because (a) it is not single-valued (b) it is infinite over a finite range (c) it is not continuous (d) its slope is not continuous 21 Examples of boundary conditions PARTICLE IN A BOX The particle is confined between two impenetrable walls, the only acceptable wavefunctions are those that fit between the walls (like the vibrations of a stretched string). Although an infinite number of solutions of the Schrödinger equation exist, (infinite waves between the walls) not all of them are physically acceptable. RED: unacceptable solutions GREEN: acceptable solutions Because each wavefunction corresponds to a characteristic energy, and the boundary conditions rule out many of the waves (solutions), only certain energies are permissible between the walls. 22 Examples of boundary conditions PARTICLE ON A RING The circumference has been opened out into a straight line; the points at f = 0 and 2 are identical. Although an infinite number of solutions of the Schrödinger equation exist, (infinite wavs between the walls) not all of them are physically acceptable. Three solutions of the Schrödinger equation for a particle on a ring are shown on the left. RED: unacceptable solutions because they have different values after each circuit and so interfere destructively with themselves. GREEN: the only acceptable solution because it reproduces itself on successive circuits. 23 Electron can be considered as a waves around the nucleus The circular wave consists of wavelengths that are multiples of whole numbers otherwise destructive interference occurs Only certain orbits have a circumference into which a whole of wavelengths can fit 5th Solvay Conference (1927), Brussels, Belgium The most prominent physicists gathered to discuss the new quantum theory (electrons and photons) E. Schrödinger W. Pauli W. Heisenberg N. Bohr M. Planck E = h M. Curie radioactivity L. de Broglie p = h M. Born 2 A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. De Donder, E. Schrödinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin; P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr; I. Langmuir, M. Planck, M. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, O.W. Richardson Summary & key equations ✓ De Broglie equation: p = h Objects have wave properties (wave-particle duality) ✓ Experiment that provides evidence of waves properties of the matter (electron diffraction) ✓ Heisenberg indicated that it was impossible to know exactly both the position and momentum of the electron in his ‘uncertainty principle’ ✓ Therefore, electrons cannot be regarded as classical particles, with pathways or trajectories described by Newtonian mechanics but may be described by wave mechanics. 26 Summary & key equations ✓ The electrons in a atom can be described by mathematical functions known as wavefunctions, which are solutions of the Schrödinger equation H = E Simple form for a particle moving in one direction:  2 − 2 + E PotentialEnergy  = ETotalEnergy  2 8 m x h2 The Schrödinger equation: 1) only has a finite number of physically acceptable solutions (wavefunctions ) obtained under specified boundary conditions, reflecting a finite number of allowed energy values also known as energy levels (quantisation). 2) can only be solved exactly for simple systems such as H. Assumptions allow solutions for other atoms. Wavefunctions are equations! 27