Quantum Chemistry Evolution of Quantum mechanics 1922: Neils Bohr won Nobel Prize in physics 1905: Albert proposed the idea photoelectric effect. 1918: 1925: Wolfgang Pauli spin number 1900: 1913: Max Planck black-body radiation Neils Bohr atomic structure Won the Nobel Prize in Physics photoelectric effect. Pauli proves Spin-Statistic Theorem 1932: Werner Heisenberg won Nobel Prize in Schrödinger's physics Equation 1936: Max Planck won Nobel prize in physics 1921: 1940: Heisenberg Uncertainty principle 1924: Louis De Broglie electron waves. 1929: 1933: Louis De Otto Stern Broglie won measures the the Nobel magnetic Prize. moment of the proton 1944: Pauli won Nobel Prize in physics 1943: Otto Stern Nobel prize in Physics Structure of the Atom • In 1909, the prevailing theory of atomic structure was the plum pudding model. • In this model, proposed by J. J. Thomson, the electrons were thought to be floating around in a soup of positive charge. count rate of incident particles The Experiment: Interpretation: α Rutherford’s Atomic Model • Since the vast majority of αEngland, particles pass through the Au foil undeflected, the Ernest Rutherford (Cambridge University, 1871-1937) studied α emission from newlyCount rate = α particles / min are mostly Au atoms discovered radioactive elements. • "It was almost as massive incredible as atom if youand fired a fifteen-inch shell A very tiny percentage of α particles hit something in the at a pieceindicates of tissuethat paper it came and hit you" backscatter (bounce back). This observation mostand of the mass back of the atom is concentrated in a very small volume relative to the volume of the entire atom. We now call this the NUCLEUS. α • New model for α particles Count rate = α particles / min It appeared that all of the α particles passed through the Au foil. A detector was built that could swing around to the front side and measure any potential back scattered particles. • Rutherford proposed the charge on the nucleus to be positive, since electrons are negatively charged and atoms are neutral. Charge of the electrons in an atom = - Ze Z = atomic number e = absolute value of electron’s charge α Charge of the nucleus of an atom = • Rutherford calculated the diameter of the nucleus to be 10 m. His calculation (see below) related the probability of backscattering to the diameter of the nucleus based on the size of the atom and the thickness of the foil. on. Drawback Rutherford’s Atomic Model • According to classical theory, atoms in the model for Rutherford’s are not stable. • The motion of moving electrons should cause them to radiate energy in their orbits and to quickly execute a death spiral and collapse into the nucleus. • The time taken for this collapse is estimated around 10−8 s, which is very small. According to the model, every atom will crumple in 10−8 s, but we know this is not reality. • If the electrons are not revolving but stationary, still the electrons will fall into the nucleus by the electrostatic force between both. The predicted death spiral of the electron! Hydrogen Atomic Spectrum Hydrogen Atomic Spectrum • When a high-energy discharge is passed through a sample of H2 gas, the H2 molecules absorb energy, causing some of the H-H bonds to break. • The resulting hydrogen atoms are excited; they contain excess energy, which they release by emitting light of various wavelengths to produce what is called the emission spectrum of the hydrogen atom., called a line spectrum. • Do the electrons in atoms and molecules obey Newton’s classical laws of motion? • It turns out that the laws of classical mechanics no longer work at this size scale. • A new kind of mechanics was needed to describe this and other “unsettling” observations. Classical Mechanics • Classically, particles and waves are distinct: – Particles – characterised by position, mass, velocity. – Waves – characterised by wavelength, frequency. !" = $ tor ew) associated wavelength is so small that it is not observed. Very small “pieces” of matter, such as photons, while showing some particulate properties through Classical Mechanics (b) Constructive interference (c) Destructive interference Trough Waves in phase (peaks on one wave match peaks on the other wave) Peak Waves out of phase (troughs and peaks coincide) Increased intensity (bright area) Decreased intensity (dark spot) gnetic radiation is scattered from a regular array of objects, such as the ions in e spot in the center is from the main incident beam of X rays. (b) Bright areas onstructive interference of it waves. The waves are in phase; that is, their peaks By the 1920s, however, was becoming apparent that sometimes matter (classically particles) can ructive interference of waves. The waves are out of phase; the peaks of one otherbehave wave. like waves and radiation (classically waves) can behave like particles. The Photoelectric Effect A beam of light hitting a metal surface can cause electrons to be ejected from the surface. Classical Paradigm: the energy of the ejected electrons should be proportional to the intensity (I) of the light and independent of the frequency (!) of the light. Experiment: the energy of the ejected electrons is independent of the intensity (I) and depends directly on the frequency (!) of the light. https://phet.colorado.edu/sims/cheerpj/photoelectric/latest/photoelectric.html?simul%20ation=photoelectric The Photoelectric Effect The following observations characterize the photoelectric effect. 1. No electrons are emitted by a given metal below a specific threshold frequency !0. 2. For light with frequency lower than the !0, no electrons are emitted regardless of the intensity of the light. For light with frequency greater than the !0, the kinetic energy of the emitted electrons increases linearly with the frequency of the light. 3. The number of electrons emitted per second (i.e. the electric current) is independent of light frequency above the !0 and zero below. 4.For light with frequency greater than the !0, the number of electrons emitted increases with the intensity of the light. are emitted regardless of the intensity of the light. 3. For light with frequency greater than the threshold frequency, the number These data were in of direct opposition to the predictions of the classical mechanics. 1905 Quantization ofintensity Light electrons emitted increases with of theInlight. Einstein analyzed plots of K.E. as a function of frequency for different metals and found 4. fit For light withform frequency greater than the threshold frequency, the kinetic that all of the data into a linear energy of the emitted electrons increases linearly with the frequency of the light. Rb K Na K.E. These observations can be explained by assuming that electromagnetic radiation is quantized (consists of photons) and that the threshold frequency represents the minimum energy required to remove the electron from the metal’s surface. ν0(Rb) ν0(K) ν0(Na) ν Minimum energy required to remove an electron ! E0 ! h!0 y = mx + b -hν0(Rb) -hν0(K) Because a photon with energy less than E0 (! " !0) cannot remove an electron, The slope slope (mof KE vs n is h light with a frequency less than the threshold frequency produces no electrons. On the other hand, for light where # -34!0Js , the energy in excess =of that required 6.626 ! x 10 = Planck’s constant to remove the electron is given to the electron as kinetic energy (KE): -hν0(Na) y-intercept (b) = - hν0 KEelectron 5 12 mv2 5 hn 2 hn0 p h Einstein could rewrite the equation of the line: Workfunction, ɸ h r Mass of Velocity Energy of Energy required electron to remove electron y = of mx + b incident electron photon from metal’s surface Quantization of Light 12.2 The Nature of Matter lculated from Einstein’s equation. Also, photons do seem to be affected by cangeneral be theory explained by assuming avity,• as These Einsteinobservations postulated in his of relativity. However, that it is portant electromagnetic to recognize that the photon is inquantized no sense a(consists typical particle. A phoradiation of photons) n has mass a relativistic sense—it has norepresents rest mass. the minimum andonly thatin the threshold frequency can describe this mathematically: We can summarize the important conclusions from the work Planck energy required to remove the electron from theofmetal’s . Einstein as follows: or Ei d Light as a wave phenomenon surface. units E called quanta. Minimum energy required to remove discrete an electron, 0 = hn 0 te: these just different It forms the equation K.E = hν - hν0)in ergy isare quantized. canof be transferred only • s try some example radiation, problems. ectromagnetic which was previously thought to exhibit only wave properties, seems to show certain characteristics of particulate matter as well. This phenomenon, illustrated in Fig. 12.6 ▶, is sometimes referred to # of electrons ejected from the surface of a metal is proportional to the as theabsorbed dual nature of and light. hotons by the metal not the energy of the photons (assuming Ei ≥ φ). Thus light, which was previously thought to be purely wavelike, was und to have certain characteristics of particulate matter. But is the opposite o true? That is, does matter that is normally assumed to be particulate exbit wave properties? This question was raised in 1923 by a young French ysicist named Louis de Broglie (1892–1987), who derived the following ationship for the wavelength of a particle with momentum mv: us, the intensity (I) of the light (energy/sec) is proportional to the # of photons orbed/sec and the # of electrons emitted/ sec h 443 Light as a stream of photons Figure 12.6 Electromagnetic radiation exhibits wave properties and particulate properties. The energy of each photon of the radiation is related to the wavelength and frequency by the equation Ephoton !h" ! hc/!. E= nhc l (n = 0,1, 2,3,.......) Do not confuse " (frequency) with ____________________________________________________________________________ Topics: I. Light as a particle continued II. Matter as a wave III. The Schrödinger equation ____________________________________________________________________________ Quantization of Light I. LIGHT AS A PARTICLE CONTINUED A) More on the Photoelectric Effect e- ejected e- ejected Three photons, each with an energy equal to φ/2 e- NOT ejected eject an electron! Waves behaving as Particles Terminology tips to help solve problems involving photons and electrons: • photons: also called light, electromagnetic radiation, etc. Example • What is the energy in joules and electron volts of a photon of 420-nm violet light? • What is the maximum kinetic energy of electrons ejected from calcium by 420-nm violet light, given that the workfunction for calcium metal is 2.71 eV? Example • What is the longest-wavelength electromagnetic radiation that can eject a photoelectron from silver? Is this in the visible range? Given that the workfunction of silver is 4.72 eV. • Only photons with wavelengths lower than 263 nm will induce photoelectrons. • This is ultraviolet and not in the visible range. Wave Nature of Particles • Louis de Broglie postulated that as light has wave-like and particle-like properties, matter must also be both particle-like and wave-like. • A particle, of mass m, travelling at velocity v, has linear momentum p = mv. • By analogy with photons, the associated wavelength of the particle (l) is given by: de Broglie wavelength h h λ= = p mv de Broglie’s equation The fact that particles can behave as waves but also as particles, depending on which experiment you perform on them, is known as the particle-wave duality. INTERACTIVE EXAMPLE 12.2 Calculations of Wavelength Compare the wavelength for an electron (mass ! 9.11 " 10#31 kg) traveling at a speed of 1.0 " 107 m/s with that for a ball (mass ! 0.10 kg) traveling at 35 m/s. Solution We use the equation ! ! h/mv, where h ! 6.626 " 10#34 J s or 6.626 " 10#34 kg m2/s since 1 J ! 1 kg m2/s2 For the electron, kg m2 6.626 3 10 s le 5 5 7.3 3 10211 m 231 7 19.11 3 10 kg2 11.0 3 10 m /s2 234 For the ball, kg m2 6.626 3 10 s lb 5 5 1.9 3 10234 m 10.10 kg2 135 m/s2 ence Source 234 • Bullets are far more massive than the electrons. One can observe them as long as one likes but it would not make any difference to them. • The interference wiggles in the case of bullets are so crowded that it is physically impossible to resolve them, one sees an average behaviour The wave properties of matter are only apparent for very small masses of matter. efficiently when the spacing between the scattering points is about the same as the wavelength. Thus, if electrons actually do have an associated wavelength, a crystal should diffract electrons. An experiment to test this idea was carried out in 1927 by Davisson and Germer at Bell Laboratories. When they directed a beam of electrons at a nickel crystal, they observed a diffraction pattern According to classical physics, electrons should behave like similar to that seen from the diffraction of X rays. This result verified de Broglie’s relationship, at least for electrons. Larger chunks of matter, such particles - they travel in straight lines and do not curve in as balls, have wavelengths (Example 12.2) too small to verify experimentally. flight unless acted on by an external agent. However, we believe that all matter obeys de Broglie’s equation. In this model, if we fire a beam of electrons through a Now we have come full circle. Electromagnetic radiation, which at the turn of the twentieth century was thought to be a pure waveform, was found double slit onto a detector, we should get two bands of to exhibit particulate properties. Conversely, electrons, which were thought to "hits”. be particles, were found to have a wavelength associated with them. The significance of these results is that matter and energy are not distinct. Energy is really a form of matter, and all matter shows the same types of properties. That is, all matter exhibits both particulate and wave properties. Large “pieces” of matter, such as baseballs, exhibit predominantly particulate properties. The associated wavelength is so small that it is not observed. Very small “pieces” of matter, such as photons, while showing some particulate properties through Wave Nature of Particles: Validation Can be explained only in terms of waves. (b) Constructive interference Waves in phase (peaks on one wave match peaks on the other wave) Increased intensity (bright area) (c) Destructive interference Trough Peak Waves out of phase (troughs and peaks coincide) Decreased intensity (dark spot) • is, all matter exhibits both particulate and wav Wave Nature of matter, Particles: such Validation as baseballs, exhibit predominan associated wavelength is so small that it is not Davisson and Germer showed that a beam of electrons be diffracted from the surface while of a nickel crystal. some of could matter, such as photons, showing • Scattered light can interfere constructively (the peaks and troughs of the beams are in phase) to produce a bright area or destructively (the peaks and troughs are out of phase) to produce a dark spot. • A diffraction pattern can be explained only in terms of waves. (a) Diffraction (b) Constructive interference (c) Dest Trou Waves in phase (peaks on one wave match peaks on the other wave) Peak Waves o (troughs coincide X rays NaCl crystal Detector screen Diffraction pattern on detector screen (front view) Increased intensity (bright area) Figure 12.7 F : A NB: Other “particles” (e.g. neutrons, protons, He atoms) can also be diffracted by crystals. (a) DiffractionDoccurs;when electromagnetic radiation is U S G S , scattered from aUregularS arrayDof objects, s Wave-Particle Duality • Now we have come full circle. • Electromagnetic radiation, which at the turn of the twentieth century was thought to be a pure waveform, was found to exhibit particulate properties. • Conversely, electrons, which were thought to be particles, were found to have a wavelength associated with them. • Energy is really a form of matter, and all matter shows the same types of properties. • All matter exhibits both particulate and wave properties. Putting it All together: The Bohr Atom • Bohr proposed a model that included the idea that the electron in a hydrogen atom moves around the nucleus only in certain allowed circular orbits. • E4 Bohr assumed that the hydrogen electron could exist only in stationary, non-radiating orbits. E5 E3 E2 E1 Putting it All together: The Bohr Atom Solving Paradox I: The unstable orbiting electron of Rutherford’s atom. • • • Make the orbiting electron stable by assuming that the electron’s orbit and energy is quantized to certain values and for these values the orbiting electron does not radiate. The electron is stable in these orbits. Thus, the orbits and energies of electrons are quantized. Solving Paradox II: The line spectra of emitting or absorbing atoms. • • • Since only certain energies are allowed for orbiting electrons, only jumps between orbits can be observed. These jumps correspond to discrete frequencies and wavelengths. Thus, line spectra as expected because of the quantized energies of the orbits. 12.4 The The Bohr Model Bohr 447 Atom n=5 Figure 12.10 n=4 Electronic transitions in the Bohr model for the hydrogen atom. (a) An n energy-level diagram for electronic transitions. (b) An orbit-transition diagram, which accounts for the experi5 (Note that the orbits mental spectrum. 4 shown are schematic. They are not 3 drawn to scale.) 2 (c) The resulting line spectrum on a photographic plate. Note that the lines in the visible region of the spectrum correspond to transitions from E higher levels to the n ! 2 level. n=3 n=2 n=1 n=5 Figure 12.10 n=4 Electronic tra model for the energy-level d transitions. (b gram, which mental spectr shown are sch drawn to scal spectrum on Note that the of the spectru tions from hi level. n=3 n=2 n=1 (b) (b) Line spectrum Line spectrum Wavelength (c) • 1 (a) Wavelength (c) Light is emitted when an electron jumps from a higher orbit to a lower orbit and absorbed when it jumps from a he valuelower of n, the is the orbit radius) to larger higher orbit. 1 for hydrogen). The negative sign in Equa• The and frequency of energy of theenergy electron bound to the nucleus lectron e.g., were at an infinite distance (n ! ") o interaction and the energy is zero: where n is an integer (the larger the value of n, the larger is the orbit radius) light emitted oratomic absorbed is given the two energies, and Z is the number (Z !by1 the for difference hydrogen). between The negative sign orbit in Equa- tion (12.1) simply means that the energy of the electron bound to the nucleus • E(photon) = E2 - E1 (Energy difference) h! it would be if the electron were at an infinite distance is lower=than $&(n ! ") 2 Z !" = $% = from the nucleus, where there is no interaction and the energy is zero: 8 3 10218 Ja b 5 0 ` Z 2 ' The Bohr Atom E 4 E5 5 4 E2 - E1 = hn E 3 E2 E1 3 Photon Absorbed 2 1 410.1 434.0 nm nm 486.1 nm 656.3 nm Bohr atom: Light absorption occurs when an electron absorbs a photon and makes a transition for a lower energy orbital to a higher energy orbital. Absorption spectra appear as sharp lines. The Bohr Atom E2 - E1 = hn E4 E5 E3 E2 E1 5 4 3 Photon Emitted 2 1 410.1 434.0 nm nm 486.1 nm 656.3 nm Bohr atom: Light emission occurs when an electron makes a transition from a higher energy orbital to a lower energy orbital and a photon is emitted. Emission spectra appear as sharp lines. The Bohr Atom Limitations of Bohr Model • At first, Bohr’s model appeared to be very promising. • The energy levels calculated by Bohr closely agreed with the values obtained from the hydrogen emission spectrum. • However, when Bohr’s model was applied to atoms other than hydrogen, it did not work at all • The model only works for hydrogen (and other one electron ions) – ignores e-e repulsion. • Does not explain fine structure of spectral lines. • Note: The Bohr model (assuming circular electron orbits) is fundamentally incorrect. Heisenberg’s Uncertainty Principle • Bohr orbit: The electron in the orbital is moving around the nucleus in a circular orbit. • We can predict the motion of the electron, similar to the motions of particles in the macroscopic world. • For example, when two billiard balls with known velocities collide, we can predict their motions after the collision. • Werner Heisenberg, discovered a very important principle in 1927—the Heisenberg uncertainty principle. closely before we discard the theory. er Heisenberg, whoHeisenberg’s was also Uncertainty involved in the development of the Principle mechanical model for the atom, discovered a very important prin• The that Heisenberg Uncertainty Principle is a fundamental theory in meaning quantum mechanics that defines why a 1927 helps us to understand the of orbitals—the scientist cannot measure multiple quantum variables simultaneously. g uncertainty principle. Heisenberg’s mathematical analysis led him • Heisenberg made the bold proposition that there is a lower limit to this precision making our knowledge of rising conclusion: There is a fundamental limitation to just how a particle inherently uncertain. we can know both the position and the momentum of a particle at • There is a fundamental limitation to just how precisely we can know both the position and the momentum me. the uncertainty principle is ofStated a particle mathematically, at a given time. U # Dx Dp $ 2 ℏ = ℎ/2( is the uncertainty in a particle’s position, !p is the uncertainty in a where Δx is the uncertainty in a particle’s position, Δp is the uncertainty in a particle’s momentum. momentum, and " is Planck’s constant divided by 2! (" # h/2!). minimum the product !x $ !p is h/4!. This relationNote: There uncertainty is no restriction onin the precision in simultaneously knowing/measuring the position along a (x) and the momentum along another, perpendicular direction (z): ns given thatdirection the more precisely weΔpknow a particle’s position, the less z. Δx = 0 we can know its momentum, and vice versa. This limitation is so Making Sense of the Uncertainty Principle • To localize a particle in space (i.e. to specify the particle’s position accurately, small Δx) many waves of different wavelengths (l) must be superimposed Þ large Δpx (p = ℎ/l). • The Uncertainty Principle imposes a fundamental (not experimental) limitation on how precisely we can know (or determine) various observables. https://www.youtube.com/watch?v=KT7xJ0tjB4A An electron is confined to the size of a magnesium atom with a 150 pm radius. What is the minimum uncertainty in its velocity? Determine the uncertainty in the position of the ball (163 gm) bowled at with velocity of 111 kph. The hydrogen atom has a radius on the order of 0.05 nm. Assuming that we know the position of an electron to an accuracy of 1% of the hydrogen radius, calculate the uncertainty in the velocity of the electron using the Heisenberg uncertainty principle. Then compare this value with the uncertainty in the velocity of a ball of mass 0.2 kg and radius 0.05 m whose position is known to an accuracy of 1% of its radius. Thus the uncertainty principle is negligible in the world of macroscopic objects but is very important for objects with small masses, such as the electron. • This limitation is so small for large particles such as baseballs or billiard balls that it is unnoticed. • However, for a small particle such as the electron, the limitation becomes quite important. • Applied to the electron, the uncertainty principle implies that we cannot know the exact path of the electron as it moves around the nucleus. • It is therefore not appropriate to assume that the electron is moving around the nucleus in a well-defined orbit as in the Bohr model.
