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Basic Quantum Theory • Why He and Ne are both non-reactive atoms? • Why H2 (g) + Cl2 (g) → 2HCl (g)? To answer these questions we have to understand the behavior of the e − in atoms When atoms react, it is the e − that interact Electronic structure of atoms: management of e − in atoms ↓ − Distribution of e around nucleus and energies Quantum Mechanics is the Physics that has been developed in order to describe atoms correctly → → Light: consists of waves of oscillating electric field ( E ) and magnetic field ( B ) that are → perpendicular to each other and to the direction of propagation ( k ) Wavelength (λ): distance between two successive peaks or troughs provided that the distance is reproducible from peak to peak. Amplitude: height of the wave Frequency (ν): is the number of cycles that pass the observer in a given time. Hertz (Hz) is the unit of frequency, and just means how many cycles per second. υ=λν Frequency and wavelength c= λν speed of light in vacuum: 2.99792458 x 108 m s-1 monochromatic light: light of single frequency or at a single wavelength Light travels more slowly in a transparent material than it does through a vacuum due to interaction of the light with the electrons in the material. 1 The electromagnetic spectrum Example A dental hygienist uses x-rays (λ = 1.00 Ǻ) to take a series of dental radiographs while the patient listens to a radio station (λ = 325 cm) and looks out the window at the blue sky (λ = 473 nm). What is the frequency (in s-1) of the electromagnetic radiation from each source? (Assume that the radiation travels at the speed of light, 3.00 x 108 m/s.) c= λν⇒ν=c/λ ν = (3.00x108 m/s/1.00 x 10-10m = 3x1018 s-1 ν = (3.00x108 m/s/325 x 10-2 m = 9.23x107 s-1 ν = (3.00x108 m/s/473 x 10-9 m = 6.34x1014 s-1 2 The Paradoxes of Classical Physics 1st Paradox – The Blackbody Radiation A black body is a theoretical object that absorbs 100% of the radiation that hits it. Therefore it reflects no radiation and appears perfectly black. In practice no material has been found to absorb all incoming radiation, but carbon in its graphite form absorbs all but about 3%. It is also a perfect emitter of radiation. At a particular temperature the black body would emit the maximum amount of energy possible for that temperature. This value is known as the black body radiation. It would emit at every wavelength of light as it must be able to absorb every wavelength to be sure of absorbing all incoming radiation. A bar of iron when it is heated first it becomes red, then orange, and as the temperature increases it becomes firstly yellow and secondly white. In other words, a bar of iron when it is heated, it emits radiation and the distribution of frequencies change with temperature 8πR T I = 4 N λ (Classical Physics) Classical Physics predict that a heated body emits radiation approaching infinite intensity at shorter wavelengths. But this doesn’t happen. (1st paradox) ULTRAVIOLET CATASTROPHE 3 Max Planck (1900) solved the paradox of the blackbody radiation. Classical Physics assumed that atoms and molecules could emit (or absorb) any arbitrary amount of radiant energy. He proposed that this energy could be emitted or absorbed only in discrete quantities. He gave the name of quantum to the smallest quantity of energy that can be emitted or absorbed in the form of electromagnetic radiation. E = hν (h = 6.62608 x 10-34 J s – Planck’s constant) Energy is emitted in multiples of hν , ( hν, 2 hν , 3 hν …) but never as 1.67 hν … In light of the new theory, at any given temperature, there is only a fixed amount of thermal energy that is available to excite a given electromagnetic oscillation. In Classical Physics one can put an arbitrary amount of energy that it can be distributed evenly among the oscillations regardless of frequency. In Planck’s model there is a minimum amount of energy that can be transferred into an EM oscillation from the object and this minimum energy (quantum) increases with increasing frequency. For low-frequency EM waves, the quantum energy is much smaller than the average amount of energy available for the excitation of the EM wave and this energy can be evenly distributed among these oscillation modes as in Classical Physics. However, for higher frequencies the quantum of energy is greater than the average available thermal energy and excitation into high frequency modes is inhibited. Black body radiation curves showing peak wavelengths at various temperatures As the temperature increases, the peak wavelength emitted by the black body decreases As temperature increases, the total energy emitted increases, because the total area under the curve increases Stefan’s Law: M = σ T4 M = emittance σ = 5.67 x 10-8 W m-2 K-4 (Stefan – Boltzmann’s constant) 4 Therefore the Power radiated is proportional to T4 for an identical body which explains why the area under the black body curves (the total energy) increases so much for a relatively small increase in temperature. Sun at 6000K 2nd Paradox – The Photoelectric Effect (1905) • If the incident wave frequency (ν) is smaller than a certain value (νo), there is no current flows. • If ν > νo, the current flows instantaneously. Thus, there is minimum required photon energy (hνo,) to overcome the work function of the material, Φ Φ = hν o If the incident light energy is less than the work function, the electron will not be freed from the surface, and no photoelectric effect will be observed. Tmax = 1 meυ 2 = h(ν − ν o ) = hν − Φ 2 Observations: 1) The energy of photons is determined by the light frequency, not intensity 2) Material-specific “red boundary” νo exists: no photocurrent at ν < νo At ν < νo (hν < Φ) the photon energy is insufficient to extract an electron from metal Einstein suggested that light consists of a stream of packets of energy called photons Therefore, light behaves like a particle (a stream of photons) and as a wave! 5 3rd Paradox – Stability of Atoms In Classical Physics atoms are constructed according to Rutherford’s model and they are not stable. The motion of electrons in their orbits would cause them to radiate energy and so quickly to spiral into the nucleus. Different behaviors of waves and particles 6 Atomic Spectra Johann Balmer (1885) observed for excited H atoms: 1 4 ν = − 1 (3.29 x1015 s −1 ) n = 3, 4, 5 … These lines are called Balmer series. 2 n Other spectroscopists discovered additional series in the H-atom as the spectrographs were improved. 1 1 ν = 2 − 2 (3.29 x1015 s −1 ) n1 > n2, n1 and n2, are integers n2 n1 n2 = 1 Lyman (UV) n2 = 2 Balmer (Vis) n2 = 3 Paschen (IR) n2 = 4 Brackett (IR) n2 = 5 Pfund (IR) 7 Atoms emit at discrete frequencies in regions of the EM spectrum quite remote from the visible if excited properly. Henry Moseley (1913): metallic elements emit X-rays of characteristic frequency when excited with high-energy electrons ν = ( Z − 1)(4.98 x10 7 s −1/ 2 ) , Z is an integer, different value for each element, same as Rutherford’s atomic number Squaring both sides of Moseley’s equation we get, ν = ( Z − 1) 2 (2.48 x1015 s −1 ) 15 But 2.48 x10 = 3 3 1 1 (3.29 x1015 ) and = 2 − 2 4 4 1 2 The Bohr Model was an early attempt to formulate a quantum theory of matter Niels Bohr made the following postulates for a single e − orbiting a nucleus of Z protons: 1. The e − moves in a circular orbit around the nucleus. 2. The energy of the e − can take on only certain well-defined values, that is quantized. 3. The only allowed orbits are those in which the magnitude of the angular momentum of the e − is equal to an integer multiple of ħ. → → → The angular momentum is: L = r x p (cross product) → → Linear momentum: p = m υ 4. The e − can only absorb or emit EM radiation when it moves From one allowed orbit to another. The emitted radiation has energy of hν equal to the difference in energy between the two orbits. 8 L = meυr = n h 2π n = 1, 2, 3…. n2 ao , Z is the atomic number and ao, is The allowed radius for each value of n: rn = Z Bohr’s radius, the predicted distance of the electron from the nucleus in the state n = 1 of the H-atom. ao = 5.29177 x 10-11 m or 0.529 Å Z2 Total energy of the e − in the allowed orbits: En = − 2 n h2 2 2 n = 1, 2, 3…. 8 π m a e o 1 Rydberg (Ry) = 2.179872 x 10-18 J (non-SI unit of energy) Z2 Z2 En = − 2 R y = − 2 (2.18 x10 −18 J ) n = 1, 2, 3…. n n Ground-state: n = 1, state of lowest energy for the system of nucleus plus electron. The Bohr explanation of the three series of spectral lines Bohr’s Model introduces two important ideas: • Electrons exist only at certain discrete energy levels ⇒ described by quantum numbers 9 • Energy is involved by moving an electron from one level to another Limitations to Bohr’s Model • Good explanation of H-atom; cannot explain spectra of other atoms • It describes electron merely as a small particle circling around the nucleus. However, the electron exhibits properties of waves (we will discuss this in details) Ionization Energy: is the minimum energy needed to remove an e − from the atom when it is in its ground state and send it into infinity. 12 12 En = − 2 R y − 2 R y = 0 − (−1) R y = R y = 2.18 x10 −18 J ∞ 1 If we multiply this value by NA we get the ionization energy per mole of H atoms: IE = (2.18x10-18J)(6.02x1023 mol-1) = 1.31 x 106 J mol-1 = 1310 kJ mol-1 Atomic Spectra Z2 1 Z2 1 En = − 2 R y − − 2 R y == − Z 2 2 − 2 R y = hν n2 n1 n2 n1 1 1 (3.29 X 1015 s −1 ) n2 > n1 (absorption) − 2 2 n1 n2 1 1 ν == Z 2 2 − 2 (3.29 X 1015 s −1 ) n1 > n2 (emission) n2 n`1 ν == Z 2 The wave nature of e − and the particle nature of photons Albert Einstein: E = mc (energy equivalent to a given amount of mass) Max Planck: E = hν 2 mc 2 = hν = h c λ ⇒λ= Louis de Broglie: λ = h mc h (wavelength of any particle of mass m moving at velocity mυ υ) – wave – particle duality of matter and energy. ( we will come back!) 10 λ= h h h = ⇒ p= λ Shorter λ-photons have greater momentum mυ p Alfred Compton Effect (1923): directed a beam of X-ray photons at a sample of graphite and the λ of the reflected photons increased ⇒ photons transferred some of their momentum to the e − in the graphite ⇒ photons behave as particles with momentum. Uncertainty Principle (1924): it is impossible to know exactly the position and the momentum of a particle simultaneously. (∆x )(∆p x ) ≥ h h and since p = mυ (∆x )m(∆υ x ) ≥ 4π 4π (∆x ) = uncertainty in position (∆p ) = uncertainty in momentum (∆υ ) = uncertainty in velocity From this principle we cannot assign fixed paths for e − (such as the circular paths in Bohr’s model). Therefore, we need a better theory. This theory is the Quantum Mechanical Model of Atom. 11 Wave motion in restricted systems The condition on the allowed wavelengths: nλ =L 2 n = 1, 2, 3…. Fundamental or 1st harmonic: n = 1 2nd harmonic: n = 2 Regions of no vibration: nodes (The ends do not count as nodes) The higher the number of harmonics ⇒ the greater the number of nodes The shorter the wavelength, the larger the frequency and the higher the energy, E = hν = h c λ For a circular standing wave on a closed loop (Fig. 16-16 of textbook) 2πr = nλ n = 1, 2, 3…. h h ⇒ 2πr = n 2π meυ h h h = nλ ⇒ λ = = --- de Broglie Consequently, n meυ meυ p Bohr: meυr = n 12 The diffraction pattern caused by light passing through two adjacent slits Davisson and Germer (1927): a crystal diffracts e − and de Broglie’s relationship correctly predicts their wavelengths. Comparing diffraction patterns of x-rays and electrons X-ray diffraction of aluminum Electron diffraction of aluminum 13 Summary of the major observations and theories leading from classical theory to quantum theory The Schrödinger Equation (1925) A wavefunction (Ψ) can describe in its entirety any physical system. Ψ incorporates both wavelike and particlelike behavior of the e − . 14 Max Born (1927) Ψ2 gives the probability density (at any point in space it represents the probability that the e − will be found in space at that location. To calculate the wavefunction for any particle we use Schrödinger Equation: h 2 d 2ψ d 2ψ − + V ( x )ψ = Eψ The term can be thought of as a measure of how dx 2 2m dx 2 sharply the wavefunction will be. The left hand side term of the equation is written as h2 d 2 − + V ( x ) = H and the SE becomes Hψ = Eψ . The H is called Hamiltonian 2m dx 2 and it is a set of mathematical operations that represent the total energy (kinetic and potential) of the electron within the atom. The symbol E represents the total energy of the electron. 2 A plot of ψ represents an orbital, a position probability distribution of the electron Spherical Polar Coordinates x = r cos φ sin θ y = r sin φ sin θ z = r cos θ The wavefunction can be written as a function that depends only on r and another function that depends on θ and φ, that is ψ ( r,θ ,φ ) = R( r )Y (θ ,φ ) The function R(r ) is called the radial wavefunction and it tells us how the wavefunction varies as we move away from the nucleus in any direction. The function Y (θ ,φ ) is called the angular wavefunction and it tells us how the wavefunction varies as the angles θ and φ change. When the SE is solved it yields many solutions - many possible wavefunctions. The wavefunctions are fairly complicated mathematical functions that we will not examine 15 them here. Instead we will discuss graphical representations (or plots) of the orbitals that correspond to these wavefunctions. An orbital is a one electron wavefunction. Each orbital is specified by three interrelated quantum numbers, the principal quantum number (n), the angular quantum number (l) – sometimes it is called azimuthal quantum number and the magnetic quantum number (ml). Electron probability density in the ground-state H atom The principal quantum number (n) is an integer that determines the overall size and energy of an orbital. Its possible values are n = 1, 2, 3,… For a H-atom, the energy of an electron with quantum number n is given by: 12 En = − 2 ( 2.18 x10−18 J ) n = 1, 2, 3, … n The angular quantum number (l) is an integer that determines the shape of an orbital, with possible values 0, 1, 2, … (n-1) Value of l 0 1 2 3 orbital s p d f The magnetic quantum number (ml) is an integer that specifies the orientation of an orbital in space and it has values ranging from –l ….. +l Collection of orbitals with the same value of n is called a shell, (n = 3 third shell) Collection of orbitals with the same values of l and ml is called a subshell 16 Example What values of the angular momentum (l) and magnetic (ml) quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals exist for n = 3? For n = 3, l = 0, 1, 2 For l = 0, ml = 0 For l = 1, ml = -1, 0, +1 For l = 2, ml = -2, -1, 0, +1, +2 There are 9 ml values and therefore, 9 orbitals with n = 3. The number of ml values are 2l + 1 and the total number of orbitals is n2 Example Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers (a) n = 3, l = 2 (b) n = 2, l = 0 (c) n = 5, l = 1 d) n = 4, l = 3 Combine the n value and l designation to name the sublevel. Knowing l, we can find ml and the number of orbitals Question n, l Orbital(sublevel) Number of orbitals ml a 3,2 3d -2,-1,0,+1,+2 5 b 2,0 2s 0 1 c 5,1 5p -1,0,+1 3 d 4,3 4f -3,-2,-1,0,+1,+2,+3 7 17 The 1s, 2s, and 3s orbitals The 2p orbitals 18 The 3d orbitals One of the seven possible 4f orbitals Radial Nodes: n – l - 1 Angular Nodes: l Total Nodes: n – 1 19 Observing the effect of electron spin: the Stern-Gerlach Experiment 20