20th Century Atomic Theory- Hydrogen Atom

Transcription

1 Background for (mostly) Chapter 12 of EDR 20th Century Atomic Theory- Hydrogen Atom EDR Section 12.7 Rutherford's scattering experiments (Raff ) in 1910 lead to a "planetary" model of the atom where all the positive charge and most of the mass were concentrated in a small nucleus. Electrons were pictured as revolving around the nucleus in a volume whose radius was - 100,000 times that of the nucleus. Classical physics dictates that motion be in a straight line unless some force exists to change the direction. For circular motion there must be a constantly changing force. For an electron orbiting a nucleus the force is the coulombic force of attraction between the nucleus of charge Ze (number of protons = Z) and the electron of charge -e. Consider force F, velocity v, and acceleration a to be vectors having both magnitude and direction. Then the coulombic force is radially directed inwards towards the nucleus and it changes its direction as does the velocity. (Note: for formula simplicity Gaussian units are used with charge in statcoulombs. To obtain Raff's formulas in SI units replace every e 2 with e 2 /4:rre 0 where 0 is the vacuum permittivity, seep. 22 in Raff.) Fcoul = (magnitude of F) (1) Stability requires that all of the forces acting upon the electron balance (i.e., Newton- the resultant of the forces is zero). Thus the coulombic force must just be balanced by the centrifugal (fictitious) force of the orbiting electron's motion which is radially directed outwards (let go a stone tied to a string which you twirl around your head and it flies off away from you): mv 2 Fcent = - r (magnitude) (2) There was a huge flaw to this otherwise very appealing planetary model - the model was unstable (p. 558). 308 Classically an accelerating charged particle radiates energy. Viewed vectorally, an electron orbiting a nucleus has a constantly changing acceleration. It is easy to picture that this motion gives rise to an oscillating electron. Picture yourself at the nucleus laying down in the plane of the electron's orbit. As the electron completes one cycle of its orbit you observe the electron to appear to oscillate up when it is closest to your head and then down when it is closest to your feet:

2 -2- Such an oscillating electron induces oscillating electric and magnetic fields and generates an electromagnetic wave. By so doing it emits electromagnetic radiation whose frequency corresponds to the number of revolutions the electron makes about the nucleus per second. In emitting radiation the atom loses some energy and the electron must move in closer to the nucleus. To see that this is so let us find the energy (classically) of a hydrogen atom and see how it depends upon r, the distance between the nucleus and the electron. Balancing the coulombic (Eq. 1) and centrifugal (Eq. 2) forces: (3) => r = mv 2 (4) The total energy is the sum of the kinetic and potential energies Etot = Ekin + Epot where the potential energy is the coulomb potential due to the coulombic force of Eq. 1 Ze 2 Epot = Vcoul = -I Fcouldr = I ;:2 dr and the kinetic energy is the familiar => Vcoul r (5) (6) where Eq. 3 was used to express mv 2 in terms of r. So Hence if the system loses energy, f).e < 0, Ze 2 Ze 2 E = tot - 2r r ze 2 ( ze2) f).e = E f - E; = - 2r f - - 2r; = ze 2 ( 1 1 ) '; rr The new radius r 1 must be less than the original radius r;. Note from Eq. 4 that the velocity increases as r decreases so the electron spirals into the nucleus faster and faster, emitting radiation of increasing frequency. In 1913 Bohr was able to reconcile the stability of the planetary model by simply saying that classical physics was wrong in its prediction. Certain stationary orbits were stable. These orbits are characterized by a particular radius and energy which he found by arbitrarily postulat- -. ing that the angular momentum L of the electron was quantized. The general idea of quantization was not new: 1) in 1900 Planck quantized the material oscillators in a solid - those that could emit blackbody radiation in multiples of hv (11.2.1) and in 1905 Einstein quantized light in his explanation of the photoelectric effect (11.2.2). However, Bohr was the first to apply quantization to the structure of the atom and his theory correctly predicted the emission spectral lines of hydrogen ( and ). 12.7, 19.1 (7)

3 -3- To understand angular momentum consider the figure accompanying Eq. 1 and view it side on... L So for counterclockwise circular motion in the x,y-plane (this piece of paper) the angular --+ momentum vector L would point in the positive z direction (coming out of the paper towards you). Angular momentum is defined as the cross product ( L = r x p = r x mv -+) = m (--+ r X v -+) = m(rv sin 8) (magnitude) = mvr (when sine= 1) where e is the angle between rand the momentum p (90 for circular motion). Bohr's condition nh L = mvr = - = nn n = 1, 2, (8) 27r We can use the quantization condition of Eq. 8 with Eq. 4 to find Bohr's stable orbits where a 0 is the Bohr radius. Ze 2 Ze 2 mr 2 Ze 2 mr 2 Ze 2 mr 2 r= =----= = mv 2 mv 2 mr 2 (mvr)2 n2n 2 => rn = ao = -2- = em = z n = 1, 2, Chapter 15, above Eq. (15.4) on p. 358 Raff's Eq in SI units To find the quantized energy levels we need to only substitute this final value for r into the total energy of Eq. 7 = _ 2 2 (_!} ) n 2 2a 0 (9) (eliminating e 2 with Eq. 9, in SI units) z2 = - - Ry n2 n = 1, 2, (10) ( e - n2 87re 0 a ' 0 ) Raff's Eq in SI units

4 where Ry is the Rydberg constant in joules and RH in wavenumbers, cm- 1 (Ry = hcrh ). -4- In explaining the emission spectrum of hydrogen, Bohr postulated that the hydrogen atom could only make a transition to a lower energy quantized state by emitting a photon whose energy exactly matched the difference in energy between the two states: photon = hvphoton = he..1photon = - 11E = - (E f - E;) = -[- z2 Ry- (- z2 Ry)~ = z2 Ry(-1-2_] n} n1 ~ n} n1 => hv ZRy--- 2 ( 1 1 J n} n1 n 1 =n 1 +l,n 1 +2, or Chapter 12, Eq. (12.8) _!_ = Z 2 RH[~ - ~J' Raff's equation in cm n 1 n; (11) Summary of Bohr's postulates for hydrogenic atoms: 1. An atom can exist in only certain discrete states of constant energy - stationary states 2. These stationary states do not emit electromagnetic radiation 3. Transitions occur between stationary states only upon absorption or emission of a photon where = hvphoton 4. An electron in a stationary state travels in a circular orbit and obeys classical mechanics 5. The electron's angular momentum in a stable orbit is an integral multiple of 1i Bohr's theory was immensely successful in explaining hydrogenic spectra (Eq. 11) and predicting the energy levels (Eq. 10). However it is now known that the orbits are not circular and the angular momentum is not integral (though it is still quantized). Furthermore, his theory could not be applied to any atom having more than one electron. Perhaps its greatest shortcomings were that it provided no rationale for. covalent chemical bonding or reason for quantization or why the atom should not radiate its energy. Nevertheless, his first three postulates are consistent with quantum mechanics and his theory was the first quantization of matter. It only remained to replace the requirement of classical particle behavior with wave behavior. Modern quantum theory was developed during the 1920's. In 1924 de Broglie proposed that any particle which has linear momentum p has wave-like properties and a wavelength..1 associated with it. The de Broglie relation gives their mathematical relationship (Eq ). Section 12.5 p..1 = h (12) In 1925 Heisenberg, Born, and Jordan presented the first formulation of quantum theory using matrices. At the time this "matrix mechanics" appeared somewhat obscure as it involved the mathematics of matrices. Later in 1925 Dirac introduced a theory based on Hamilton's classical equations of motion which were developed a century earlier ( ). Dirac translated his

5 -5- Chapter 13 Chapter 14 theory into a series of postulates, creating a versatile system of quantum mechanics though somewhat abstract saw the introduction of Schrodinger's "wave mechanics", based upon differential equations (11.5), still popular today. He took de Broglie's wave idea and Bohr's stationary states and concluded that the equation of motion must be a wave-like equation with boundary conditions which fix the energy levels Gust like a differential equation). The mathematical statement of his equation is HVI = EVI where E is the energy, fi is the operator for kinetic and potential energy, and VI is the solution to the equation - the wavefunction. To see a simple fi for the "particle-in-a-box" model (infinite potential well) that we will encounter in Chapter 12 (12.1.2), consider a one-dimensional particle of mass m confined to move in a one-dimensional box on the x-axis. It's Schrodinger equation is ~ n 2 d 2 1f!(x) HV1(X) = - 2m dx2 = EVI(X) (13) where d 2 VIIdx 2 is the second derivative of the wavefunction VI(X). The solutions to this equation, the VI, are standing waves. We interpret the wavefunction as suggested by Born: the square of the wavefunction, IVI(x)l 2, is proportional to the probability of finding the particle at x. Notice that we never know exactly where the particle is only its probability. This accords very nicely with Heisenberg's uncertainty principle of 1927 ( ) Section pl1x ~ which states that we can never precisely and simultaneously know (or measure) the momentum and position of a particle. There will always be an uncertainty of at least n/2 in the product of the uncertainty in p and the uncertainty in x. So Bohr's stable orbit does not exist. Since we know the VI solutions to Eq. 13, how about the energies? Without solving the Schrodinger equation we can still solve for E like we did in class. Let L be the length of the box that the particle moves in. An integral number of half wavelengths must fit into the length L in order to have the constructive interference necessary to create a standing wave (or to produce a pleasant sounding harmonic on a stringed instrument) na 2 n 2 (14) = L n=1,2,... (15) From the de Broglie relationship of Eq. 12 we can relate this wavelength A to the particle's lin~ar momentum p h p = A nh 2L = mv Using the formula for kinetic energy in terms of p 1 2 Ekin = 2 mv = and substituting into Eq. 16 for the momentum Eki" ~ :: ~ L ( ~~ )' n = 1, 2, (16) n = 1, 2, (17) The last expression for the energy is exactly what we would have obtained had we solved the

6 Schroding equation, Eq Finishing the development of quantum theory: in 1926 Dirac and Jordan, working independently, formulated quantum mechanics in an abstract version called "transformation theory" that is a generalization of matrix mechanics and wave mechanics. Schrodinger shows that his approach is identical to those of Heisenberg and Dirac. In 1928 Dirac develops relativistic quantum mechanics where the spin quantum number m 5 emerges. In 1948 Feynman devised the "path integral" formulation of quantum mechanics. In the two cases that we have looked at so far, Bohr's hydrogen atom with the electron traveling on the circumference of a circle (Eq. 10) or the particle in a one-dimensional box (Eq. 17), a quantum number n appears upon which the energy is dependent. An electron in a real hydrogenic atom moves in three-dimensional space so three quantum numbers (13.1.2) are necessary to specify the state of the electron (one quantum number for each degree of freedom or dimension). When fully treated by considering relativistic effects, a fourth quantum number becomes necessary to fully specify the state. n = 1, 2, is the principle quantum number which determines the energy of the electron. n indicates the effective volume in space in which the electron moves. If n = 1 the electron is more likely to be closer to the nucleus than if n = 2, 3, l = 0, 1,..., n - 1 is the angular momentum (azimuthal) quantum number which determines the magnitude of the electron's angular momentum. l designates the shape of the volume or region in space that the electron occupies. The integer specifying l is generally replaced by a letter s => l = 0 p => l 1 d => l = 2 f => l = 3 g => l = 4 Section 15.2 m 1 = -l, -l + 1,..., 0,..., l- 1, lis the magnetic quantum number which determines the orientation in space of the angular momentum and hence the magnetic moment. m 1 indicates the orientation in space of the volume or region that the electron occupies. The wavefunction for a hydrogenic atom depends upon n, l, and m 1 and is referred to as an orbital. The three 2p orbitals correspond to three different values for m 1 (-1,0,1). Dirac's relativistic treatment showed that a fourth quantum number was needed to fully specify the state of an electron, the spin quantum number m 5 with values -112 or+ 1/2 corresponding to two different orientations of spin (13.2.3). Section 15.7