0th Centuy Atomic Theoy - Hydogen Atom Ruthefod s scatteing expeiments (Section.5, pp. 53-55) in 1910 led to a nuclea model of the atom whee all the positive chage and most of the mass wee concentated in a small nucleus. Electons wee pictued as evolving aound the nucleus in a volume whose adius was 100,000 times that of the nucleus. Classical physics dictates that motion is in a staight line unless some foce exists to change the diection. Fo cicula motion thee must be a constantly changing foce. Fo an electon obiting a nucleus the foce is the coulombic foce of attaction between the nucleus of chage Ze (numbe of potons = Z) and the electon of chage -e. (Conside foce F,velocity u,and acceleation a to be vectos having both magnitude and diection.) The coulombic foce is adially diected inwads towads the nucleus and it changes its diection as does the velocity u > > a O+ > O F F coul = Ze (magnitude of F) (1) Stability equies that all of the foces acting upon the electon balance (i. e., Newton - the esultant of the foces is zeo). Thus the coulombic foce must just be balanced by the centifugal (fictitious) foce of the obiting electon s motion which is adially diected outwads (let go a stone tied to a sting which you twil aound you head and it flies off away fom you): F cent = mu (magnitude) () Thee was a huge flaw tothis othewise vey appealing nuclea model - the model was unstable (p. 309). Classically an acceleating chaged paticle adiates enegy. Viewed vectoally, anelecton obiting a nucleus has a constantly changing acceleation. It is easy to pictue that this motion gives ise to an oscillating electon. Pictue youself at the nucleus laying down in the plane of the electon s obit. As the electon completes one cycle of its obit you obseve the electon to appea to oscillate up when it is closest to you head and then down when it is closest to you feet: O
-- Such an oscillating electon induces oscillating electic and magnetic fields and geneates an electomagnetic wave. By so doing it emits electomagnetic adiation whose fequency coesponds to the numbe of evolutions the electon makes about the nucleus pe second. In emitting adiation the atom loses some enegy and the electon must move in close to the nucleus. To see that this is so let us find the enegy (classically) of a hydogen atom and see how it depends upon, the distance between the nucleus and the electon. Balancing the coulombic (Eq. 1) and centifugal (Eq. ) foces: Ze + mu = 0 (3) => = Ze mu (4) The total enegy is the sum of the kinetic and potential enegies E tot = E kin + E pot whee the potential enegy is the coulomb potential due to the coulombic foce of Eq. 1 E pot = V coul = F could = Ze d and the kinetic enegy is the familia => V coul = Ze (5) E kin = 1 mu = Ze whee Eq. 3 was used to expess mu in tems of. So (6) E tot = Ze Hence if the system loses enegy, E 0, E = E f E i = Ze f Ze = Ze Ze i = Ze 1 1 i f The new adius f must be less than the oiginal adius i. Note fom Eq. 4 that the velocity inceases as deceases so the electon spials into the nucleus faste and faste, emitting adiation of inceasing fequency. In 191 Boh was able to econcile the stability of the nuclea model by simply saying that classical physics was wong in its pediction. Cetain stationay obits wee stable. These obits ae chaacteized by a paticula adius and enegy which he found by abitaily postulating that the angula momentum L of the electon was quantized. The geneal idea of quantization was not new: 1) in 1900 Planck quantized the mateial oscillatos in a solid - those that could emit blackbody adiation in multiples of hν and ) in 1905 Einstein quantized adiation in his explanation of the photoelectic effect (pp. 308-311, 30-31). Howeve, Boh was the fist to apply quantization to the stuctue of the atom and his theoy coectly pedicted the emission spectal lines of hydogen (Section 7.3, pp. 306-309). (7)
-3- on To undestand angula momentum conside the figue accompanying Eq. 1 and view itside > L O+ > O mu > Fo counteclockwise cicula motion in the x,y-plane (this piece of pape) the angula momentum vecto L would point in the positive z diection (coming out of the pape towads you ). Angula momentum is defined as the coss poduct L = (m u) = m( u) = m(u sin θ ) (magnitude) = mu (when sin θ = 1) whee θ is the angle between and the momentum m u (90 o fo cicula motion). Boh s condition L = mu = nh n = 1,,... (8) π We can use the quantization condition of Eq. 8 with Eq. 4 to find Boh s stable obits = Ze = Ze m = Ze m = 4π Ze m mu mu m (mu) n h n h => n = 4π Ze m = n Z a o n = 1,,... (9) whee a o is the Boh adius. To find the quantized enegy levels we need to only substitute this final value fo into the total enegy of Eq. 7 E n = Ze Z = Z n a o n e a o = Z h n 8π ma o (eliminating e with Eq. 9) = Z n Ry n = 1,,... (10*) whee Ry is the Rydbeg constant. In explaining the emission spectum of hydogen, Boh postulated that the hydogen atom could only make a tansition to a lowe enegy quantized state by emitting a photon whose enegy exactly matched the diffeence in enegy between the two states: E photon = hν photon = hc λ photon = E atom = (E f E i )
-4- = Z Ry Z n i Ry = Z Ry 1 1 n i o, asinthe week 10 lab, => hν = Z Ry 1 λ 1 1 n i = Z Ry hc 1 Summay of Boh s postulates fo hydogenic atoms: n i = n f + 1, n f +,... (11*) 1. Electon tavels in a cicula obit. A set of discete obits exist which ae stable 3. Angula momentum is an integal multiple of h/π 4. Tansitions occu between stable obits only upon absoption o emission of a photon whee E =hν photon Boh s theoy was immensely successful in explaining hydogenic specta (Eq. 11) and pedicting the enegy levels (Eq. 10). Howeve it is now known that the obits ae not cicula and the angula momentum is not integal (though it is still quantized). His theoy could not be applied to any atom having moe than one electon. Futhemoe it povided no ationale fo covalent chemical bonding o eason fo quantization o why the atom should not adiate its enegy. Nevetheless Boh opened the way fo the quantization of the enegy of atoms and molecules. Moden quantum theoy was developed duing the 190 s. In 194 de Boglie poposed that any paticle which has linea momentum mu has wave-like popeties and a wavelength λ associated with it. The de Boglie elation gives thei mathematical elationship (p. 313) 1 n i (mu)λ = h (1*) In 195 Heisenbeg pesented a consistent theoy based on matix mechanics. At the time it appeaed somewhat obscue as it involved the mathematics of matices. Late in 195 Diac intoduced a theoy based on Hamilton s classical equations of motion which wee developed a centuy ealie. Diac tanslated his theoy into a seies of postulates, ceating a vesatile system of quantum mechanics though somewhat abstact. 196 saw the intoduction of Schödinge s wave mechanics (Section 7.5), still popula today. He took de Boglie s wave idea and Boh s stationay states and concluded that the equation of motion must be a wave-like equation with bounday conditions which fix the enegy levels (just like a diffeential equation). The mathematical statement of his equation is Ĥψ = Eψ whee E is the enegy, Ĥ is the Hamiltonian opeato (the opeato fo kinetic and potential enegy) and ψ is the solution to the equation - the wavefunction. To see a simple Ĥ, conside a one-dimensional paticle of mass m confined to move in a one-dimensional box on the x-axis. It s Sch ödinge equation is
Ĥψ (x) = h 8π m -5- d ψ (x) dx = Eψ (x) (13) whee d ψ /dx is the second deivative of the wavefunction ψ (x) The solutions to this diffeential equation, the ψ, ae standing waves. We intepet the wavefunction as suggested by Max Bon: the squae of the wavefunction, ψ (x),ispopotional to the pobability of finding the paticle at x. Notice that we neve know exactly whee the paticle is only its pobability. This accods vey nicely with Heisenbeg s uncetainty pinciple of 197 (pp. 314-315) (mu) x h (14*) 4π which states that we can neve pecisely and simultaneously know (o measue) the momentum and position of a paticle. Thee will always be an uncetainty of at least h/4π in the poduct of the uncetainty in mu and the uncetainty in x. SoBoh s stable obit does not exist. Since we know the ψ solutions to Eq. 13, how about the enegies? Without solving the Schödinge equation one can solve fo E. Let L be the length of the box that the paticle moves in. An integal numbe of half wavelengths must fit into the length L in ode to have the constuctive intefeence necessay to ceate a standing wave (o to poduce a pleasant sounding hamonic on a stinged instument p. 36) nλ = L n = 1,,... (15) Fom the de Boglie elationship of Eq. 1 we can elate this wavelength λ to the paticle s linea momentum mu mu = h λ = nh L Using the fomula fo kinetic enegy n = 1,,... (16) E kin = 1 mu and substituting into Eq. 16 fo the momentum E kin = (mu) m = 1 m nh L = n h 8mL n = 1,,... (17) The last expession fo the enegy is exactly what we would have obtained had we solved Eq. 13. Finishing the development of quantum theoy: in 196 Schödinge shows that his appoach is identical to those of Heisenbeg and Diac. In 198 Diac develops elativistic quantum mechanics whee the spin quantum numbe m s emeges. In the two cases that we have looked at so fa, Boh s hydogen atom with the electon taveling on the cicumfeence of a cicle (Eq. 10) o the paticle in a one-dimensional box (Eq. 17), a quantum numbe n appeas upon which the enegy is dependent. An electon in a eal hydogenic atom moves in thee-dimensional space so thee quantum numbes (pp. 317-318) ae necessay to specify the state of the electon (one quantum numbe fo each degee of feedom o dimension). When fully teated by consideing elativistic effects, a fouth quantum numbe (spin) becomes necessay to fully specify the state. n = 1,,... is the pincipal quantum numbe which detemines the enegy of the electon. n indicates the effective volume in space in which the electon moves. If n = 1the electon
-6- is moe likely to be close to the nucleus than if n =, 3,... l = 0, 1,..., n 1isthe angula momentum (azimuthal) quantum numbe which detemines the magnitude of the electon s angula momentum. l designates the shape of the volume o egion in space that the electon occupies. The intege specifying l is geneally eplaced by a lette: l = 0 => s l = 1 => p l = => d l = 3 => f l = 4 => g m l = l, l + 1,..., 0,...,l 1, l is the magnetic quantum numbe which detemines the oientation in space of the angula momentum and hence the magnetic moment. m l indicates the oientation in space of the volume o egion that the electon occupies. The wavefunction fo a hydogenic atom depends upon n, l, and m l and is efeed to as an obital such as the thee p obitals of Figue 7.7 (n =, l = 1). The thee p obitals coespond to thee diffeent values fo m l (-1,0,1). The five 3d obitals ae shown in Figue 7.8 coesponding to m l values of -, -1, 0, 1, and. Diac s elativistic teatment showed that a fouth quantum numbe was needed to fully specify the state of an electon, the spin quantum numbe m s with values -1/ o +1/ coesponding to two diffeent oientations of spin (p. 319).