Extracting Atomic and Molecular Parameters From the de Broglie Bohr Model of the Atom

Transcription

1 Extracting Atomic and Molecular Parameters From te de Broglie Bor Model of te Atom Frank ioux Te 93 Bor model of te ydrogen atom was replaced by Scrödingerʹs wave mecanical model in 96. However, Borʹs model is still profitably taugt today because of its conceptual and matematical simplicity, and because it introduced a number of key quantum mecanical ideas suc as te quantum number, quantization of observable properties, quantum jump and stationary state. In addition it provided realistic values for suc parameters as atomic and molecular size, electron ionization energy, and molecular bond energy. In is ʺplanetaryʺ model of te ydrogen atom Bor began wit a Newtonian analysis of te electron executing a circular orbit of radius about a stationary nucleus, and ten arbitrarily quantized te electronʹs angular momentum. Finally, by fiat e declared tat te electron was in a non radiating stationary state because an orbiting (accelerating) carge radiates energy and will collapse into te oppositely carge nucleus. In 94 de Broglie postulated wave particle duality for te electron and oter massive particles, tereby providing te opportunity to remove some of te arbitrariness from Borʹs model. For example, an electron possessing wave properties is subject to constructive and destructive interference. As will be sown tis leads naturally to quantization of electron momentum and kinetic energy, and consequently to a stable ground state for te ydrogen atom. Te de Broglie Bor model of te ydrogen atom presented ere treats te electron as a particle on a ring wit wave like properties. Te key equation is wave particle duality as expressed by te de Broglie equation. Te particle concept momentum and te wave concept are joined in a reciprocal relationsip mediated by te ubiquitous Planckʹs constant. p λ Tis equation will be used wit te Bor model of te ydrogen atom to explain atomic stability and to generate estimates of atomic size and electron binding energy in te atom. In te de Broglie version of te Bor ydrogen atom we say tat te electron occupies a ring of radius. It is not orbiting te nucleus, it is beaving as a stationary wave. In order to avoid self interference te following wavelengt restriction must be obeyed for te ground state of te ydrogen atom. λ π

2 Wen combined wit te de Broglie equation it reveals te following restriction on te electronʹs particle property, linear momentum. p π Tis means tere is also a restriction on te electronʹs kinetic energy. Use of tis equation in te classical expression for kinetic energy yields te quantum mecanical kinetic energy or more accurately electron confinement energy. T p m 8π m In tis equation we ave moved from te classical definition of kinetic energy to te quantum mecanical version expressed on te rigt in atomic units. π m e 4πε 0 Te electrostatic potential energy retains its classical definition in quantum mecanics. V e 4πε 0 Te total electron energy, E H ( ) T( ) V( ), is now minimized wit respect to te ring or orbit radius, te only variational parameter in te model. Te total energy, and kinetic and potential energy are also displayed as a function of ring radius..5 E H ( ) Minimize E H.000 E H ( ) E H () r 0.5 r r r From tis simple model we learn tat it is te wave nature of te electron tat explains atomic stability. Te electronʹs ring does not collapse into te nucleus because kinetic (confinement) energy goes to positive infinity (~ ) faster tan potential energy goes to negative infinity (~ ). Tis is seen very clearly in te grap. Te ground state is due to te sarp increase in kinetic energy as te ring radius

3 decreases. Tis is a quantum effect, a consequence of de Broglieʹs ypotesis tat electrons ave wave like properties. As Klaus uedenberg as written, ʺTere are no ground states in classical mecanics.ʺ Te minimization process above te figure provides te ground state ring radius and electron energy in atomic units, a 0 and E, respectively. a pm gives us te bencmark for atomic size. Tables of atomic and ionic radii carry entries ranging from approximately alf tis value to rougly five or six times it. Te ground state (binding) energy, E 0.5 E 3.6 ev 3 kj/mol, is te negative of te ionization energy. Tis value serves as a bencmark for ow tigtly electrons are eld in atoms and molecules. A more compreensive treatment of te Bor atom utilizing te restriction tat an integral number of wavelengts must fit witin te ring, n, were n,, 3,... reveals a manifold of allowed energy states ( 0.5 E /n ) and te basis for Borʹs concept of te quantum jump wic ʺexplainedʺ te ydrogen atom emission spectrum. Here for example is te n 4 Bor atom excited state. udimentary estimates of some molecular parameters, te most important being bond energy and bond lengt, can be obtained using te following Bor model for H. Te distance between te protons is, te electron ring radius is, and te bond axis is perpendicular to te plane of te ring. Tere are eigt contributions to te total molecular energy based on tis model: electron kinetic energy (), electron proton potential energy (4), proton proton potential energy () and electron electron potential energy. E H ( ) 4

4 Minimization of te energy wit respect to ring radius and proton proton distance yields te following results. Minimize E H E H ( ).00 Te H H bond energy is te key parameter provided by tis analysis. We see tat it predicts a stable molecule and tat te energy released on te formation of H is 0. E or 63 kj/mol, compared wit te experimental value of 458 kj/mol. Te model predicts a H H bond lengt of 58 pm (. 5.9 pm), compared to te literature value of 74 pm. Tese results are acceptable given te primitive caracter of te model. H H H ΔE bond E H ( ) E H ( ) ΔE bond 0.00 In addition to tese estimates of molecular parameters, te model clearly sows tat molecular stability depends on a balancing act between electron proton attraction and te ʺrepulsiveʺ caracter of electron kinetic energy. Just as in te atomic case, it is te / dependence of kinetic (confinement) energy on ring radius tat prevents molecular collapse under electron proton attraction. As te energy profile provided in te Appendix sows, te immediate cause of te molecular ground state is a rise in kinetic energy. Potential energy is still declining at tis point and does begin to rise until 0.55 a 0, well after te ground state is reaced at.0 a 0. Altoug te model is a relic from te early days of quantum teory it still as pedagogical value. Its matematical simplicity clearly reveals te importance of te wave nature of matter, te foundational concept of quantum teory. Two relatively recent appraisals of Borʹs models of atomic and molecular structure ave been appeared in Pysics Today: ʺNiels Bor between pysics and cemistry,ʺ by Helge Krag, May 03, ʺBorʹs molecular model, a century later,ʺ by Anatoly Svidzinsky, Marlan Scully, and udley Herscbac, January 04,

5 Appendix. Energy Given Energy E H ( ) d E H d ( ) 0 Energy ( ) Find( Energy) T( ) Energy( ) 0 V ( ) 4 Energy( ) 0 Energy( ) Energy( ) T ( ) V ( )