CHE1112 Lecture I-part II PDF

The Bohr Model of the Atom Which series releases most energy? The larger the fall the {ExcitedElectrons*} greater the energy Bohr’s model for hydrogen atom  The Bohr model of hydrogen (original argument) Bohr exp...

The Bohr Model of the Atom Which series releases most energy? The larger the fall the {ExcitedElectrons*} greater the energy Bohr’s model for hydrogen atom  The Bohr model of hydrogen (original argument) Bohr explained the line spectrum of hydrogen with a model in which the single Ln=rp=m vn rn hydrogen electron can only be in certain definite orbits. In the nth allowed orbit, the electron has orbital angular momentum nh/2π. Bohr proposed that angular momentum is quantized. Copyright © 2012 Pearson Education Inc.  The Bohr model of hydrogen Let’s use a different argument based on deBroglie waves to obtain the same conclusions. Think of a standing wave with wavelength λ that extends around the circle. 2p rn = nln Q: How is the momentum of the atomic electron related to its wavelength ? (remember the Prince) h 2p rn = nln = n mvn h Same as the Bohr mvn rn = n quantization 2p condition Copyright © 2012 Pearson Education Inc.  The Bohr model of hydrogen (The mass m is that Now let’s use a Newtonian h of the electron.) mvn rn = n argument for a planetary 2p model of the atom but use the Bohr quantization condition. e mvn2 2 Fe = = Fc = Balance electrostatic 4pe 0 rn 2 rn and centripetal forces e2 mvn 2 rn 2 (mvn rn )2 e2 (nh / 2p )2 = = Þ = 4pe 0 rn rn 4pe 0 rn Here we used the Bohr quantization condition Copyright © 2012 Pearson Education Inc.  The Bohr model of hydrogen (Bohr radius) e 0 (nh) 2 rn = p me2 e 0 (nh)2 rn = Þ r = n 2 a0 p me 2 n Here n is the “principal quantum number” and a0 is the “Bohr radius”, which is the minimum radius of an electron orbital. e 0h 2 a0 = Þ r = n 2 a0 a0 = 5.29 ´10-11 m p me 2 n Copyright © 2012 Pearson Education Inc. The Bohr model of hydrogen (Energy levels, derivation) h e2 mvn rn = n Þ vn = 2p e 0 (2nh) 1 me 4 K n = mvn 2 = 2 2 2 8n h e 0 (nh) 2 -me4 rn = En = K n +Un = 2 2 2 p me2 e 0 8n h -me-e 2 4 Un = = 2 2 2 4pe 0 rn e 0 4n h Note that E and U are negative (1/8-1/4=-1/8) This expression for the allowed energies can be rewritten and used to predict atomic spectral lines ! Copyright © 2012 Pearson Education Inc. The Bohr model of hydrogen (Energy levels) -me4 -hcR me4 En = 2 2 2 = 2 ; R = e 0 8n h n 8 e 0 2 h 3c -hcR Here R is the “Rydberg En = 2 n constant”, R=1.097 x 107 m-1 Also hcR = 13.60 eV is a useful result. Question: How can we find the energies of photon transitions between atomic levels ? hc 1 1 = EUpper - ELower = hcR( - ) l n 2 Upper n 2 Lower Copyright © 2012 Pearson Education Inc. The Bohr model of hydrogen (It works) 1 1 1 = R( - ) l n 2 Upper n 2 Lower Here R is the Rydberg constant, R=1.097 x 107 m-1 If nupper=3, nlower=2, let’s calculate the wavelength. 1 1 1 -1 1 1 = R( 2 - 2 ) = (1.097 ´10 m )( - ) = 656.3nm 7 l 2 3 4 9 Balmer Hα line, agrees with experiment within 0.1% Copyright © 2012 Pearson Education Inc.  Hydrogen spectrum (also has other spectral lines) The line spectrum at the bottom of the previous slide is not the entire spectrum of hydrogen; it is just the visible-light portion. Hydrogen also has series of spectral lines in the infrared and the ultraviolet. Copyright © 2012 Pearson Education Inc.  Hydrogen-like atoms The Bohr model can be applied to any atom with a single electron. This includes hydrogen (H) and singly-ionized helium (He+). See the Figure below. -hcR me4 En = 2 ; R = n 8 e 0 2 h 3c Question: How should this formula be modified for singly-ionized helium ? Ans: He has 2p. If an atom is singly ionized, then rnrn/ZEnZ2E Ze2 (nh / 2p )2 = But the Bohr model does not 4pe 0 rn work for other atoms; need QM Copyright © 2012 Pearson Education Inc. Copyright © 2012 Pearson Education Inc. Copyright © 2012 Pearson Education Inc. Atomic Spectra and the Bohr Atom Notice that the wavelength calculated from the Rydberg equation matches the wavelength of the green colored line in the H spectrum.   4.865 x 10-7 m E  0.4088 x 10-18 J  Limitations of Bohr Theory theory fails to explain : could not account the spectra of atoms more complex than hydrogen. to give any information regarding the distribution and arrangement of electrons in atoms. does not explain the experimentally observed variations in intensity of the spectral lines of an element.  the transitions of electrons from one level to another, the rate at which they occur or the selection rules which apply to them. the fine structure of spectral lines. Bohr’s theory does not throw any light on it. when electric Stark effect or magnetic fieldZeeman effec is applied to the atom, each spectral line is splitted into several lines.  Extensions of Bohr Theory Finite nuclear mass Elliptical orbits (Bohr-Sommerfeld model)s we assumed that the nucleus is so massive that it is effectively at rest. But the mass of the nucleus is finite and the classical model of the atom therefore envisages that the electron and the nucleus both revolve around their common centre of mass.