The Hydrogen Atom According to Bohr

The Hydrogen Atom According to Bohr

The atom We ve already talked about how tiny systems behave in strange ways. Now let s s talk about how a more complicated system behaves. The atom! Physics 9 4

Early models of the atom The basic concept of atoms goes back to the Greeks. John Dalton was pretty much the first to formulate a modern concept of the atom. He formulated a basic model of the elements. Physics 9 5

Early models of the atom Dmitri Mendeleev Physics 9 6

Early models of the atom A more sophisticated model was developed by J.J. Thompson, who also discovered the electron. His model involved a neutral atom. Physics 9 7

Early models of the atom Thompson imagined the atom as being a positive sphere, with electrons embedded in the sphere. This gave an overall neutral atom. Physics 9 8

Early models of the atom This is the plum pudding model. A more modern interpretation Physics 9 9

Ernest Rutherford Of course, it s important to test theories in science. Rutherford wanted to confirm or discredit Thompson s s model via direct experiment. Physics 9 10

Rutherford s experiment The Geiger-Marsden experiment Physics 9 11

Rutherford s model Rutherford formulated the familiar solar system model of the atom. He imagined a central nucleus with the electrons orbiting it like tiny planets. This is a very pretty and intuitive model of the atom. There s s only one problem Physics 9 12

This atom is unstable!! Rutherford s model has electrons orbiting around the nucleus. They re accelerating. Accelerating charges radiate energy! Physics 9 13

" = h p Remember de Broglie We ve already met de Broglie. The de Broglie wavelength. What does this tell us? Physics 9 14

Particle in a box A particle is confined to a small region of space by BIG forces Particles in confined spaces form standing waves. Physics 9 15

Particle in a box The energy of this particle is quantized. E n = h 2 8mL 2 n 2 Particles in confined spaces form standing waves. Physics 9 16

This idea might be useful After all, the electrons are confined to orbits. We ll come back to this idea. But, there s s another interesting problem with the hydrogen atom that also needed solving. Physics 9 17

Atomic spectra Pass an electric current through an atomic gas. Look at it through a diffraction grating. We get a discrete set of lines. This is the emission spectrum for iron. Physics 9 18

The hydrogen spectrum Hydrogen is much simpler. Any good theory of the hydrogen atom should explain where these lines come from. No one had a good theory, but there was a formula that worked Physics 9 19

The Rydberg formula Johannes Rydberg Examining the hydrogen atom he found a formula that predicted the spectra. 1 " = R $ 1 H 2 n # 1 ' & 2 ) % 1 n 2 ( R H =1.097 "10 7 m #1 This IS NOT a theory! Physics 9 20

Niels Bohr In 1913, Bohr proposed a model to explain the hydrogen atom, which explained the emission spectra. The model was formulated in terms of classical quantum mechanics. Physics 9 21

The main ideas Bohr suggested that the electrons could have only certain classical properties. The electrons can only travel in certain discrete orbits. Their orbits don t t decay! The electrons can only change orbits by absorbing or emitting light. This changes their energy discretely and the change is carried away (added) by radiating (absorbing) photons. Physics 9 22

The main ideas How can this happen? Bohr said that the angular momentum of the electron was quantized. h " h 2# =1.05 $10%34 Js L = mvr = nh This was reinterpreted by de Broglie! Physics 9 23

Let s see how this works Bohr L = mvr = nh de Broglie " = h p mvr = pr = hr nh = nh = " 2# $2#r = n" This is part of what gave de Broglie the idea Physics 9 24

The mathematics The electron is held to the proton by electrical forces. F E = ke2 r 2 But, it s s going around in a circle, too! F = mv 2 r Physics 9 25

The mathematics If the electron is in a stable orbit, then ke 2 = mv 2 r 2 r This gives the speed of the electron v = ke2 mr Physics 9 26

The mathematics Recall Bohr s quantization condition mvr = nh Plug in the expression we found for the velocity v = ke2 mr Physics 9 27

The mathematics This gives the allowed orbital radii. They re quantized! r n = h2 m e ke 2 n 2 This form can be re-expressed in terms of the Bohr radius: r n = a B n 2 a B = h2 m e ke 2 " 0.0528nm Physics 9 28

The mathematics So, now we know the quantized radii. Now we can get the velocities! v n = ke2 mr n Then v n = ke2 h This is often expressed as a fraction of c. 1 n v n c = ke2 hc 1 n " # n " # 1 137.03599 Physics 9 29

The mathematics So, we now have the velocities, and the radii. What about the energy? The energy is made up of two pieces, kinetic and potential. E = 1 2 mv 2 " ke2 r Both the velocity and radius are quantized. Plug in these values! Physics 9 30

The mathematics Plugging in the values: The first term is a constant! E n = " E 0 n 2 E 0 = m ek 2 e 4 "13.6eV 2h 2 1eV =1.602 #10 $19 J E n = 1 2 mv 2 n " ke2 r n = " ke2 2r n = " m ek 2 e 4 1 2h 2 The energy is quantized! n 2 Physics 9 31

The mathematics So, what s s the picture? r n = a B n 2 v n = "c n E n = " E 0 n 2 Physics 9 32

Bohr tackles the H spectrum Electrons jump from one orbit to another. The energy changes via photons. Physics 9 33

Bohr tackles the H spectrum The energy changes by a discrete amount. "E n1 #n 2 = $ m ek 2 e 4 % 1 2h 2 2 n $ 1 ( ' 2 * & 1 n 2 ) The energy is carried away by a photon. "E n1 #n 2 = $ m k 2 e e 4 % 1 2h 2 2 n $ 1 ( ' 2 * = E + = hf & 1 n 2 ) Physics 9 34

Bohr tackles the H spectrum The wavelength of a photon " = c f The energy of a photon Plug this back in! E " = hf # hc $ Physics 9 35

Bohr tackles the H spectrum Plugging it back in 1 " = # $m ek 2 e 4 % 1 ' h 3 2 n # 1 2 & 1 n 2 Recall Rydberg: 1 " = R $ 1 H 2 n # 1 ' & 2 ) R =1.097 H "107 m #1 % 1 n 2 ( But "me k 2 e 4 h 3 =1.097 #10 7 m $1 ( * ) Physics 9 36

Bohr tackles the H spectrum So, we understand the hydrogen spectrum! 1 " = # $m ek 2 e 4 % 1 ' h 3 2 n # 1 2 1 n 2 & ( * ) Physics 9 37

It s pretty - This can be extended to any nucleus with a single electron (i.e., singly ionized He, doubly ionized Li, etc.). The formulae change slightly. For a nucleus with charge q = Ze, just change ke 2 to kze 2! h 2 r n = m e kze n 2 2 v n c = kze2 hc 1 n " #Z n E n = " m ek 2 Z 2 e 4 2h 2 1 n 2 Physics 9 38

It s pretty - but it s wrong! Bohr s s model fails miserably when applied to atoms with more than one electron! People tried to fix it up, for example by letting the electrons have elliptical orbits, etc., but nothing worked. It doesn t t matter how pretty the theory is - if it disagrees with experiment, it has to be thrown away! The basic model is wrong - it has too much classical physics in it! Physics 9 39

We need quantum mechanics! The atom just doesn t t behave in a classical way! The electrons don t t have well- defined orbits. It would be another 13 years before the correct answer was found. Physics 9 40

We need quantum mechanics! The electron orbits for the s, p, d orbitals. The electron is really described by the Schrodinger equation (non-( relativistically). " h2 2m # 2 $ + V$ = ih %$ %t Physics 9 41