Bohr’s model of the atomQuantum Physics Lecture 20A

7/28/2019 Bohrs model of the atomQuantum Physics Lecture 20A

1/21

PHYS 2040Quantum Physics

A brief history of atomic structure

Rutherfords scattering experiment

The problems with Rutherfords model

Bohrs postulates

Bohrs 1st postulate A closer look

Developing Bohrs model

Optical spectroscopy

Two types of spectroscopy

Spectroscopy as an analytical tool

The spectrum of Hydrogen

The Rydberg-Ritz equation

PHYS2040 Lecture 20A Bohrs model of the atom

Updated: 19/5/2008 8:42 PM

Lecture 20A: Bohrs model of the atomLecture 20A: Bohrs model of the atom


2/21


One of the first models of the structure of the atom (beyond just being the smallestpossible piece of some material) was known as Thompsons plum-pudding model. Itrelied upon a few pieces of knowledge that were emerging from experiments at thetime, namely:

A brief history of atomic structureA brief history of atomic structure

1. Atoms contain particles with negative charge than can come out (i.e., electrons) asfound in the photoelectric effect.

2. Overall, an atom is neutral, so therefore there must also be a compensating amountof positive charge in the atom.

3. The mass of an atom is ~1800 times that of anelectron, therefore an atom must be mostly positivecharge by mass (not by charge though, since thetotal charge must be neutral see 2).

If you take these three observations, the simplestmodel you can generate is the plum-pudding model,where the atom is a ball of mostly positive charge,with small negatively charged particles embeddedinside it.

~1


3/21


Rutherfords scattering experimentRutherfords scattering experiment

In around 1910, a New Zealander Ernest Rutherford and col leagues were looking atthe scattering of alpha particles (helium nuclei with 2 protons and 2 neutrons therefore it has charge 2+) from a very thin leaf of gold foil .

The expectation was that almost all of the -particles should be transmitted,because in the plum-pudding model, a material is very homogeneous and there isno free charge to backscatter the positively charged -particles.

Thompsons model predicted that ~1 in 103500 -particles would be backscattered.

Foil is relatively

homogeneous and has nofree charge, so s shouldgo roughly straight through


4/21


Rutherfords scattering experimentRutherfords scattering experiment

Rutherfords result was quite surprising. He found that significantly more -particleswere reflected, approximately 1 in 104.

It was the most incredible event that ever happened to me in my li fe. It was as

incredible as i f you f ired a 15-inch artil lery shell at a piece of tissue paper, and

it came back and hit you." Ernst Rutherford.

This can only be explained if therelatively massive positive charge andrelatively massless negative chargeare separate in the atom.

Then, the enhanced backscattering isdue to the -particles experiencingCoulomb repulsion off the positivenuclei.

In this model, the nucleus is small,and the rest of the volume of the atomis made up of a cloud of electronsorbiting the nucleus.


5/21


Rutherfords model of the atomRutherfords model of the atom

Based on his experiments, Rutherford arrived at a new model for the atom, based onthe idea that the nucleus and the electrons are separate, distinguishable objects.

The key idea behind Rutherfords model is that an electron in an atom moves in a

circular orbit about the nucleus under the influence of the Coulomb attractionbetween the electron and the nucleus, obeying the laws of classical mechanics.

Rutherfords model is basically this

picture of electrons in planetaryorbits around a nucleus that we oftensee drawn inside/outside science.

Despite the fact that Rutherfordsmodel is an improvement onThompsons plum-pudding model, italso has its problems


6/21


The problems with the Rutherford modelThe problems with the Rutherford model

The problems with the Rutherford model are actually quite simple The negativelycharged electron should be drawn slowly in towards the posit ive nucleus,continuously emitting energy until i t crashes into the nucleus (and eventually endingup as the plum pudding model again).

And in the process it should emit radiation of frequency f 1/r3/2, which shouldincrease continuously as the electron spirals inwards, but experimentally theemission spectrum for hydrogen is discrete (as we will see later).


7/21


Bohrs PostulatesBohrs Postulates

In 1913, Bohr developed a new model. It is st il l based on Rutherfords idea that theelectron fol lows a circular orbit due to Coulomb attraction between the electron andthe nucleus, but Bohr added on three simple postulates to avoid the problems withRutherfords model of the atom. These postulates are:

1. Instead of the infinity of orbits allowed classically, it is only possible for an electronto move in an orbit for which its orbital angular momentum L is an integer multipleofh. In other words, L = nh where n = 1, 2, 3, is an integer. We typically call n aquantum number (we will see more of this in the coming weeks).

2. Despite the fact that it is constantly accelerating, an electron moving in such anorbit does not radiate electromagnetic energy. Thus its total energy E remainsconstant, unless it changes to another orbit .

3. Electromagnetic radiation is emitted if an electron, initially moving in an orbit oftotal energy Ei, discontinuously changes its motion so that it moves in an orbit of

total energy Ef. The frequency of the emitted radiation fis equal to (Ei Ef)/h.

Postulate 1 actually has some hidden meaning in i t, lets take a closer look at itbefore moving on


8/21


Bohrs 1st postulate A closer lookBohrs 1st postulate A closer look

If we consider that L = mvr= nh, then we can rearrange to obtain the radius r= nh/mv

1. Instead of the infinity of orbits allowed classically, it is only possible for an electronto move in an orbit for which its orbital angular momentum L is an integer multipleofh. In other words, L = nh where n = 1, 2, 3, is an integer

The circumference or length l of an orbit is given by l = 2r= 2nh/mv = nh/mv

Remembering back to the de Broglie relation: p = mv = h/, then = h/mv

And if we feed this back into our earlier result l = nh/mv= n, in other words, the number of electronwavelengths around an orbit must be an integer. This,as we saw earlier, was part of the motivation for deBrogl ie in establishing his relations.

n = 3These sorts of logical loops are very useful in science,they often reassure you that you are on the right track.

Also, as a student, they give you more than one way toremember useful facts, good back-up for exams.


9/21


Developing Bohrs ModelDeveloping Bohrs Model

So now lets start with Rutherfords model work our way through Bohrs postulates,and see if we can calculate what the allowed radii of the electron orbi ts will be, usethis to obtain the corresponding electron energies, and then try to calculate theexpected atomic absorption/emission spectra. We can then compare this toexperimental results.

Rutherfords model: An electron in an atom moves in a circular orbit about the

nucleus under the influence of the Coulomb attraction between the electron and thenucleus, obeying the laws of classical mechanics.

This means that the centripetal force on the electron that makes it take a circularpath will be provided by the Coulomb force acting between the negative electron (e)and the positive nucleus (+Ze, where Z is the atomic number), therefore:

Were going to start here with the simplest case a hydrogen atom - with one protonand one electron. It turns out Bohrs model starts to fail as we make much largeratoms, but that doesnt stop it from being a good place to start.

(19.1)2

0

22

4 r

Ze

r

mv

=


10/21



If we now consider postulate 1 again:

and then substitute Eqn. 19.3 back into Eqn. 19.1, we find:

(19.4)2

0

2

3

222

4 r

Ze

mr

n

mr

n

r

m

==

hh

(19.2)hnmvrL ==

(19.3)mr

nv

h=

(19.5)2220

324 rnmrZe h=

(19.6)22

2

0

2

22

04

nmZe

h

mZe

nr

==

h

cross-multiply to get:

and then isolate r:


11/21



We can now calculate the energy E as the kinetic energy K plus potential energy V:

if we now go back to Eqn. 19.1 and multiply both sides by r/2, then:

(19.9)

(19.7)r

ZemvVKE

0

2

2

42

1

=+=

(19.10)

(19.11)

substi tut ing 19.8 into 19.7:

and then substituting in r:

(19.8)r

Zemv

0

22

82 =

rZe

rZe

rZeE

0

2

0

2

0

2

848 ==

22

0

2

0

2

8

nh

mZeZeE

=

222

0

42

8 nh

emZ

=


12/21


Collecting our resultsCollecting our results

We now have expressions for the allowed radii and corresponding energy levels:

where: a0

= 0h2/me2 = 0.53 is called the Bohr radius

(19.6)202

2

2

0 nZ

an

mZe

hrn ==

(19.11)

2

0

2

222

0

42

8 n

EZ

nh

emZEn ==

E0 = e2/80r0 = me4/80h2 = 13.6eV is called the Rydberg energy


13/21


The fine-structure constantThe fine-structure constant

It is also interesting to calculate the electrons speed as it travels around an orbi t. Wecan do this by going back to:

(19.11)

(19.3)mr

nv

h=

And then substitute in appropriate values forn and rn (from Eqn. 19.6).

For hydrogen in the ground state, n = 1, Z = 1 and r= a0, and if you use these values,you get v1 = 2.18 106 ms1, which is 0.73% of the speed of light, justi fying the factthat we havent used relativity for this problem, but indicative of the fact that ifr


14/21


Emission and AbsorptionEmission and Absorption

(19.12)

(19.13)

=

=

2232

0

411

8 mnh

me

h

EEf mn

And using postulate 3, we can now calculate the frequency/wavelength of radiationemitted by an electron going from level n to level m as (we wil l set Z = 1 from hereonwards):

=

2232

0

411

8

1

mnch

me

where: R = me4/802h3c = 1.097373 107 m1 is called Rydbergs constant

(19.14)

= 22

111

mnR

Fantastic! We now have something observable that we can use to test Bohrs model.We can take a gas of hydrogen atoms, and see what wavelengths of photons areemitted/absorbed, and these should hopefully obey Eqn. 19.14. So lets do this


15/21


Optical spectroscopyOptical spectroscopy

A spectrometer basically uses Newtons discovery that a glass prism can be used todisperse light based on its wavelength (you can also use a diffraction grating toachieve the same effect). You can then measure the spectral content of the light, inother words, its intensity as a funct ion of wavelength.


16/21


Two types of spectroscopyTwo types of spectroscopy

The two main types of spectroscopy are emission and absorption spectroscopy. Inemission spectroscopy, a gas of excited atoms emit photons at frequenciescorresponding to transit ions to lower electronic energy level n. In absorptionspectroscopy, a gas absorbs photons of frequencies corresponding to transitions tohigher electronic energy level n.


17/21


Spectroscopy as an analytical toolSpectroscopy as an analytical tool

Different atoms have different electronic energy level structures, and thus differentemission and absorption spectra, and hence this can be used as a tool to identify thecomposition of gases. This is particularly important in astronomy!


18/21


The spectrum of HydrogenThe spectrum of Hydrogen


19/21


The spectrum of HydrogenThe spectrum of Hydrogen


20/21


The Rydberg-Ritz equationThe Rydberg-Ritz equation

The wavelengths of the spectral lines in hydrogen can be fitted by the followingequation:

(19.15)

= 22

111

mnR

which is known as the Rydberg-Ritz equation (and is the same equation as Eqn 19.14).Before Bohrs model no physical explanation could be given for this relationship, itsimply worked.

The measured R was 10967757.6 1.2 m1 and the Bohrs model calculation ofR,corrected for the finite mass of the hydrogen atom, gives R = 10968100 m1.

These values agree to with in 3 parts in 100,000! This gave impressive confirmation ofthe Bohr model, and earned Bohr the Nobel pr ize in Physics in 1922.


21/21


The earliest model for the structure of the atom was provided by J.J. Thompson, andwas called the plum-pudding model. It consists of a ball of positive charge, withnegatively charged electrons embedded in it. It relies on the knowledge that atomscan eject negatively charged electrons, atoms are generally neutral and have a massroughly 1800 times that of an electron.

Rutherfords work scattering alpha-particles (He2+ ions) off thin gold f ilms showed farmore backscattering than can be explained by the plum-pudding model. Hisconclusion was that a cloud of electrons orbit a very small posit ive nucleus, but faceda significant problem in such a model, the electron should spiral back in towardsthe nucleus.

Bohr aimed to improve this model by adding three postulates that the angularmomentum of electron orbits is quantised in integer multiples of h, an electron in afixed orbit doesnt radiate energy, and an electron changing orbits emits/absorbs aphoton with a frequency corresponding to the energy dif ference between levels.

Bohrs model does an excellent job of predicting the emission/absorption spectra of

simple one-electron atoms such as hydrogen (and other group I elements if youcorrect for finite mass), but fails for larger, multi-electron atoms. The relationshipbetween the emission/absorption wavelength and the transitions is given by theRydberg-Ritz equation.

SummarySummary

= Brooks/Cole - Thomson

Documents

Bohr’s model of the atomQuantum Physics Lecture 20A