pa 114 waves and quanta unit 4: revision pa1140 waves and quanta unit 4: revision dr matt burleigh...

PA 114 Waves and Quanta Unit 4: Revision

PA1140

Waves and Quanta

Unit 4: Revision

Dr Matt Burleigh (S4)

http://www.star.le.ac.uk/~mbu/lectures.html


PA1140 Waves and Quanta

Previous Lecture Slides for Unit 4:

http://www.star.le.ac.uk/~mbu/lectures.html

•Bohr theory•Atomic size and shape•Mass and binding energy•Radioactivity, fission and fusion


21

22

11=

1

nnR

Atomic Spectra

Ch. 37

Rydberg-Ritz empirical formula: the wavelengths of lines in a spectrum of H are given by:

Where n1 and n2 are integers and R is the Rydberg constant


Where the Rydberg constant, R, is:

Know Bohr’s postulates

Derive frequency/wavelength of lines:

Bohr Model of Atom

Derive energy of Bohr orbits:

Understand energy level diagrams


Nuclear Physics

Radioactivity

Ch. 40

Derive number of nuclei N remaining after time t:

where is the decay constantand N0 is the number of nuclei at t=0

Derive decay rate R:

where R0= = rate of decay at t=0

Derive half life:

Average lifetime:


Nuclear size and Shape

Ch. 40

• Volume is proportional to A, so density constant

• Nucleus looks like a liquid drop• For light nuclei N~Z• For heavier nuclei the number of

neutrons increases• The extra uncharged neutrons act to

stabilize heavy nuclei from repulsive electrostatic forces

• Atomic number (Z) and mass number (A)

• Radius of nucleus:

• Mass and binding energy


Nuclear reactions

-Decay -Decay -Decay

Q value, exothermic & endothermicUnderstand fission & fusion


Z, the number of protons, the atomic number of the atom.A, the mass number of the nucleus, the total number of nucleons, A=N+Z, where N, is the number of neutrons.

239 23993 94Np P eu

15 158 7 eO N

59 4 6127 2 29Co He 2nCu

63 6429 3

201 Cu n nH Z

3 21 1

21p H H H

235 93 14192 37 55U Rb Cn s 2n

238 23992 93n U Np e

11

1

1

11

1

1

1

1


= 121.6 nm E = hc/=10.2 eV

Maximum energy which can be absorbed is equal to the electron energy, 12.9 eVEnergy states for Hydrogen are En=-E1/n

2=-13.6/n2 eV

(from memory or use given formula for transition energies )

So energy states available are

n=1 -13.6eV, n=2 -3.4eV, n=3 -1.51eV, n=4 -0.85eV, n=5 -0.544eV

Transition energies then are (1->2) 10.2eV, (1->3) 12.1eV, (1->4) 12.75eV, (1->5) 13.056eVSo we can reach n=4.

Longest wavelength will then correspond to the smallest transition from this state, n=4 -> n=3, Back to Rydberg formula

-2 12 2

1 1 1=1.096776 x 10 nm

1 2

-2 12 2

1 1 1=1.096776 x 10 nm

3 4

Substitute into given Rydberg formula for

ni = 3, nf = 2 H = 656.3 nm

ni = 4, nf = 2 H = 486.1 nm

= 1875 nm

PA 114 Waves and Quanta Unit 4: RevisionB6. Describe what is meant by the decay constant of a radioactive nucleus.

[2]Describe what is meant by the half-life of a radioactive source.

[2]Write down an equation relating the half-life to the decay constant.

[2]

A radioactive nucleus with decay constant is produced in a nuclear reactor at a rate R0 nuclei per second. Assuming that the number of nuclei

initially present is zero, show that the number of nuclei N after time t is given by the expression:

[8]The rate of production of 22Na in a reactor is 1015 nuclei s–1. Production continues for a period of one year. What is the decay rate of the 22Na samplea further one year after the completion of the irradiation? [6]

The half-life of 22Na is 2.6 years. There are 3.15 107 seconds in a year.


Describe what is meant by the decay constant of a radioactive nucleus:

If radioactive decay is a random process, we expect the number of nuclei that decay after time dt to be proportional to N and t. The constant of proportionality is called the decay constant.

Describe what is meant by the decay constant of a radioactive nucleus

The half life t1/2 is defined as the time it takes the number of nuclei and the decay rate to decrease by half 2

Write down an equation relating the half-life to the decay constant

2

2


h

EEf fi

Bohr’s postulates – Bookwork!(1) Bohr proposed that certain “magical” circular orbits existed, called

“stationary states”, which did not radiate, and that electrons could only exist in these states, with radiation occurring when they made the transition from one to the other.

(2) He also postulated that the frequency of the radiation from spectral lines was determined by energy conservation during transitions from one stationary state to the other. i.e.

(3). Trial and error led Bohr to his third postulate, that angular momentum is quantized, specifically that

From E=hf, where h is Planck’s constant

n is the quantum number of the state

6


So angular momentum quantization IS given by standing wave condition

Substituting in n, n-1 for large n

In the radial direction then there is no uncertainty in r. momentum is this direction is zero, and also has no uncertainty. So the Bohr model clearly violates the uncertainty principle.

2

2

4

2

11

2

pa 114 waves and quanta unit 4: revision pa1140 waves and quanta unit 4: revision dr matt burleigh...

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