PA 114 Waves and Quanta Unit 4: Revision PA1140 Waves and Quanta Unit 4: Revision Dr Matt Burleigh (S4)

PA 114 Waves and Quanta Unit 4: Revision PA1140 Waves and Quanta Previous Lecture Slides for Unit 4: Bohr theory Atomic size and shape Mass and binding energy Radioactivity, fission and fusion

PA 114 Waves and Quanta Unit 4: Revision Atomic Spectra Ch. 37 Rydberg-Ritz empirical formula: the wavelengths of lines in a spectrum of H are given by: Where n 1 and n 2 are integers and R is the Rydberg constant

PA 114 Waves and Quanta Unit 4: Revision

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

PA 114 Waves and Quanta Unit 4: Revision Nuclear Physics Radioactivity Ch. 40 Derive number of nuclei N remaining after time t: where is the decay constant and N 0 is the number of nuclei at t=0 Derive decay rate R: where R 0 =  = rate of decay at t=0 Derive half life: Average lifetime:

PA 114 Waves and Quanta Unit 4: Revision 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

PA 114 Waves and Quanta Unit 4: Revision Nuclear reactions  -Decay  -Decay  -Decay Q value, exothermic & endothermic Understand fission & fusion

PA 114 Waves and Quanta Unit 4: Revision

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

PA 114 Waves and Quanta Unit 4: Revision

 = nm E = hc/ =10.2 eV Maximum energy which can be absorbed is equal to the electron energy, 12.9 eV Energy states for Hydrogen are E n =-E 1 /n 2 =-13.6/n 2 eV (from memory or use given formula for transition energies ) So energy states available are n= eV, n=2 -3.4eV, n= eV, n= eV, n= eV Transition energies then are (1->2) 10.2eV, (1->3) 12.1eV, (1->4) 12.75eV, (1->5) eV So 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 Substitute into given Rydberg formula for n i = 3, n f = 2 H   = nm n i = 4, n f = 2 H   = nm  = 1875 nm

PA 114 Waves and Quanta Unit 4: Revision B6.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 R 0 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 22 Na in a reactor is nuclei s –1. Production continues for a period of one year. What is the decay rate of the 22 Na sample a further one year after the completion of the irradiation? [6] The half-life of 22 Na is 2.6 years. There are 3.15  10 7 seconds in a year.

PA 114 Waves and Quanta Unit 4: Revision 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 t 1/2 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

PA 114 Waves and Quanta Unit 4: Revision

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

PA 114 Waves and Quanta Unit 4: Revision 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

PA 114 Waves and Quanta Unit 4: Revision