CH301 Fall 2009 Worksheet 2 Answer Key 1. Calculate wavelength from the following frequencies.a) 625 kHzλ = c/νλ = (3.00 E8 m/s)/(625 E3 Hz) = 480. mb) 734 MHzλ = (3.00 E8 m/s)/(734 E6 Hz) = .409 mc) 8.4 E14 Hzλ = (3.00 E8 m/s)/(8.4 E14 Hz) = 3.6 E-7 md) 92 GHzλ = (3.00 E8 m/s)/(92 E9 Hz) = 3.3 E-32. Given the following energies, calculate the frequencies of the photons.a) 17 kJν = E/hν = 17000 J/6.63 E-34 J·s = 2.6 E37 s-1b) 564 E-25 Jν = 564 E-25 J/6.63 E-34 J·s = 8.51 E10 Hzc) 98 pJν = 98 E-12 J/6.63 E-34 J·s = 1.5 E23 Hzd) 230 Jν = 230 J/6.63 E-34 J·s = 3.5 E35 s-13. Rank the wavelengths of the photons in question 2 from longest to shortest. Use A, B, C,or D to identify the photon. (Hint: No calculations necessary.)B C D ARemember that wavelength and energy are inversely related (E = h·c/λ) – the higher theenergy, the shorter the wavelength.4. The double slit experiment is a famous demonstration of the wave nature of light: whenlight of a single frequency is passed through two parallel sits in a barrier, it produces aninterference pattern on a surface placed on the other side. Draw an example of aninterference pattern (as amplitude of light received vs. position). Compare the frequencyand phase of the light waves coming from the two slits at the dark spots and at the lightspots on the interference pattern.An interference pattern looks something like this:At the dark spots, the two waves still have the same frequency as the original light source,but have opposite phase causing them to cancel (destructive interference). At the lightspots, the frequency is also the same but the phase is identical, causing constructiveinterference.5. When passed passed through very fine gratings (or even regular crystals), electrons andneutrons form a distinctive pattern of ridges and troughs because of their wave nature.What is this effect called?The scattering of waves by lattices (the fancy name for a grating) is called diffraction. Funfact: X-rays were first identified as light waves in the 19th century by the fact that theywere diffracted by crystals. Knowing what we do now about the wave nature of matter,would we necessarily draw the same conclusion from that evidence?6. The description of light as a collection of particles goes back to long before quantummechanics. Who are two examples of people who proposed such a description before 1800?The Greek philosopher Democritus proposed that light was composed of particles in the 5thcentury B.C (!), and Sir Isaac Newton gave a "Corpuscular Theory of Light" at the end of the17th century.7. Classical physics assumed that the intensity of radiation emitted was a function of λ andthat at a given temperature as λ increases the intensity of radiation decreased. Therefore,since UV radiation has a short wavelength its intensity distribution should be large. Describehow Plank solved the ultraviolet catastrophe.Plank proposed a radical idea, that the exchange of energy between matter and radiationoccurs in quanta. A radiation with a certain frequency can be generated only if an oscillator(electron, atom, or molecule) of that frequency has acquired the minimum energy, E = hvto start he oscillation. At low temperatures there is not enough energy to cause highfrequency oscillations, hence the intensity of the curve is zero. A peak of intensity isreached at a certain wavelength and the intensity decreases again because there's notenough energy to stimulate higher frequency oscillators. No high energy radiation is emittedand the UV catastrophe is no more!8. Before Einstein won the Nobel Prize for discovering the photoelectric effect, classicalphysics suggested that there is no minima of energy needed to eject electrons from a metal.Explain the principles of the photoelectric effect and how Einstein concluded that light isparticle-like.Classical physics failed because there's no relation to energy of incidient light and that ofthe ejected electrons. Einstein explained that electromagnetic radiation consists of particles(later called photons) and that each photon is a quanta of energy related to the wavelengthof its radiation (E = hv or E = hc/λ). If the energy of the incident photon is less than thework function of the metal no electron will be ejected no matter how long you sit there andshine radiation on it. If the energy of the incident light is higher than the work function thanthe electron is ejected with a certain kinetic energy. These observations are expressed as:KE = ½ mv2= hv – Φ. Einstein's work backed Plank's theory that energy exchange occursin quanta.9. What is the deBroglie wave equation? In your own words, what is its significance?The deBroglie wave equation is λ = h/(m*v) = h/p. Matter, like EMR, has a wave particleduality, such that wavelength and mass have an inverse relationship.10. Use the deBroglie equation to calculate the following values.a) the wavelength of a 70.0 kg person traveling at 6.0 mi/hr.λ = h/(m*v)v = 6.0 mi/hr * 1600 m/mi * 1 hr/3600 s = 2.67 m/sλ = 6.63 E-34 J·s /(70 kg * 2.67 m/s) = 3.6 E-36 mb) the velocity of an e-(mass 9.1 E-31 kg) with a wavelength of 450 pm.v= h/(m*λ)v = 6.63 E-34 J·s /(9.1 E-31 kg * 450 E-12 m) = 1.6 E6 m/sc) the mass of a particle traveling at the speed of light with a wavelength of 12 nm.m= h/(v*λ)m = 6.63 E-34 J·s / (3 E8 m/s * 12 E-9 m) = 1.8 E-34 kg11. What is the energy difference between the n = 1 and n = 2 energy levels for an electronin a 1 Å box? What is wavelength of a photon with this energy?E2-E1= (22-12)·h2/ 8·m·L2= 3·(6.6E-34 J·s)2/ 8·(9.1E-31 kg)·(1E-10 m)2= 1.8E-17 JThis corresponds to a photon of wavelength 11 nm, which is in the X-ray region.12. Could the particle in a box solution be used to approximate atomic energy levels? Whyor why not?Since the energy levels of the particle in a box grow increasingly further apart as nincreases while atomic energy levels grow increasing closer together, the particle in a boxcould not be used as a valid approximation.13. One of the conditions at which quantum mechanics simplifies to classical mechanics iswhen the quantum number, n, approaches infinity. What is the probability density for theparticle, Ψ(x)2, for the particle in a box in this limit? Does this agree with the classicalprediction?As n increases, the peaks and troughs in Ψ(x)2get closer and closer together. At n equalsinfinity, these peaks merge into a flat line, Ψ(x)2= 1/L (so that the total probability in thebox is equal to 1). Since a
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