AP Phyic Quantum Wrap Up Not too many equation in thi unit. Jut a few. Here they be: E hf pc Kmax hf Thi i the equation for the energy of a photon. The hf part ha to do with Planck contant and frequency. The other part, the pc bit, how that the energy i a function of the photon momentum (p) and the peed of light (c). Thi i the equation for the maximum kinetic energy of an electron that ha been expelled from a metal by a photon. hf i the energy of the photon and the part i the work function for the metal. Note that the electron can have le energy than thi, it jut repreent a maximum. h p Thi i an equation for the wavelength of a photon a a function of it momentum and Planck contannt. E mc Thi i a ridiculouly complicated way to write Eintein famou E mc equation. What thi one i aying i that energy and ma are equivalent to each other. Thi mean that when a ytem undergoe change in energy, there will be a change in ma a well the ma being converted into energy, ee? v f Thi equation i not part of the modern phyic equation et, but i needed to olve the problem. Uing it, one can relate frequency and wavelength. For many of the problem you will be taked to olve, you will be given wavelength and not frequency of the photon. Uing thi equation, you can properly work thing out. Modern Phyic A. Atomic Phyic and Quantum Effect 1. You hould know the propertie of photon and undertand the photoelectric effect o you can: a. Relate the energy of a photon in joule or electron-volt to it wavelength or frequency. 6
Jut ue the E hf pc equation. Actually you want the E hf part of it. To get the wavelength thing in there, ue the v f o c f equation. Solve for f hc and plug that into the firt equation. You get E. b. Relate the linear momentum of a photon to it energy or wavelength, and apply linear momentum conervation to imple procee involving the emiion, aborption, or reflection of photon. Ue the E hf pc equation. Mainly though, you ue the E pc part of it. c. Calculate the number of photon per econd emitted by a monochromatic ource of pecific wavelength and power. Power i work divided by time, which mean that it i alo energy divided by time. Ue the E hf equation to calculate the energy for one photon. You know the total amount of energy in one econd (it jut the power) o you can divide it by the energy of a photon to get the number of photon per econd. d. Decribe a typical photoelectric effect experiment, and explain what experimental obervation provide evidence for the photon nature of light. Thi would require you to decribe a typical photoelectric tube. You would want to talk about the emitter and the collector. Alo talk about the topping potential that i applied to the collector and how it affect the photoelectron. The key thing for the photon nature of light (by thi they mean the particle nature of light) i that the photon are having colliion with electron and knocking them out of the metal, giving them (the electron) kinetic energy. The photoelectric equation allow you to calculate the maximum kinetic energy one of thee photoelectron can have. e. Decribe qualitatively how the number of photoelectron and their maximum kinetic energy depend on the wavelength and intenity of the light triking the urface, and account for thi dependence in term of a photon model of light. The wavelength of the light determine the amount of energy each photon ha. If the energy i too mall (the wavelength i not hort enough or the frequency i too mall) then no photoelectron are emitted. The photon don t have enough energy to break the electron looe. The work function can be thought of a the energy that bind the electron to the metal. If the photon ha more energy than the work function, electron will be knocked out of the metal. If the photon energy i le than the work function no photoelectron are produced. The energy of the photon i a function of it wavelength. The horter the wavelength the greater the energy. It i alo a function of it frequency. The bigger the frequency the bigger the energy of the photon. 63
The intenity of the light i a meaure of the number of photon per econd that are incident on the metal urface. If the wavelength i too large to produce photoelectron, the intenity of the light will have no effect. No photoelectron will be produced. If the wavelength i uitable to produce photoelectron, the intenity will determine the number that are releaed. The intenity will have no effect on the energy of the photoelectron however, jut their number. f. When given the maximum kinetic energy of photoelectron ejected by photon of one energy or wavelength, determine the maximum kinetic energy of photoelectron for a different photon energy or wavelength. Ue the Kmax hf the photon (or wavelength) o you can calculate the work function,. Then uing the work function and the new photon energy you can work out the maximum kinetic energy of the photoelectron for the new wavelength. equation. You are given the maximum kinetic energy and the energy of g. Sketch or identify a graph of topping potential veru frequency for a photoelectriceffect experiment, determine from uch a graph the threhold frequency and work function, and calculate an approximate value of h/e. Here i a typical graph. The f C term i the threhold frequency. It i V imply where the curve hit the old x axi. Photon that have a higher frequency than the cutoff frequency will produce photoelectron that have enough energy to reach the collector, overcoming the electric field etablih on the thing. Photon at the threhold frequency will f c have a maximum kinetic energy that i equal to the potential energy of the field. f To find the work function, okay. Let ee. At the threhold frequency the maximum kinetic energy of the photoelectron i equal to the potential energy in the field o: K hf and U qv o qv hf at the threhold frequency, the hf and hf. So the work function i max topping potential i zero, o 0 imply the threhold frequency time Plank contant. The energy of a photon i E hf and the potential energy of the field i U qv. Set them equal and you get hf qv q i the charge of an electron, o we can plug in e for C the electron charge and write the equation a: hf ev olving for h like they want u e to do we get: h V Thu, h/e i imply the lope of the graph. e f. You hould undertand the concept of energy level for atom o you can: a. Calculate the energy or wavelength of the photon emitted or aborbed in a tranition between pecified level, or the energy or wavelength required to ionize an atom. Okay. Thi i the one where you have the energy diagram. They look like thi: 64
0 ev Ionized Atom (continuou energy level) 0 ev -1.0 ev Ionized Atom (continuou energy level) -3.0 ev -5.0 ev Initial State -5.0 ev Initial State The one on the left you would ue to determine the energy needed to ionize an atom. The one the right give you the pecified energy level that are available for the electron to make their quantum leap thing to. (The Phyic Teacher recognize that that wa a really wretched entence but, he doen t care.) Electron aborb photon that have energy that i equal to the energy difference between the energy level. The wavelength or the frequency of the photon can be calculated becaue you know the energy that i needed to make the quantum leap. We did everal of thee problem and the whole thing wa wonderfully explained in the handout. What a lucky AP tudent you are to have uch a plendiferou reource! b. Explain qualitatively the origin of emiion or aborption pectra of gae. The qualitatively thing i nice. It mean that you don t have to calculate anything, jut explain it On econd thought, maybe that not o good. It i alway eaier to calculate omething than to explain it. Ga atom that are excited will give off an emiion pectrum. Thee are photon of pecific wavelength that repreent the energy level within the atom. The excited electron jump up to higher energy level, are not table, and fall back down to lower energy level. The energy difference between the energy level i given off a a photon by the electron. SO the only color given off correpond to the energy level difference in the atom. The pectrum given off appear a different colored line. You will recall having viewed jut uch a thing. The aborption pectra ha to do with white light (all color) hining through a ga ample. The photon of light that have wavelength that correpond to the energy level in the atom of the ga are aborbed. The pectrum i continuou goe from red to violet, but the aborbed wavelength how up a dark line where there i no light. c. Given the wavelength or energie of photon emitted or aborbed in a two-tep tranition between level, calculate the wavelength or energy for a ingle-tep tranition between two level. Thi i pretty imple - jut calculate the energy of the photon from it wavelength and then convert it into ev. You can then tranlate that into the energy level in the atom. We did a bunch of problem like that. Check em out. 3. You hould undertand the concept of DeBroglie wavelength o you can: 65
a. Calculate the wavelength of a particle a a function of it momentum. The DeBroglie wavelength i the flip ide of the coin for matter. The idea i that moving particle uch a electron have wave characteritic frequency, wavelength, &tc. Alo that the particle can undergo wave interaction uch a contructive and detructive interference, producing interference pattern. h Jut ue the equation. It give you the wavelength a a function of momentum. The p equation i for a wave, but it alo work for particle a well. Of coure the momentum of a particle i imply p mv. b. Decribe the Davion-Germer experiment, and explain how it provide evidence for the wave nature of electron. Okay, thi i a claic experiment that took place in 197. Two guy, Davion and Germer, meaured the wavelength of electron that were cattered off a nickel target in a vacuum. The crytal tructure of the nickel acted like a diffraction grating. The cattered electron exhibited interference pattern you know the whole minima and maxima deal at pecific angle. Thi wa confirmation of DeBroglie theory of particle wave behavior. 4. You hould undertand the nature and production of x-ray o you can calculate the hortet wavelength of x-ray that may be produced by electron accelerated through a pecified voltage. x-ray are produced when high velocity electron are incident on a dene metal uch a tungten. If we aume that the energy of the electron i converted into the energy of the x-ray photon produced, then we get: hc U qv E hf vf c o qv o hc qv Where q i the charge of the electron and V i the potential difference that the electron wa accelerated through. 5. You hould undertand Compton cattering o you can: a. Decribe Compton experiment, and tate what reult were oberved and by what ort of analyi thee reult may be explained. The Compton thing i that a beam of x-ray are flahed onto a graphite crytal. The x-ray beam i made up of a tream of photon. The photon, acting like particle, collide with electron in the carbon atom in the graphite. The photon give ome of their energy to the electron. Now, having le energy than before, the photon end up with a longer wavelength. For a photon le energy mean longer wavelength. Thi wa good evidence for the particle nature of electromagnetic wave. 66
b. Account qualitatively for the increae of photon wavelength that i oberved, and explain the ignificance of the Compton wavelength. Oop. The Phyic Teacher already did thi in the deal up above. Oh well. The important thing i that increae in photon wavelength i due to the lo of photon energy in the colliion between the photon and electron. B. Nuclear Phyic 1. You hould undertand the ignificance of the ma number and charge of nuclei o you can: a. Interpret ymbol for nuclei that indicate thee quantitie. It jut like the thing over here to the right. Ma number, atomic number, and the element chemical ymbol. b. Ue conervation of ma number and charge to complete nuclear reaction. Okay, thi i imple. The baic idea i that the atomic number and ma number are conerved in a nuclear reaction. Thi mean that the total atomic number on one ide of the equation mut equal the total atomic number on the other ide. Same deal for the ma number. c. Determine the ma number and charge of a nucleu after it ha undergone pecified decay procee. Three kind of decay to worry about alpha decay, beta decay, and gamma decay. In alpha decay, the nucleu loe an alpha particle, which i a He-4 nucleu. In Beta decay the nucleu loe an electron. Gamma decay i uually a byproduct of either alpha or beta decay. In alpha decay, the ma number of the parent nuclei goe down by four and atomic number goe down by two. In beta decay the ma number tay the ame but the atomic number increae by one. The gamma photon ha no effect on either ma number or atomic number. c. Decribe the proce of,, decay and write a reaction to decribe each. Darn, the Phyic Teacher ha already done that! (See above). But here are a couple of example equation. 38 34 4 9 90 U Th He Alpha decay ma number atomic number 34 34 0 90Th 91Pa 1e Beta decay e. Explain why the exitence of the neutrino had to be potulated in order to reconcile experimental data from decay with fundamental conervation law. C ymbol 67
A particle accelerator wa ued to bombard an atomic nucleu with a high-energy particle. Energy and momentum would have to be conerved, but when the propertie of the particle were examined after the colliion, the energy and momentum did not add up. There had to be another particle that had the miing energy and momentum. A new particle had to exit and wa potulated. Later it wa given the name neutrino by Enrico Fermi. It wan t until the mid 1950 that the neutrino wa actually detected. Here an example of a reaction that produce a neutrino (thi would be beta decay, right?): 14 14 0 6 7 1 C N e v The ymbol for the neutrino i v.. You hould know the nature of the nuclear force o you can compare it trength and range with thoe of the electromagnetic force. The nuclear force bind the nucleon together within the nucleu. It i many order of magnitude tronger than the electromagnetic force. It effective range, however, i very mall eentially the particle have to be really cloe together, like almot touching, before the trong nuclear force i greater than the electromagnetic force. So it trength i much greater than the electromagnetic force but it i a much horter range deal. 3. You hould undertand nuclear fiion o you can: a. Decribe a typical neutron-induced fiion and explain why a chain reaction i poible. By uing the phrae decribe, the College Board folk are telling you that you don t need to write out an actual fiion equation, but imply to tate what happen in one. Thi i pretty imple. Okay here what happen: a heavy nucleu, like your baic U-35, aborb a neutron. Thi make it untable, and it plit apart. Thi i the fiion part of the deal. It become two new nuclei. Stuff like barium and krypton. Now why i a chain reaction poible? The Phyic Teacher leave you with thi one to reearch on your own. b. Relate the energy releaed in fiion to the decreae in ret ma. E mc equation. The energy releaed (the E deal) i equal to the change in the ret ma (the m deal) time the peed of light quared. You jut ue the equation. Jut ue the From 001: 68
Conider the following nuclear fuion reaction that ue deuterium a fuel. 4 1 1 1 1 0 3 H He + H + n (a) Determine the ma defect of a ingle reaction, given the following information. 1 H.0141 u 4 He 1 4.006 u 1 H 1.0078 u 1 0 n 1.0087 u Okay, thi i imple, jut add up all the mae and then compare it to the ma of a helium nuclei plu a proton and a neutron. We will ubtract the individual part from the ma of the three deuterium nuclei. 3.0141u 4.006 u1.0078 u1.0087 u 0.03 u (b) Determine the energy in joule releaed during a ingle fuion reaction. 7 1.66 x10 kg 7 9 0.03 u 0.0385 x10 kg 3.85 x10 kg 1 u 9 8 m 13 1 E mc 3.85 x10 kg3.0 x10 34.65 x10 J 3.46 x10 J (c) The United State require about 10 0 J per year to meet it energy need. How many deuterium atom would be neceary to provide thi magnitude of energy? 0 1 nuclei 3 31 10 J 0.89 x 10 nuclei.89 x 10 nuclei 1 3.46 x10 J (d) Aume that 0.015% of the hydrogen atom in eawater (H O) are deuterium. The atomic ma number of oxygen i 16. About how many kilogram of eawater would be needed per year to provide the hydrogen fuel for fuion reactor to meet the energy need of the United State? 31 1molecule 1mol 18 g 1kg 5.89 x10 nuclei 4.3 x10 kg 3 3 nuclie 6.0 x10 molecule 1mol 10 g 5 1 9 4.3 x10 kg 3 x10 kg 0.00015 From 1997: 69
A monatomic ga i illuminated with viible light of wavelength 400 nm. The ga i oberved to aborb ome of the light and ubequently to emit viible light at both 400 nm and 600 nm. 0 ev Ionized Atom (continuou energy level) 0 ev Ionized Atom (continuou energy level) 400 nm 600 nm -5.0 ev Initial State -5.0 ev Initial State a. In the box, (above) complete an energy level diagram that would be conitent with thee obervation. Indicate and label the oberved aborption and emiion. b. If the initial tate of the atom ha energy - 5.0 ev, what i the energy of the tate to which the atom were excited by the 400 nm light? c hc E hf f E hc 1.4 x10 ev nm E 400 nm 3 3.1eV E 5.0 ev 3.1eV 1.9 ev e c. At which other wavelength() outide the viible range do thee atom emit radiation after you are excited by the 400 nm light? 3 hc 1.4 x10 ev nm E.1eV 600 nm E 3.1eV.1eV 1.0 ev e hc hc 1.4 x10 ev nm E E 1.0 ev 3 140 nm From 1996: 630
An untable nucleu that i initially at ret decay into a nucleu of fermium-5 containing 100 proton and 15 neutron and an alpha particle that ha a kinetic energy of 8.4 MeV. The atomic mae of helium-4 and fermium-5 are 4.0060 u and 5.0849 u, repectively. a. What i the atomic number of the original untable nucleu? Z 10 b. What i the velocity of the alpha particle? (Neglect relativitic effect for thi calculation.) 19 6 1.6 x10 J 1 K 8.4 x10 ev 1.35 x10 J 1 ev 7 1.67 x10 kg 7 m4.0 u 6.68 x10 kg 1 u 1 kg m 1.35 x10 1 K K mv v 7 m 6.68 x10 kg 15 14 7 m m m v 0.404 x10 4.04 x10.01 x10 c. Where doe the kinetic energy of the alpha particle come from? Explain briefly.where doe the kinetic energy of the alpha particle come from? Explain briefly. Ma Equivalence: The original nucleu decay into the product particle and energy. The energy how up primarily a the kinetic energy of the daughter nucleu and the particle that are emitted. Energy Conervation i alo involved the potential or binding energy of the nucleu wa converted into kinetic energy of the product of the reaction. d. Suppoe that the fermium-5 nucleu could undergo a decay in which a - particle wa produced. How would thi affect the atomic number of the nucleu? Explain briefly. Atomic number increae by one. A neutron convert into a proton and an electron. 5 5 0 100 101 1 Fm X e Note you would not have acce to a periodic table, o you probably wouldn t know what element had an atomic number of 101. From 1995: 631
A free electron with negligible kinetic energy i captured by a tationary proton to form an excited tate of the hydrogen atom. During thi proce a photon of energy E a i emitted, followed hortly by another photon of energy 10. electron volt. No further photon are emitted. The ionization energy of hydrogen i 13.6 electron volt. a. Determine the wavelength of the 10. ev photon. c hc E hf c f f E 15 8 m 4.14 x10 ev 3 x10 hc E 10.eV b. Determine the following for the firt photon emitted. i. The energy E c of the photon ii. E 13.6 ev 10. ev 3.4 ev c The frequency that correpond to thi energy 1. x10 E 3.4 ev 14 E hf f 8. x10 Hz 15 h 4.14 x10 ev c. The following diagram how ome of the energy level of the hydrogen atom, including thoe that are involved in the procee decribed above. Draw arrow on the diagram howing only the tranition involved in thee procee. d. The atom i in it ground tate when a 15 ev photon interact with it. All the photon' energy i tranferred to the electron, freeing it from the atom. Determine the following. i. The kinetic energy of the ejected electron. E E E 15 ev 13.6 ev Photon electron Energy (ev0-1.5-3.4-5 7 m Ionized atom (Continuou Energy Level) E 1.4 ev -10-13.6 Ground State ii. The de Broglie wavelength of the electron. 63
-19 1.60 10 1 1.4 ev x K K mv v 31 m 9.11 x10 kg 1eV 1 6 5 kg m m m m v 0.4618 x 10 0.6796 x 10 6.796 x 10 34 kg m 6.63 x10 h h pmv o p mv 31 5 m 9.11 x10 kg 6.796 x 10 8 9 0.107 x10 m 1.07 x10 m From 1994: A erie of meaurement were taken of the maximum kinetic energy of photoelectron emitted from a metallic urface when light of variou frequencie i incident on the urface. a. The table below lit the meaurement that were taken. On the axe, plot the kinetic energy veru light frequency for the five data point given. Draw on the graph the line that i your etimate of the bet traight-line fit to the data point. b. From thi experiment, determine a value of Planck' contant h in unit of electron voltecond. Briefly explain how you did thi. 633
Plot the data point and obtain a curve. Then determine the lope of the curve. From 1985: Kmax hf Equation for a line. h i the lope of the line. K K1 1.9 ev 0 ev 15 h 3.6 x10 ev f f1 14 1 10 4.6 x10 An energy-level diagram for a hypothetical atom i hown to the right. a. Determine the frequency of the lowet energy photon that could ionize the atom, initially in it ground tate. Energy ev b. Aume the atom ha been excited to the 0 tate at -1.0 electron volt. Second Excited State - 1.0 i. Determine the wavelength of the photon for each poible Firt Excited State pontaneou tranition. ii. Which, if any, of thee wavelength are in the viible range? Ground State c. Aume the atom i initially in the ground tate. Show on the following diagram the poible tranition from the ground tate when the atom i irradiated with electromagnetic radiation of wavelength ranging continuouly from.5 x 10-7 meter to 10.0 x 10-7 meter. (a) The lowet energy photon that could ionize the atom would be one that kick the electron to the 0 ev energy tate. Thi would be an energy of 5.0 ev. E 5.0 ev 15 E hf f 1.1 x10 Hz 15 h 4.14 x10 ev - 3.0-5.0 634
(b) i. Here i the diagram howing all poible tranition for the electron that ha reached the econd excited tate: Energy ev 0 Second Excited State - 1.0 Firt Excited State - 3.0 Ground State - 5.0 The poible energy tranition are: 1.0 ev 5.0 ev 4.0 ev 1.0 ev 3.0 ev.0 ev 3.0 ev 5.0 ev.0 ev Now we can find the wavelength of the photon that have thi energy: E hf and v f v v hv f plug in for f E h hv hv hc E E E v c, right? Firt tranition: hc 5 1 1eV 6 1.99 x10 J m 0.31 x10 m 19 E 4.0 ev 1.60 x10 J 9 6 10 nm 3 0.31 x10 m 0.31 x10 nm 1 m 310 nm Second (and third) tranition: hc 5 1 1eV 6 1.99 x10 J m 0.6 x10 m 19 E.0 ev 1.60 x10 J 9 6 10 nm 3 0.6 x10 m 0.6 x10 nm 1 m 6 nm 635
ii. Which, if any, of thee wavelength are in the viible range? The econd and third tranition; from 1.0 ev to 3.0 ev and from 3.0 ev to 5.0 ev. c. Aume the atom i initially in the ground tate. Show on the following diagram the poible tranition from the ground tate when the atom i irradiated with electromagnetic radiation of wavelength ranging continuouly from.5 x 10-7 meter to 10.0 x 10-7 meter. The light incident on the atom would have a wavelength of 50 nm to 1000 nm. Wavelength that would caue a tranition would be at 6 nm and at 310 nm. The light needed to ionize the atom ha a wavelength of 49 nm, the incident light i longer than that. So only the three tranition are poible. Second Excited State Energy ev 0-1.0 From 199: Firt Excited State Ground State 6 nm - 3.0 310 nm 6 nm - 5.0 The ground-tate energy of a hypothetical atom i at - 10.0 ev. When thee atom, in the ground tate, are illuminated with light, only the wavelength of 07 nanometer and 146 nanometer are aborbed by the atom. (1 nanometer = 10-9 meter). a. Calculate the energie of the photon of light of the two aborption-pectrum wavelength. 636
hc 1.4 x10 ev nm E 07nm 3 5.99 ev 3 hc 1.4 x10 ev nm E 8.49 ev 146 nm b. Complete the energy-level diagram hown above for thee atom by howing all the excited energy tate. c. Show by arrow on the energy-level diagram all of the poible tranition that would produce emiion pectrum line. d. What would be the wavelength of the emiion line correponding to the tranition from the econd excited tate to the firt excited tate? E 8.49 ev 5.99 ev.5 ev E hc hc 1.4 x10 ev nm E.5 ev 3 496 nm e. Would the emiion line in (d) be viible? Briefly jutify your anwer. Ye. The viible pectrum extend from 400 to 700 nm. 637