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PHY–309 L. Solutions for homework set # 10. Textbook question Q.21 ant the end of chapter 18: The gold atoms — just like all the other atoms — are mostly empty space containing a few electrons. But the electrons are so much lighter than the α particles — m(e) : m(α) ≈ 1 : 7300 — that when an α particle collides with an electron, it hardly notices the collision and keeps moving in almost the same direction with almost the same speed. For comparison, think of an M1 Abrams tank colliding with a grocery-store cart. So when Rutherford, Geiger, and Marsden aimed a beam of α particles at a very thin (less than a micrometer) gold foil, they expected all the α particles to go through the foil with very little deflection. And indeed, most of the α particles did precisely that. The surprise was that a small fraction of the α particles were somehow deflected through large angles, and some even bounced back. In Rutherford’s words, “It was almost as incredible as if you fired a 15–inch shell at a piece of tissue paper and it came back and hit you”. For more details, please see Wikipedia page on Geiger–Marsden experiment. Non-textbook problem #I: (a) To find if the particles flying along some tube are electrically charged or neutral, we may use a magnetic field in a direction ⊥ to the particle beam. We do not even need to open up the tube, we may simply put a strong magnet outside the tube. If the particles are neutral, the magnetic field would not exert any force on them, so the particles would keep moving in the same direction as before. However, if the particles are electrically charged, they would fill the Lorentz force ~ F~ = q~v × B (1) in a direction ⊥ to their velocity ~v , so their motion would be deflected sideways. Moreover, the direction of the force (1) depends on the sign of the electric charge q, so by observing the side towards which the particles are deflected we may find if they are positively or negatively charged. For example, in a magnetic field directed vertically up, the positively charged particles would turn right while the negatively charged particles would turn left. 1 (b) The acceleration due to Lorentz force (1) is proportional to the particles’ charge/mass ratio, F~ q ~ = ~v × B. m m ~a = (2) Consequently, the angle through which the particles are turned as they fly through the magnetic field depends on the charge/mass ratio q/m. Specifically, θturn = a×t q = × Bt v m (3) where t is the time it takes a particle to cross the magnetic field; it depends of the magnet’s geometry and on the particles’ speed, but we do not need the exact formula here. Thus, measuring the angle through which the particles change their direction in the magnetic field, we may find their charge-to-mass ratio as θturn q = . m Bt (4) Non-textbook problem II: Note: Modern flat-screen plasma or LCD TVs or monitors do not use tubes. Instead, the brightness of each pixel on the screen is controlled by its own electronic circuit. But from the invention of electronic TV by Zworykin in 1929 until a about 2005, all TV sets and even computer monitors used Cathode Ray Tubes (CRT). At the back end of a CRT tube, an electron gun produces a beam of electrons that fly to the screen (which acts as the anode) and hit it at high speed (see homework set 2, problem II). Along the way, there are magnetic coils that deflect the beam horizontally and vertically. The currents in those coils change in repetitive pattern, which makes the beam to scan the phosphorescent pixels on screen. When the beam hits a pixel, it lights up with a brightness proportional to the beam’s intensity. The beam intensity varies with time according to the signal received by the TV, so different pixels light up with different brightness, and that’s how the picture appears on the screen. 2 In color TVs and monitors, there are three electron guns producing three electrons beams. Just before the screen, there is a mask that allows each beam to strike only the phosphors emitting a particular color — red for one beam, green for another, and blue for the third. This wikipedia page shows a diagram of a color CRT tube. In any CRT tube — a TV, a monitor, an oscilloscope, or an X-ray tube — a beam of electrons hits the anode (a screen, or just a piece of metal) at high speed. When the atoms in the anode are hit by fast electrons, sometimes the inner electrons of the atom change their orbits. When the orbits change back to the lowest-energy state of the atom, the inner electrons emit electromagnetic radiation of very high frequency — the X-rays. Thus, all CRT tubes — including the old TVs and monitors — emit X-rays. But the intensity of the X-rays depend on the speed of the electrons, on the anode material they hit, and on the intensity (i.e., the electric current) of the beam itself. Modern X-ray tubes use very short pulses of very intense beams of electrons accelerated to speeds up to 190,000 km/s (63% of light speed) by very high voltages (up to 150 kilovolt). When such electrons hit tungsten atoms in the anode, they produce a lot of X-rays. In a TV or monitor, the electrons move slower, hit lighter atoms, and the beam intensity is much lower, so they produce much fewer X-rays. But they do produce some X-rays, and that could be hazardous to the health of a person — especially a child — who sits too close to a TV or a monitor for too many hours. This was a particular concern with the earliest models of color TV made in your grandparent’s time. The later models were engineered to emit fewer X-rays, so the health damage came more from the programming than from the sets themselves. Non-textbook problem #III: (a) Let’s start with the isotope naming conventions. A complete name of an isotope has form A Eℓ, Z for example 16 1 H, 1 O, 35 Cl, 17 63 Cu, . . . 29 (5) Here ‘El’ is the chemical symbol for the element — ‘H’ for hydrogen, ‘O’ for oxygen, ‘Cl’ for chlorine, ‘Cu’ for copper, etc. In addition, to the left of the chemical symbol there are two 3 numbers describing the nucleus of the isotope. The upper left number A is the net number of nucleons — i.e., protons and neutrons, — while the lower left number Z is the number of protons. The number of neutrons in the nucleus is not written, but it can be easily obtained as the difference N = A − Z. 63 Cu 29 In particular, is a copper isotope whose nuclei have 63 protons and neutrons, including 29 protons; the remaining 63 − 29 = 34 nucleons are neutrons. Likewise, 65 Cu 29 is another copper isotope whose nuclei have 65 neutrons and protons, including 29 protons and 65 − 29 = 36 neutrons. The number of protons is the nucleus controls its electric charge Qnucleus = N × 0 + Z × (+e) = +Ze (6) and hence the number of electrons E in a neutral atom: Qatom = Qnucleus + E × (−e) = 0 =⇒ E = Z. (7) This is the only number that matters for chemistry, so all isotopes with the same Z belong to the same chemical element. Conversely, all isotopes of the same chemical element have the same proton number Z. In fact, the serial number of an element in the Periodic Table — usually called the atomic number — is precisely the proton number Z. Thus, all isotopes of hydrogen have one proton in the nucleus and one electron in the neutral atom, all isotopes of oxygen have 8 protons in the nucleus and 8 electrons in the neutral atom, etc. In particular, all isotopes of copper have 29 protons in the nucleus and 29 electrons in the neutral atom. Since the proton number Z is implicit in the chemical element symbol, it is usually omitted from the isotope’s name. Thus, instead of 63 Cu, 29 1 people write simply any copper isotope has proton number Z = 29. Likewise, H means isotope has Z = 1, 16 O means 16 H 8 1 H 1 63 Cu since since any hydrogen since any oxygen isotope has Z = 8, etc., etc. Please remember this rule for the textbook problem SP2. If an isotope’s name is written down without a left subscript and you need to find its proton number, look up the chemical 4 element in the Periodic Table: The atomic number of the element — it’s sequential number in the Table — is the proton number of any isotope of that element. (b) Let’s start with atomic weights of pure isotopes. All atoms of a pure isotope have exactly the same mass Matom = Z × (Mproton + Melectron ) + (N = A − Z) × Mneutron − binding energy . (8) c2 To a very good approximation — three or four significant figures — this mass is proportional to the net number of nucleons A = Z + N, and the atomic mass unit is chosen such that Matom ≈ A amu. (9) In other words, the atomic weight of a pure isotope is very close to the net number A of protons and neutrons in the nucleus. In particular, the copper isotope isotope 65 Cu 29 63 Cu 29 has atomic weight 62.929, 597 ≈ 63.0, while the has atomic weight 64.927, 789 ≈ 65.0. (The high-precision atomic weights are taken from the NIST web page.) But the natural sources (ores, etc.) for many chemical elements provide mixtures of different isotopes, and since the isotopes of the same element have similar chemical properties, they don’t get separated when the element is refined from its ore or participates in chemical reactions. Although there are ways to separate isotopes in a lab, or even on the industrial scale, this is very difficult and expensive, so it’s done only when the element is used for its nuclear properties (for example, uranium). Thus, most chemical elements and compounds you can find on Earth do not contain pure isotopes but rather natural mixtures of several ⋆ isotopes. In particular, all copper on this planet that hasn’t been through a nuclear lab is a mixture of two isotopes: about 69% of 63 Cu and 31% of 65 † Cu. ⋆ Some elements — beryllium, fluorine, sodium, aluminum, phosphorus, scandium, manganese, cobalt, arsenic, yttrium, niobium, rhodium, iodine, cesium, gold, bismuth, and some lantanides — have only one stable isotope. But all the other elements are present on Earth as isotope mixtures. † Actually, there are small differences between isotope ratios of elements coming from different sources. 63 For example, copper from different mines can have from 68.98% to 69.38% of Cu. 5 As far as chemistry is concerned, the atomic weight of an element with multiple isotopes is the weighted average of the isotopes’ atomic weights, weighted by their relative abundances, µ(element) = X µ(isotope)×fraction(isotope) ≈ isotopes X A(isotope)×fraction(isotope). isotopes (10) In particular, µ(copper) = µ 63 63 65 65 Cu × fraction Cu + µ Cu × fraction Cu (11) ≈ 63 × 0.69 + 65 × 0.31 = 63.62 ≈ 63.6. To see how this works, note that to a chemist, the atomic weight of an element — or a molecular weight of a compound — is simply the mass (in grams) of 1 mol of the substance. And 1 mol is the the Avogadro’s number 6.02 · 1023 of atoms (or molecules), regardless of their isotopes. Indeed, one mol of copper — 6.02 · 1023 copper atoms — is what combines with 1 2 mol of oxygen O2 to make copper oxide CuO, or with 1 mol of sulfuric acid to make copper sulfate, etc., etc., and it does not make any difference what isotopes do those copper atoms belong to, as long as they are all copper atoms and there are 6.02 · 1023 of them. On the other hand, the net mass of those 6.02 · 1023 copper atoms depends on the isotope mixture. One mol of natural copper comprises 0.69 mols of 63 Cu and 0.31 mols of 65 Cu, so its net mass is M(1 mol of natural Cu) = M 0.69 mol of Cu + M 0.31 mol of 63 65 = 0.69 × µ Cu g + 0.31 × µ Cu g 63 65 Cu (12) ≈ 0.69 × 63 g + 0.31 × 65 g ≈ 63.6 g. To a chemist, thus mass (in grams) of 1 mol of natural copper is its atomic weight, thus µ(natural copper) ≈ 63.6. 6 (13) Non-textbook problem #IV: Note: the atom number Z is the sequential number of the chemical element in the periodic table; it is equal to the number of electrons in a neutral atom, which in turn is equal to the number of protons in the atom’s nucleus. The atom number is the same for all isotopes of the same chemical element. OOH, the mass number A is different for different isotopes. Specifically, A = Z + N (14) is the net number of nucleons — protons and neutrons — in the nucleus. It’s called the mass number because it governs the atomic mass of a pure isotope; to a 3-digit or 4-digit accuracy, µ ≈ A. (a) In an α decay mother nucleus → daughter nucleus + α particle, (15) the net number of protons is conserved, i.e., Znet after the decay is the same as before the decay. Since the α particle has 2 protons, this means Zmother = Zdaughter + 2 (16) Zdaughter = Zmother − 2, (17) and therefore Similarly, the net number of neutrons is conserved in the α decay, and since the α particle has 2 neutrons, Ndaughter = Nmother − 2. (18) Finally, the mass number A = Z + N follows from the numbers of protons and neutrons, Adaughter = Amother − 4. (19) Altogether, in an alpha decay, the daughter nucleus loses 2 units of atom number and 4 units of mass number relative to the mother nucleus. 7 (b) In a β decay, the weak forces turn one of the neutrons in the mother nucleus into a proton, an electron, and an antineutrino, n → p + e + ν̄. (20) The new proton remains in the nucleus while the electron and the antineutrino fly away; the antineutrino is usually undetected while the electron shows up as the β particle. In the nuclear context, the whole reaction has form mother nucleus → daughter nucleus + e + ν̄, (21) where the daughter nucleus has one more proton but one less neutron than the mother nucleus, Zdaughter = Zmother + 1, Ndaughter = Nmother − 1, (22) Adaughter = Amother . Thus, in a beta decay, the daughter nucleus has the same mass number as the mother nucleus, but its atom number increases by 1. (c) In a γ decay, the protons and the neutrons do not leave the mother nucleus or change into each other. They merely change the way they move inside the nucleus lower their net kinetic + potential energy; the energy ‘saved’ by this reconfiguration is radiated away in the form of γ-rays, which are very-high-frequency electromagnetic waves. Altogether, a γ decay mother nucleus → daughter nucleus + γ (23) does not change the net numbers of protons and neutrons, thus Zdaughter = Zmother , Ndaughter = Nmother , (24) Adaughter = Amother , and the daughter nucleus has the same mass number and the same atom number as the mother nucleus. 8 Textbook question Q.5 at the end of chapter 19: In terms of the atomic nucleus, the atom number Z is the number of protons. This number determines the electric charge Q = +Ze of the nucleus and hence the number of electrons #e = Z in the neutral atom. The nuclear charge also affects the electron’s orbits around the nucleus, Which are very important for the way atoms combine with other atoms into molecules. Thus, the atom number Z is very important for the chemical properties of an element. On the other hand, the mass number A = Z + N — the net number of protons and neutrons in the nucleus — governs the atom’s mass, but it has almost no effect on the electrons. Indeed, the Coulomb force between the electrons and the nucleus depend only on the nucleus’s electric charge +Ze and don’t care about its mass as long as it’s much heavier than the electrons (which is always true). Consequently, the chemical properties of the element do not depend on the mass number A but only on the atom number Z. And that’s why isotopes with the same atom number Z but different mass numbers A all belong to the same chemical element. Indeed, the number of that element in the periodic table is Z, regardless of A. PS: While most kinds of chemical bonds depend only on the electrons of the atoms involved, the so-called hydrogen bonds depend on the relatively low mass of the hydrogen atoms. 2 Consequently, the heavier isotopes of hydrogen — the deuterium D = 1 H and the tritium 1 3 T = 1 H — have slightly different chemical properties from the main hydrogen isotope 1 H. This difference is important in biochemistry, so drinking too much heavy water (D2 O instead of H2 O) would be bad for your health. But this mass effect is limited to hydrogen isotopes. For all the other elements, isotopes with similar atom numbers but different mass numbers are chemically indistinguishable. Textbook problem SP.2 at the end of chapter 19: (a) The chemical symbols in the thorium decay chain stand for the following elements: Th is thorium, Z = 90; Ra is radium, Z = 88; Ac is actinium, Z = 89; Rn is radon, Z = 86; Po is polonium, Z = 84; Pb is lead, Z = 82; Bi is bismuth, z = 83. 9 (b) The α and β decays have different effects on the atomic number Z: In an α decay it decreases by 2, while in a β decay, it increases by 1. Thus, comparing the atomic numbers of a parent isotope and its decay product will immediately tells us if the decay in question is α or β. In particular, in the thorium decay chain in question, in the decays Th → Ra, Ra → Rn, Rn → Po, Po → Pb (25) the atomic number decreases by 2, so these are α decays, while in the decays Ra → Ac, Ac → Th, Pb → Bi, Bi → Po (26) the atomic number increases by 1, so these are β decays. (c–d) Now that we know which decay is α and which is β, we may determine the mass numbers A of all the isotopes involved by following a simple rule: In an α decay A decreases by 4, while in a β decay A does not change. Thus: (α) (β) (β) (α) (α) (α) (α) (β) (β) (α) 232 Th 90 228 Ra 88 228 Ac 89 228 Th 90 224 Ra 88 220 Rn 86 216 Po 84 212 Pb 82 212 Bi 83 212 Po 84 → → → → → → → → → → 228 Ra 88 228 Ac 89 228 Th 90 224 Ra 88 220 Rn 86 216 Po 84 212 Pb 82 212 Bi 83 212 Po 84 208 Pb 82 10 4 + 2 He, + e− + ν̄, + e− + ν̄, 4 + 2 He, 4 + 2 He, 4 + 2 He, 4 + 2 He, + e− + ν̄, + e− + ν̄, 4 + 2 He. (27) And the whole decay chain can be summarized as β 228 β 228 232 α α 220 α 224 α 228 Th → 88 Ra → 89 Ac → 90 Th → 88 Ra → 86 Rn → 90 β 212 β 212 α 216 α 216 α 212 α 208 → 84 Po → 84 Po → 82 Pb → 83 Bi → 84 Po → 82 Pb. (28) There is a Walter Fendt applet showing this decay chain step by step. It also shows several other decay chains, in particular 238 U 92 → ··· → 11 206 Pb 82 and 235 U 92 → ··· → 207 Pb. 82