IB Physics E.4: Fission worked solutions

In C a heavy nucleus absorbs a neutron and splits into two nuclei of medium mass, with more neutrons released: that is neutron-induced fission. A joins two light nuclei (fusion), B is alpha decay, and D is neutron capture: the nucleus gets heavier and does not split.

Nucleons: 235+1=132+A+4, so A=100. Protons: 92=50+Z, so Z=42. A forgets the absorbed neutron, C forgets the four emitted neutrons, D swaps the two numbers.

The products are more tightly bound than the parent, so the total binding energy goes up; the energy released has come from rest mass, so the total rest mass goes down by E/c^2. "Energy is released, so binding energy falls" (A, C) mixes up the energy given out with the energy that would be needed to pull the nuclei apart.

For a steady output exactly one neutron per fission must go on to cause another fission. 2.5-0.6-0.2-1=0.7. B leaves out the neutron that must cause the next fission, C ignores both losses, D ignores the capture without fission.

Natural uranium is less than 1% uranium-235; most slow neutrons meet uranium-238, which captures some without splitting. Raising the uranium-235 proportion means more of the neutrons cause fission, so the chain can be kept going. A reverses the two isotopes; the energy per fission (C) and the neutrons per fission (D) belong to the nucleus, not to the fuel mixture.

Capture gives {}^{239}_{92}\mathrm{U} (nucleon number up by one, proton number unchanged). Each \beta^- decay keeps the nucleon number and raises the proton number by one: 92\to93\to94. So the middle nucleus is {}^{239}_{93}\mathrm{Np}. A is the nucleus before the first decay, C loses the captured neutron, D lowers the proton number.

Each collision passes some of the neutron's kinetic energy to a moderator nucleus, so the neutron slows down. A uranium-235 nucleus captures a slow neutron, and then splits, much more readily than a fast one. B is the intuitive guess that slower means less effective; it is the other way round for U-235.

A neutron loses the largest share of its kinetic energy when it hits a nucleus of similar mass, so light nuclei (hydrogen in water, carbon in graphite) slow it in few collisions; the moderator must not absorb the neutrons it is slowing. A describes a control-rod material, B a shield, D a fuel-cladding property.

Control rods absorb neutrons, and moving them in or out sets the fission rate. The heat exchanger passes energy from the hot primary coolant to the water of the secondary circuit, which becomes steam. The moderator slows neutrons and the shielding absorbs radiation; neither does (i) or (ii).

E=200\times10^{6}\times1.60\times10^{-19}=3.20\times10^{-11}\,\mathrm{J} per nucleus; 3.20\times10^{-11}/3.90\times10^{-25}=8.2\times10^{13}\,\mathrm{J\,kg^{-1}}. A inverts the ratio, B treats the mass as grams, C keeps the energy in electronvolts.

The products are a mixture: some decay within days, others (and the plutonium formed in the fuel) have half-lives of thousands of years, which is why long-term storage is planned. The control rods stop the chain reaction, not radioactive decay (B); C ignores the long-lived products; D puts the radioactivity in the wrong place: it is in the fission products themselves.

Without the moderator the neutrons stay fast, and fast neutrons rarely cause fission of uranium-235, so fewer than one neutron per fission causes another and the chain dies away. A is the "runaway" misconception, B ignores the probability of fission, D gives the moderator a job it does not have.

Multiply each value by its nucleon number: 137\times8.3+97\times8.6-235\times7.6=1971-1786=185\,\mathrm{MeV}. Free neutrons have no binding energy. A subtracts the per-nucleon values, B gives the absorbed neutron a share of the uranium binding energy (236\times7.6), D is the products' total alone.

(a) Nucleons: 1+235=139+95+x, so x=2 (A1). (b) The left side has 92 electrons (uranium) and the right side 54+38=92, so the electron masses cancel in the subtraction (A1). (c) \Delta m=(235.043930+1.008665)-(138.918793+94.919359+2\times1.008665)=0.197113\,\mathrm{u} (M1); 0.197113\times931.5=184\,\mathrm{MeV} (A1). (d) 183.6\times10^{6}\times1.60\times10^{-19}=2.94\times10^{-11}\,\mathrm{J} (A1).

(a) Rapid rise for small A, a broad maximum of about 8.8\,\mathrm{MeV} near A\approx56 labelled "most stable" (A1); a gentle fall to about 7.6\,\mathrm{MeV} at A\approx235 (A1). (b) Medium-mass fragments have a larger binding energy per nucleon (about 8.5\,\mathrm{MeV}) than uranium (about 7.6\,\mathrm{MeV}), so the total binding energy of the products is greater (M1). The increase, roughly 0.9\,\mathrm{MeV} for each of about 235 nucleons, is released as energy, and the products have less rest mass (A1).

(a) Kinetic energy of the fission fragments (A1); smaller amounts go to the neutrons and to gamma radiation. (b) The fragments are stopped within the fuel by collisions, which raises the internal energy and temperature of the fuel; the coolant flowing past carries this thermal energy out of the core (A1). In the heat exchanger it boils water in a separate circuit; the steam turns a turbine, which drives a generator (A1).

(a) With so little uranium-235, many neutrons are captured by uranium-238 and by the water without causing fission (M1). Fewer than one neutron per fission then goes on to cause another, so the chain dies away; raising the proportion of uranium-235 lets one neutron per fission cause a further fission (A1). (b) Capture: {}^{238}_{92}\mathrm{U}+{}^{1}_{0}\mathrm{n}\to{}^{239}_{92}\mathrm{U} (A1). Two \beta^- decays: {}^{239}_{92}\mathrm{U}\to{}^{239}_{93}\mathrm{Np}+\mathrm{e}^-+\bar{\nu}, then {}^{239}_{93}\mathrm{Np}\to{}^{239}_{94}\mathrm{Pu}+\mathrm{e}^-+\bar{\nu} (A1). (c) Plutonium-239 undergoes fission with slow neutrons, so it releases energy in the reactor; it can also be extracted from used fuel and used in weapons, which is why it is tracked (A1); either point earns the mark.

(a) Mass of uranium-235 =0.040\times8.0=0.32\,\mathrm{g}=3.2\times10^{-4}\,\mathrm{kg} (M1); N=3.2\times10^{-4}/(235\times1.661\times10^{-27})=8.2\times10^{20} (A1). (b) 8.2\times10^{20}\times3.20\times10^{-11}=2.6\times10^{10}\,\mathrm{J} (A1). (c) 2.6\times10^{10}/8.0\times10^{-3}=3.3\times10^{12}\,\mathrm{J\,kg^{-1}} (A1). (d) Specific energy divides by the whole mass of fuel, and 96\% of the pellet is uranium-238, which here releases no fission energy; the result is 4\% of the pure value (A1).

(a) Containment stops radioactive material itself (fuel, fission products, contaminated coolant) from escaping into the surroundings (A1). Shielding absorbs the radiation emitted by that material, so people outside receive a low dose (A1). (b) Alpha and beta particles have short ranges and are absorbed within the fuel, its cladding and the steel vessel (M1). Neutrons and gamma photons travel much further through matter, so metres of concrete and water are needed to reduce them to a safe level (A1).

(a) In spontaneous fission the nucleus splits on its own, at random; neutron-induced fission happens only after the nucleus absorbs a neutron (A1). (b) 252=142+106+n gives n=4; protons 54+44=98 (A1). (c) Products: 142\times8.27+106\times8.56=1174.3+907.4=2081.7\,\mathrm{MeV} (M1). Energy released = products' binding energy - parent's binding energy, so the parent has 2081.7-205=1876.7\,\mathrm{MeV} (M1); per nucleon 1876.7/252=7.45\,\mathrm{MeV} (A1). (d) A free neutron is not held by any other nucleon, so no energy is needed to free it: its binding energy is zero (A1).

(a) Exactly one, on average (A1). (b) 2.4-0.5-0.3-1=0.6 (A1). (c) Now 2.4-0.5-0.3-0.5=1.1 neutrons per fission cause another, so each generation is 1.1 times the one before (M1): 1.0\times10^{6}\times1.1^{5}=1.6\times10^{6} (A1). (d) Neutrons are made throughout the volume but escape only through the surface; the volume grows faster than the surface area, so a neutron in a larger core is more likely to hit a nucleus before it gets out (M1). Below the critical mass too many escape for one neutron per fission to cause another, however the rods are set; above it the rods can hold the chain steady (A1).

(a) Input power =900/0.33=2.73\times10^{3}\,\mathrm{MW}=2.7\times10^{9}\,\mathrm{W} (A1). (b) Energy per fission =200\times10^{6}\times1.60\times10^{-19}=3.20\times10^{-11}\,\mathrm{J} (M1); rate =2.73\times10^{9}/3.20\times10^{-11}=8.5\times10^{19}\,\mathrm{s^{-1}} (A1). (c) Fissions in a day =8.52\times10^{19}\times86\,400=7.36\times10^{24} (M1); mass =7.36\times10^{24}\times3.90\times10^{-25}=2.9\,\mathrm{kg} (A1). (d) \Delta m=E/c^2=2.73\times10^{9}\times86\,400/(3.00\times10^{8})^2=2.6\times10^{-3}\,\mathrm{kg} (A1). (e) In each fission only about 0.1\% of the mass disappears as energy; the rest stays as the mass of the fragments and neutrons (A1).

(a) 17.6\times10^{6}\times1.60\times10^{-19}=2.82\times10^{-12}\,\mathrm{J}; mass =5.030\times1.661\times10^{-27}=8.355\times10^{-27}\,\mathrm{kg} (M1); 2.82\times10^{-12}/8.355\times10^{-27}=3.4\times10^{14}\,\mathrm{J\,kg^{-1}} (A1). (b) 3.20\times10^{-11}/(235.0\times1.661\times10^{-27})=8.2\times10^{13}\,\mathrm{J\,kg^{-1}} (A1). (c) 3.37\times10^{14}/8.20\times10^{13}=4.1 (A1). (d) Per reaction, fission releases 200/17.6\approx11 times more energy, but a uranium nucleus has 235/5.03\approx47 times the mass of the fusing pair, so per kilogram fusion wins (A1).

(a) The rods absorb neutrons, so fewer than one neutron per fission goes on to cause another fission and the chain dies away (A1). (b) The fission products are unstable and continue to decay (mainly \beta^- and \gamma) whatever the control rods do (M1); the radiation is absorbed in the fuel and its surroundings, raising their internal energy (A1). (c) 0.010\times3000=30\,\mathrm{MW} (A1). (d) \Delta T=Pt/C=3.0\times10^{7}\times600/4.0\times10^{8}=45\,\mathrm{K} (A1). (e) The short-lived products still release a lot of energy and radiation; the water carries the energy away and shields people, and after a few years the power and activity are low enough for dry storage (A1).

(a) Momentum: m(2.6\times10^{7})=m(-2.2\times10^{7})+12mV (M1), so V=4.8\times10^{7}/12=4.0\times10^{6}\,\mathrm{m\,s^{-1}} (A1). (b) Before: \tfrac12m(2.6\times10^{7})^2=3.38\times10^{14}m (M1). After: \tfrac12m(2.2\times10^{7})^2+\tfrac12(12m)(4.0\times10^{6})^2=2.42\times10^{14}m+0.96\times10^{14}m=3.38\times10^{14}m (A1). (c) Fraction kept =(2.2/2.6)^2=0.716, so fraction lost =0.28 (A1). (d) 0.716^{N}=0.025/2.0\times10^{6}=1.25\times10^{-8} (M1); N=\ln(1.25\times10^{-8})/\ln0.716=54.5, so about 55 collisions (A1). (e) Head-on collisions remove the most energy; most real collisions are glancing and remove less, so more are needed (A1).

(a) P=\dfrac{m}{t}c\Delta T, so \dfrac{m}{t}=3.0\times10^{9}/(5.4\times10^{3}\times35) (M1) =1.6\times10^{4}\,\mathrm{kg\,s^{-1}} (A1). (b) 1.0/3.0=0.33=33\% (A1). (c) 3.0-1.0=2.0\,\mathrm{GW} (A1). (d) Raising the pressure raises the boiling point, so the water stays liquid at 325\,{}^{\circ}\mathrm{C} and carries energy away efficiently (A1). (e) The primary water passes through the core and picks up radioactive material; the heat exchanger passes its energy to clean water in a separate circuit, so radioactivity stays inside the containment (A1). (f) Less dense water means fewer moderator nuclei, so fewer neutrons are slowed down (M1); fast neutrons rarely cause fission, so the rate and the power fall, which tends to stabilise the reactor (A1).

(a) {}^{137}_{55}\mathrm{Cs}\to{}^{137}_{56}\mathrm{Ba}+{}^{0}_{-1}\mathrm{e} (A1) +\,\bar{\nu}, the electron antineutrino (A1). (b) 150/30=5 half-lives: (1/2)^5=1/32=0.031 (A1). (c) 300/30=10 half-lives; 2^{10}=1024, so the activity is about 1/1000 of its starting value (A1). (d) Over 300 years plutonium-239 completes only about 1\% of one half-life, so almost none of it has decayed (M1); it takes about ten half-lives, roughly 240\,000 years, to fall by a factor of a thousand, so the storage has to stay secure for that long (A1). (e) For example: stable, dry rock hundreds of metres below ground, with the waste set in glass inside corrosion-resistant containers, so that groundwater cannot carry it to the surface (A1).

(a) The nucleus is at rest, so the total momentum is zero; with the neutrons ignored, the two fragments must have equal and opposite momenta (A1). (b) E_{\mathrm{k}}=p^2/2m (M1); with the same p, E_{\mathrm{k}}(\text{Rb})/E_{\mathrm{k}}(\text{Cs})=m_{\text{Cs}}/m_{\text{Rb}}=140/93 (A1). (c) 170\times140/233=102\,\mathrm{MeV} (A1). (d) E_{\mathrm{k}}=102.1\times10^{6}\times1.60\times10^{-19}=1.63\times10^{-11}\,\mathrm{J}; m=93\times1.661\times10^{-27}=1.545\times10^{-25}\,\mathrm{kg} (M1); v=\sqrt{2E_{\mathrm{k}}/m}=1.45\times10^{7}\,\mathrm{m\,s^{-1}} (A1). (e) \beta^- decay, which turns a neutron into a proton (A1).