Chemistry · Chapter 21

Nuclear Chemistry

Every reaction so far in this book moved electrons around. The nucleus sat in the middle and did not change. This chapter opens the nucleus up. You will see why some nuclei last for billions of years while others fall apart today. You will learn where a star gets its energy, and why that energy dwarfs the energy of a flame. You will date a bone from the carbon left inside it. You will follow neutrons through a reactor core. You will also meet the units that say how much radiation a body has taken in, and the ways doctors put that radiation to work.

By the end of this chapter you can

Read a nuclide symbol and count the protons, neutrons and nucleons in it.
Use the band of stability to predict whether a nuclide will decay.
Find a mass defect and turn it into a binding energy with E = mc².
Tell alpha, beta, positron, electron capture and gamma decay apart.
Use a half-life to find what is left of a sample, or how old a find is.
Balance any nuclear equation for both mass number and charge.
Compare the energy released by fission with the energy released by fusion.
Match radiation units, shielding and medical uses to the right kind of ray.

Section 1The nucleus and nuclear stability

Nucleons and nuclide symbols

The nucleus is a tiny speck at the center of an atom. It holds almost all of the atom's mass. Protons and neutrons both live there, and together they are called nucleons. A nuclide is one exact kind of nucleus. Fix the proton count and the neutron count and you have named a nuclide.

A nuclide symbol carries two numbers to the left of the element symbol. The top one is the mass number A, which counts every nucleon. The bottom one is the atomic number Z, which counts only the protons. Subtract to get the neutrons, so N = A − Z.

²³⁸₉₂U → A = 238 nucleons, Z = 92 protons, N = 238 − 92 = 146 neutrons

Isotopes share a proton count but differ in neutron count. Carbon-12 and carbon-14 each hold 6 protons. Carbon-14 carries two extra neutrons, and those two neutrons are what make it fall apart. A nuclide that falls apart on its own is radioactive.

The band of stability

Every proton carries a positive charge, so protons shove each other hard. What holds them together anyway is the strong nuclear force. It pulls any two nucleons together, but only across a very short range. Neutrons add that pull without adding any shove, so they work like nuclear glue.

Now plot every nuclide with neutrons up the page and protons across it. The stable ones crowd into one narrow stripe. That stripe is the band of stability. Light stable nuclei sit near a 1-to-1 ratio of neutrons to protons. Heavy ones need extra glue, so the band bends upward toward 1.5 to 1.

A nuclide above the band holds too many neutrons. One below the band holds too few. Either way it decays in whatever direction moves it toward the band. Nothing past bismuth is safe, because every nuclide with Z above 83 is radioactive.

Some counts are steadier than others. A nucleus with 2, 8, 20, 28, 50, 82 or 126 protons or neutrons is unusually stable. These counts are the magic numbers, and they act the way a filled electron shell does. Lead-208 holds 82 protons and 126 neutrons, so it is magic twice over.

Figure 21.1 · Each small dot is a stable nuclide. They fall in a stripe that starts along the N = Z line and then bends above it, because heavy nuclei need extra neutrons. The red strip on the right marks Z > 83, where no stable nuclide exists.

Scale check. A nucleus is about one hundred-thousandth the width of its atom. Blow the atom up to the size of a stadium and the nucleus is a pea on the center spot. All the mass of the stadium sits in that pea.

Mass defect and binding energy

Weigh a helium-4 atom. Then weigh two hydrogen atoms and two neutrons on their own and add those up. The separate parts weigh more than the finished atom does. The missing mass is the mass defect.

That mass did not vanish. It left as energy at the moment the nucleus formed. Einstein's equation ties mass and energy together, and c is the speed of light, 2.998 × 10⁸ m/s.

E = mc²

The energy that comes out is the binding energy. It is also the bill you would have to pay to pull that nucleus back apart. Divide it by the nucleon count to get the binding energy per nucleon. That is the fair way to compare nuclei of very different sizes, and it peaks near iron-56 at about 8.79 MeV.

Worked example · the binding energy of helium-4

A helium-4 atom weighs 4.002603 u. A hydrogen-1 atom weighs 1.007825 u and a neutron weighs 1.008665 u. Find the binding energy per nucleon.

Add up the loose parts

Two hydrogen atoms come to 2 × 1.007825 u = 2.015650 u, and two neutrons come to 2 × 1.008665 u = 2.017330 u. Added together, the loose parts total 4.032980 u. The two electrons ride along on the hydrogen atoms, and helium-4 has two of its own, so they cancel out.

Subtract to find the mass defect

Subtracting the finished atom from its loose parts gives 4.032980 u − 4.002603 u = 0.030377 u of missing mass.

Turn the lost mass into energy

One u is 1.6605 × 10⁻²⁷ kg, so the defect is 5.0441 × 10⁻²⁹ kg. Then E = mc² = (5.0441 × 10⁻²⁹ kg)(2.998 × 10⁸ m/s)² = 4.534 × 10⁻¹² J.

Report it per nucleon

Nuclear energies are usually quoted in MeV, and 1 u is worth 931.5 MeV. So 0.030377 u × 931.5 MeV/u = 28.30 MeV. Divide by 4 nucleons and you get 7.074 MeV per nucleon.

Check it a second way

One MeV is 1.602 × 10⁻¹³ J. So 4.534 × 10⁻¹² J ÷ 1.602 × 10⁻¹³ J/MeV = 28.30 MeV, which matches. For a whole mole of helium-4 that is 2.73 × 10⁹ kJ, which is millions of times any bond energy.

Key points

A nuclide symbol puts A on top and Z below, and the neutron count is A − Z.
Neutrons add strong-force pull without adding charge, so heavy nuclei need extra neutrons.
Stable nuclides lie inside the band of stability, and no nuclide past Z = 83 is stable.
The loose parts of a nucleus weigh more than the nucleus itself, and that mass defect is its binding energy.
Binding energy per nucleon peaks near iron-56, so iron sits at the bottom of the nuclear energy well.

Interactive Plot nuclides on the band of stability and test binding energy. Open full ↑

Section 2Types of radioactive decay

Alpha decay

An alpha particle is a helium-4 nucleus on its own. It is two protons and two neutrons, with a charge of 2+. A heavy nucleus can shrink by throwing one out. The mass number drops by 4 and the atomic number drops by 2.

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

Alpha decay is the main exit for nuclides above Z = 83. The particle is heavy and slow for its energy, so it travels only a few centimetres in air. A sheet of paper stops it.

Beta decay and positron emission

A beta particle is a fast electron fired out of the nucleus. There are no electrons stored in there waiting. Instead a neutron turns into a proton plus an electron, and the electron leaves. The mass number holds steady while the atomic number climbs by 1.

¹⁴₆C → ¹⁴₇N + ⁰₋₁e

This is the fix for a nuclide sitting above the band of stability. It swaps one of its spare neutrons for a proton, which walks the nuclide down into the band. A tiny neutral particle called an antineutrino leaves with the electron and carries off part of the energy.

Positron emission runs the trade the other way. A proton turns into a neutron plus a positron, which is an electron with a positive charge. The mass number holds steady and the atomic number falls by 1.

¹⁵₈O → ¹⁵₇N + ⁰₊₁e

A positron does not get far. It meets an ordinary electron and the two destroy each other. Their mass turns into two gamma photons of 0.511 MeV, fired in opposite directions. That event is called annihilation, and a PET scanner is built to catch both photons at once.

Electron capture

A nucleus short on neutrons has a second route. It reaches out and swallows one of its own inner electrons. That electron joins a proton, and the pair becomes a neutron. The atomic number falls by 1 and the mass number does not move, so the books balance exactly as they do for positron emission.

⁴⁰₁₉K + ⁰₋₁e → ⁴⁰₁₈Ar

An outer electron then drops into the empty inner spot, and an X-ray comes out. Heavier nuclides lean toward capture rather than positron emission. Their inner electrons sit closer in, so they are easier to grab.

Gamma emission

Alpha and beta decay often leave the new nucleus holding extra energy. Such a nucleus is excited. It dumps the extra as a gamma ray, a photon with a very short wavelength and no mass or charge. Neither the mass number nor the atomic number changes.

^99m₄₃Tc → ⁹⁹₄₃Tc + γ

The letter m marks a metastable state, one that holds its extra energy long enough to be useful. Technetium-99m is the busiest tracer in any hospital, and it gives off a clean 0.1405 MeV gamma ray.

Figure 21.2 · Five decay modes and the bookkeeping for each. The last column says how the nuclide shifts on the band of stability plot in Figure 21.1.

Decay series

One step is rarely enough for a heavy nucleus. Uranium-238 needs 14 steps to reach lead-206, which is stable. Eight of those steps are alpha and six are beta. Every step has its own daughter and its own half-life, and the whole chain is called a decay series. That series is why uranium ore always holds radium and leaks radon gas.

Particle	Symbol	Mass	Charge	Change in A	Change in Z
Alpha	⁴₂He	4 u	2+	−4	−2
Beta minus	⁰₋₁e	1/1836 u	1−	0	+1
Positron	⁰₊₁e	1/1836 u	1+	0	−1
Electron capture	⁰₋₁e in	1/1836 u	1−	0	−1
Gamma	γ	0	0	0	0
Neutron	¹₀n	1.008665 u	0	−1	0

Worked example · naming the daughter nuclide

Radium-226 has Z = 88 and emits an alpha particle. Polonium-218 has Z = 84 and does the same. Name both daughters and write both equations.

Take 4 off A and 2 off Z

Taking 4 off the mass number gives 226 − 4 = 222, and taking 2 off the atomic number gives 88 − 2 = 86. Element 86 is radon, so the daughter nuclide is radon-222.

Repeat for polonium-218

The same two subtractions give 218 − 4 = 214 for the mass number and 84 − 2 = 82 for the atomic number. Element 82 is lead, so this daughter nuclide is lead-214.

Write them out

²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He and ²¹⁸₈₄Po → ²¹⁴₈₂Pb + ⁴₂He.

Check both sides

In the radium line the mass numbers come to 222 + 4 = 226 and the atomic numbers come to 86 + 2 = 88. In the polonium line they come to 214 + 4 = 218 and 82 + 2 = 84. Both equations match the left side on each total, so both of them are right.

Key points

Alpha decay drops A by 4 and Z by 2, and it is the usual exit for very heavy nuclides.
Beta decay turns a neutron into a proton, so Z rises by 1 and A holds still.
Positron emission and electron capture both turn a proton into a neutron, so Z falls by 1.
Gamma emission changes no count at all, because only energy leaves.
Heavy nuclides run through a decay series, and uranium-238 takes 14 steps to reach lead-206.

Interactive Fire alpha, beta, positron and gamma and see what changes. Open full ↑

Section 3Half-life and dating

What a half-life means

You cannot say when one single nucleus will decay. The event is random, and the nucleus keeps no memory of how long it has waited. But a real sample holds a staggering number of nuclei. Across that crowd the behaviour is smooth and predictable.

The half-life, written t_1/2, is the time it takes for half the nuclei in a sample to decay. Each half-life halves whatever is left, not what you began with. Start at 100 percent and you pass 50, then 25, then 12.5, then 6.25. The curve bends toward zero but never arrives.

Half-lives cover an almost absurd range. Iodine-131 has a half-life of 8.02 days. Carbon-14 sits at 5,730 years. Uranium-238 takes 4.5 billion years, which is roughly the age of the Earth. Polonium-214 is done in 164 microseconds.

Figure 21.3 · The same curve fits every isotope. Only the width of one step changes, from microseconds to billions of years.

Decay kinetics

Nuclear decay is first order. The rate depends on how many unstable nuclei are present and on nothing else. Heating the sample does not speed it up. Neither does crushing it, dissolving it or bonding it into a new compound. That is a sharp break from ordinary reaction rates.

The rate constant for decay gets its own name, the decay constant k. It is tied to the half-life by the natural log of 2.

k = 0.693 ÷ t_1/2 ln(N₀ ÷ N) = kt N ÷ N₀ = (1/2)^t/t_1/2

Use the right-hand form when the elapsed time is a tidy number of half-lives. Use the log form for everything else. The activity of a sample is how many decays it produces each second, and it shrinks along the very same curve as the nuclei do.

Worked example · how much iodine-131 is left

A clinic receives 40.0 mg of iodine-131, whose half-life is 8.02 days. How much is left after 32.08 days?

Count the half-lives

Dividing the elapsed time by the half-life gives 32.08 days ÷ 8.02 days = 4.00 half-lives.

Halve it four times

Halving the amount once for every half-life runs 40.0 mg → 20.0 mg → 10.0 mg → 5.00 mg → 2.50 mg.

Check with the formula

N = 40.0 mg × (1/2)⁴ = 40.0 mg × 0.0625 = 2.50 mg. The two routes agree, so 2.50 mg is left.

Radiometric dating

Cosmic rays strike nitrogen high in the air and keep making carbon-14. Plants take that carbon in, and animals eat the plants. So every living thing carries the same small fraction of carbon-14 as the air does. Death shuts the intake off, and from that moment the carbon-14 only decays.

Measure the fraction left and you can solve for the time since death. This is radiometric dating. Carbon-14 is useful out to roughly 50,000 years, which is about nine half-lives. Past that, too little is left to measure well.

Rock needs a slower clock. Uranium-238 decays all the way to lead-206 with a half-life of 4.5 billion years. Potassium-40 becomes argon-40 in about 1.25 billion years. Counting the parent and daughter atoms in a mineral grain dates the moment the crystal locked them in.

Worked example · dating a piece of charcoal

Charcoal from an old hearth holds 54.0 percent of the carbon-14 that living wood holds. The half-life of carbon-14 is 5,730 years. How old is the charcoal?

Find the decay constant

Dividing 0.693 by the half-life gives k = 0.693 ÷ 5,730 yr = 1.21 × 10⁻⁴ yr⁻¹.

Set up the log form

The starting amount divided by the present amount is N₀ ÷ N = 1 ÷ 0.540 = 1.85, and the natural log of that ratio is ln(1.85) = 0.616.

Solve for the time

Dividing that log by the decay constant gives t = 0.616 ÷ (1.21 × 10⁻⁴ yr⁻¹) = 5,090 years.

Check that it is sensible

5,090 years is 0.888 of one half-life, so a little under half should be gone. And (1/2)^0.888 = 0.540, which is the fraction we measured. The date holds up.

Why the date needs a correction. The carbon-14 level in the air has drifted over the centuries. Labs fix this with a calibration curve built from tree rings of known age. A raw radiocarbon date is a first draft, not the final year.

Key points

One half-life halves whatever is left, so the fraction after n half-lives is (1/2)ⁿ.
Decay is first order, and k = 0.693 ÷ t_1/2.
Temperature, pressure and chemical bonding have no effect on a decay rate.
Use ln(N₀/N) = kt when the time is not a whole number of half-lives.
Carbon-14 dates things that once lived out to about 50,000 years, while uranium and potassium clocks date rock.

Interactive Decay a sample through many half-lives and date a find. Open full ↑

Section 4Nuclear equations and transmutation

Two rules and nothing more

A nuclear equation balances when two totals agree across the arrow. The mass numbers on the left must add to the mass numbers on the right. The atomic numbers must do the same, because charge is conserved. Notice what is missing from that list. Elements are not conserved, so atoms of one element really do become atoms of another.

The trick is to give every particle a nuclide symbol, even the small ones. An alpha particle is ⁴₂He. A beta particle is ⁰₋₁e and a positron is ⁰₊₁e. A neutron is ¹₀n and a bare proton is ¹₁H. A gamma ray is ⁰₀γ, so it adds nothing to either total.

Once every symbol has its pair of numbers, finding an unknown is two subtraction problems. Add the mass numbers you know and subtract to get the missing A. Do the same with the atomic numbers to get the missing Z. Then look up which element owns that Z, and you have named the particle.

Figure 21.4 · The ledger for one fission equation. Both totals come to the same number on each side, which is the only test a nuclear equation has to pass.

Worked example · finding a missing product

Fill the gap: ²³⁵₉₂U + ¹₀n → ¹³⁷₅₂Te + ? + 2 ¹₀n

Add the mass numbers on each side

The left side totals 235 + 1 = 236, while the pieces already written on the right total 137 + 1 + 1 = 139. The gap between those two totals is 236 − 139 = 97, so the missing particle has A = 97.

Add the atomic numbers on each side

The left side totals 92 + 0 = 92, while the known products on the right total 52 + 0 + 0 = 52. That leaves a gap of 92 − 52 = 40, so the missing particle has Z = 40.

Name the element

Element 40 is zirconium, so the missing product is ⁹⁷₄₀Zr.

Check the finished line

The mass numbers on the right now total 137 + 97 + 2 = 236, which matches the 236 on the left. The atomic numbers total 52 + 40 = 92, which matches the 92 on the left. Both totals agree across the arrow, so the equation is balanced.

Transmutation on purpose

Turning one element into another is transmutation. Radioactive decay does it by itself. You can also force it by firing a particle into a target nucleus. Ernest Rutherford did the first deliberate one in 1919. He fired alpha particles into nitrogen gas and got oxygen and protons back.

¹⁴₇N + ⁴₂He → ¹⁷₈O + ¹₁H

Neutrons make the best bullets. They carry no charge, so a nucleus does not push them away. A reactor core is swimming in spare neutrons, which makes it a fine factory for useful isotopes. The tracer most hospitals run on is built this way.

⁹⁸₄₂Mo + ¹₀n → ⁹⁹₄₂Mo → ^99m₄₃Tc + ⁰₋₁e

A charged bullet needs a hard push to beat the repulsion of the target. That push comes from a particle accelerator, which uses electric and magnetic fields to drive ions to huge speeds. Accelerators and reactors together built every element past uranium. Those are the transuranium elements, and none of them occurs on Earth in any real amount.

²³⁸₉₂U + ¹₀n → ²³⁹₉₂U → ²³⁹₉₃Np + ⁰₋₁e → ²³⁹₉₄Pu + ⁰₋₁e

Climbing higher takes heavier bullets still. Seaborgium came from firing oxygen-18 into californium-249. Runs like that may make only a few atoms, and those atoms live for seconds or less. Even so, the equation balances the same way any other does.

²⁴⁹₉₈Cf + ¹⁸₈O → ²⁶³₁₀₆Sg + 4 ¹₀n

Worked example · naming the particle that flew out

Aluminium-27 is hit by an alpha particle and phosphorus-30 comes out. What else comes out?

Write what you know

The line so far reads ²⁷₁₃Al + ⁴₂He → ³⁰₁₅P + ?

Balance the mass numbers

The mass numbers on the left total 27 + 4 = 31, and the phosphorus accounts for 30 of them. That leaves 31 − 30 = 1, so the missing particle has A = 1.

Balance the atomic numbers

The atomic numbers on the left total 13 + 2 = 15, and the phosphorus already accounts for all 15. That leaves 15 − 15 = 0, so the missing particle has Z = 0.

Identify it

A = 1 with Z = 0 is a neutron, ¹₀n. Both totals now read 31 and 15, so the equation is balanced.

Key points

Balance a nuclear equation on mass number and on atomic number, and nothing else.
Elements are not conserved, so one element can become another.
Give every particle a nuclide symbol, including the neutron, the beta particle and the gamma ray.
Transmutation fires a particle at a target nucleus, and neutrons work best because they carry no charge.
Every element past uranium was built in a reactor or an accelerator.

Interactive Balance mass and charge in a nuclear equation. Open full ↑

Section 5Fission and fusion

What the binding energy curve predicts

Section 1 left you with one shape worth memorising. Binding energy per nucleon climbs steeply from hydrogen, peaks near iron-56, then drifts back down. Any change that moves nucleons toward that peak releases energy. A very heavy nucleus gets closer to the peak by splitting. Very light nuclei get closer by joining up.

Figure 21.5 · Both arrows point uphill toward iron-56. That single picture is the reason fission works at the heavy end and fusion works at the light end.

Fission

Fission splits a heavy nucleus into two lighter ones. Uranium-235 fissions when it swallows a slow neutron. It does not split the same way twice, and dozens of product pairs show up across a real sample. Here is one of the common lines.

²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3 ¹₀n

One fission event releases about 200 MeV. Burning a single carbon atom releases about 4 eV. The nuclear event is roughly 50 million times larger, and that gap is the whole story of nuclear power. A fuel pellet the size of a fingertip holds the energy of a tonne of coal.

Worked example · fission energy per mole

One uranium-235 fission gives about 200 MeV. How much does a mole of fissions give, and how does that compare with burning a mole of carbon at 393.5 kJ/mol?

Turn MeV into joules

1 MeV = 1.602 × 10⁻¹³ J, so 200 MeV × 1.602 × 10⁻¹³ J/MeV = 3.204 × 10⁻¹¹ J per event.

Scale up by Avogadro's number

3.204 × 10⁻¹¹ J × 6.022 × 10²³ /mol = 1.93 × 10¹³ J/mol, which is 1.93 × 10¹⁰ kJ/mol.

Compare the two fuels

Dividing the nuclear figure by the chemical one gives 1.93 × 10¹⁰ kJ ÷ 393.5 kJ = 4.90 × 10⁷. So one mole of uranium-235 fissions releases about 49 million times the energy of one mole of carbon burned.

Chain reactions and reactors

Every fission throws out two or three fresh neutrons. Each of those can split another uranium-235 nucleus. The reaction then feeds itself, and that self-feeding loop is a chain reaction.

Whether the loop grows or dies comes down to one number. The multiplication factor k is the number of fissions in one generation divided by the number in the generation before. If k is below 1 the chain fades out. If k is exactly 1 the rate holds steady. If k is above 1 the rate climbs with every generation.

Neutrons that leak out of the fuel are neutrons that cannot split anything. So you need enough fuel, packed closely enough, to keep k at 1 or above. The smallest amount that can hold a chain reaction going is the critical mass.

Natural uranium is only 0.72 percent uranium-235, and almost all the rest is uranium-238. Power reactor fuel is enriched to about 3 to 5 percent uranium-235. That is enough to hold a chain reaction in a large core full of water.

A reactor is a machine for parking k at exactly 1. The moderator, usually water or graphite, slows the fresh neutrons down, because uranium-235 captures slow neutrons far more readily than fast ones. Control rods of boron or cadmium slide in to soak up spare neutrons, and sliding them out lets the rate rise. A coolant carries the heat away to boil water, and the steam drives a turbine. Thick shielding and a sealed containment building keep the radiation inside.

Fusion

Fusion joins light nuclei into a heavier one. The Sun runs on it. Four hydrogen nuclei become one helium-4 nucleus over several steps, releasing about 26.7 MeV and two positrons along the way.

The reaction engineers chase on Earth pairs deuterium with tritium, the hydrogen isotopes of mass 2 and mass 3.

²₁H + ³₁H → ⁴₂He + ¹₀n (17.6 MeV)

Spread 17.6 MeV over 5 nucleons and you get about 3.5 MeV each. Spread 200 MeV over 236 nucleons and you get about 0.85 MeV each. Per unit of mass, fusion is the bigger payout by a factor of four.

The obstacle is repulsion. Both nuclei are positive, so they fly apart unless they arrive at tremendous speed. That means temperatures near 100 million kelvin, plus some way to hold the hot plasma together long enough. Magnetic fields and laser compression are the two routes being built. A fusion plant feeding a power grid is still ahead of us.

Why fusion fuel is so cheap. Deuterium is already in seawater, about one hydrogen atom in 6,400. Tritium can be bred from lithium inside the reactor itself. The fuel is almost free, and it is the machine around it that is hard.

Key points

Energy is released whenever nucleons move toward the binding energy peak at iron-56.
Fission splits a heavy nucleus and releases about 200 MeV per event.
Spare neutrons keep a chain reaction alive, and the multiplication factor k says whether it grows, holds or fades.
A reactor holds k at 1 with a moderator, control rods and a coolant loop.
Fusion gives about 3.5 MeV per nucleon against 0.85 MeV for fission, but it needs temperatures near 100 million kelvin.

Interactive Split a heavy nucleus and fuse light ones, and compare the energy. Open full ↑

Section 6Radiation, detection and uses

Counting decays and counting dose

Radiation raises two separate questions. How busy is the source, and how much energy did a body take in? Each one has its own unit, and mixing the two is the commonest mistake here.

The busyness of a source is its activity, measured in becquerels, where one becquerel is one decay per second. The older unit is the curie, which is 3.7 × 10¹⁰ Bq, the activity of one gram of radium-226.

Energy taken in by tissue is the absorbed dose, measured in grays, where one gray is one joule absorbed per kilogram. A gray of alpha particles harms far more than a gray of gamma rays, because the alpha dumps its energy along a very short track. So the gray is multiplied by a quality factor to give the dose equivalent in sieverts. Beta and gamma rate a 1, neutrons about 10, and alpha a 20.

Quantity	SI unit	Meaning	Older unit
Activity	becquerel, Bq	1 decay per second	1 curie = 3.7 × 10¹⁰ Bq
Absorbed dose	gray, Gy	1 joule absorbed per kilogram	1 rad = 0.01 Gy
Dose equivalent	sievert, Sv	gray × quality factor	1 rem = 0.01 Sv

Worked example · adding two kinds of dose

A hand absorbs 2.0 mGy from an alpha emitter and 3.0 mGy from a gamma source. What is the total dose equivalent?

Weight each dose by its quality factor

Alpha rates a 20, so 2.0 mGy × 20 = 40 mSv. Gamma rates a 1, so 3.0 mGy × 1 = 3.0 mSv.

Add the weighted doses

Adding the two together gives 40 mSv + 3.0 mSv = 43 mSv of dose equivalent.

Put it in context

Natural background runs about 3.1 mSv a year, so 43 mSv ÷ 3.1 mSv is about 14 years' worth in one exposure. Notice that the alpha dose was smaller in grays yet counted for far more in sieverts.

Detecting what you cannot see

A Geiger-Müller counter is a sealed tube of gas with a high-voltage wire down the middle. A ray passing through knocks electrons off the gas atoms, those electrons rush to the wire and free more on the way, and each pulse is counted as a click.

A scintillation counter works on light instead. Radiation strikes a crystal, the crystal flashes, and a sensor counts the flashes and measures their brightness. Brightness reveals the energy of the ray, so this instrument can name the isotope. Workers also wear a dosimeter, which keeps a running total of the dose taken.

What radiation does inside tissue

These rays are called ionizing because they strip electrons off the molecules they pass. A cell is mostly water, so most hits split water into reactive fragments called free radicals, which then attack DNA, proteins and membranes nearby. A cell can usually repair one broken DNA strand, but repairs sometimes fail or finish wrong, and that is where the long-term risk sits.

Where the source sits matters enormously. Alpha emitters outside the body are almost harmless, because dead skin stops the particles before they reach anything living. Breathe them in and they become the worst case, since all their energy lands inside living tissue. Radon seeping into a basement works this way, and it is a leading cause of lung cancer.

Figure 21.6 · Penetrating power runs opposite to charge and mass. Gamma rays are never stopped completely, only cut down, so gamma shielding is quoted as the thickness that halves the beam.

Doses worth memorising. Natural background averages about 3.1 mSv a year, mostly from radon and from the potassium-40 in your own body. A chest X-ray adds about 0.1 mSv and a chest CT about 7 mSv. The yearly limit for radiation workers is 50 mSv. A single dose near 1 Sv brings on radiation sickness, and about 4 Sv is fatal to half of those exposed without treatment.

Protection comes down to three words. Spend less time near the source, keep more distance from it, and put shielding in between. Distance is the cheapest, because intensity falls with the square of the distance.

Worked example · backing away from a source

A survey meter reads 48 µSv/h at 1.0 m from a small gamma source. What will it read at 4.0 m?

Find the distance ratio and square it

Dividing the new distance by the old one gives 4.0 m ÷ 1.0 m = 4.0, and intensity falls as 1 ÷ d², so divide the reading by 4.0² = 16.

Do the division

Dividing the close-up reading by that factor gives 48 µSv/h ÷ 16 = 3.0 µSv/h.

Check the logic

Four times the distance gives one sixteenth the dose rate, so two steps back can do more for you than a slab of lead.

Radiation put to work

A radioactive tracer is a short-lived isotope stuck onto a molecule the body already handles. Technetium-99m has a half-life of 6.01 hours and gives a clean gamma ray, so it maps bone, heart and kidney function. Fluorine-18 lasts 110 minutes and emits positrons, and on a sugar molecule it shows a PET scanner which tissues burn the most fuel.

Treatment uses much larger doses, aimed to kill. Iodine-131 is swallowed, and the thyroid pulls iodine from the blood whatever isotope it is, so its beta particles destroy diseased thyroid tissue from within. Cobalt-60 gamma beams treat tumours from outside, and tiny iodine-125 seeds can be placed inside one.

Half-life choice is part of the design. The isotope must last long enough to reach its target and finish the job, then fade fast so the patient carries no source for months. Outside the clinic the same rays sterilise surgical supplies, gauge the thickness of rolled steel and trip the americium-241 smoke alarm on your ceiling.

Key points

Becquerels measure the source, grays measure energy absorbed, and sieverts measure expected harm.
Multiply grays by the quality factor to get sieverts, and alpha radiation carries a factor of 20.
Geiger-Müller tubes count rays, while scintillation counters also measure their energy.
Ionizing radiation mostly damages tissue by splitting water into free radicals that attack DNA.
Time, distance and shielding are the three defences, and intensity falls as 1 ÷ d².
Tracers use short half-lives and gamma or positron emitters, while treatment uses beta and gamma emitters at higher dose.

Interactive Shield alpha, beta and gamma and compare doses and medical uses. Open full ↑

The chapter in one breath

A nuclide is set by its proton count Z and its mass number A, and N = A − Z.
Stable nuclides lie in the band of stability, and nothing above Z = 83 is stable.
The parts of a nucleus weigh more than the nucleus, and E = mc² turns that mass defect into its binding energy.
Binding energy per nucleon peaks near iron-56 at about 8.79 MeV.
Alpha decay cuts A by 4 and Z by 2, beta raises Z by 1, and positron emission and electron capture each lower Z by 1.
Decay is first order, so one half-life always halves what is left and k = 0.693 ÷ t_1/2.
Carbon-14 at 5,730 years dates what once lived, and uranium-238 at 4.5 billion years dates rock.
Nuclear equations balance on mass number and atomic number only, which lets elements change.
Fission splits heavy nuclei for about 200 MeV each, and spare neutrons carry a chain reaction that k keeps in check.
Fusion pays about 3.5 MeV per nucleon but demands temperatures near 100 million kelvin.
Becquerels measure a source, grays measure absorbed energy, and sieverts measure expected harm.

ReferenceWords worth knowing

Twenty terms carry most of this chapter. Read down the list once now, then come back to it while you work the questions below.

Alpha particle: A helium-4 nucleus thrown out of a decaying nucleus, written ⁴₂He and carrying a 2+ charge.
Band of stability: The narrow stripe on a neutron-against-proton plot where every stable nuclide falls.
Becquerel: The SI unit of activity, equal to one decay per second. 1 curie = 3.7 × 10¹⁰ Bq.
Beta particle: A fast electron fired from a nucleus when a neutron turns into a proton. Written ⁰₋₁e.
Binding energy: The energy released when a nucleus forms, and the energy needed to take it apart again.
Chain reaction: A sequence in which neutrons from one fission trigger more fissions, so the reaction feeds itself.
Critical mass: The smallest amount of fuel, in a close enough shape, that can keep a chain reaction going.
Electron capture: Decay in which a nucleus pulls in one of its own inner electrons, lowering Z by 1 and leaving A alone.
Fission: The splitting of a heavy nucleus into two lighter nuclei plus spare neutrons.
Fusion: The joining of light nuclei into a heavier one, which is the energy source of every star.
Gamma ray: A very short wavelength photon released by an excited nucleus. It changes neither A nor Z.
Gray: The SI unit of absorbed dose, one joule of radiation energy taken in per kilogram of tissue.
Half-life: The time needed for half the unstable nuclei in a sample to decay. Written t_1/2.
Magic number: A nucleon count of 2, 8, 20, 28, 50, 82 or 126 that makes a nucleus unusually stable.
Mass defect: The difference between the mass of a nucleus and the total mass of its separated nucleons.
Nuclide: One exact kind of nucleus, named by its proton count and its neutron count.
Positron: The antiparticle of the electron, with the same mass and a 1+ charge. Written ⁰₊₁e.
Radiometric dating: Finding an age from the fraction of a radioactive parent isotope still left in a sample.
Sievert: The unit of dose equivalent, found by multiplying the absorbed dose in grays by a quality factor.
Transmutation: Any change of one element into another, whether by decay or by firing a particle at a nucleus.

Check yourself

1. Radon-222 has Z = 86 and decays by alpha emission. Write the balanced equation and name the daughter.

²²²₈₆Rn → ²¹⁸₈₄Po + ⁴₂He Mass: 222 − 4 = 218. Charge: 86 − 2 = 84, and element 84 is polonium, so the daughter is polonium-218.

2. Strontium-90 has Z = 38 and is a beta emitter. What does it become, and why does strontium-90 end up in bone?

Yttrium-90. Beta decay leaves A at 90 and raises Z from 38 to 39. Strontium sits directly below calcium in group 2, so the body treats it like calcium and files it in bone.

3. A wooden handle from a grave holds 25.0 percent of the carbon-14 of living wood. How old is it?

About 11,500 years. 25.0 percent is two halvings, since 100 → 50 → 25. Two half-lives is 2 × 5,730 yr = 11,460 yr, which rounds to 1.15 × 10⁴ years.

4. A vial holds 100.0 mg of iodine-131, half-life 8.02 days. How much iodine-131 is left after 24.06 days?

12.5 mg. 24.06 d ÷ 8.02 d = 3.00 half-lives. So 100.0 mg × (1/2)³ = 100.0 mg × 0.125 = 12.5 mg.

5. Uranium-238 has a half-life of 4.5 × 10⁹ years. Find its decay constant.

1.5 × 10⁻¹⁰ yr⁻¹. k = 0.693 ÷ t_1/2 = 0.693 ÷ (4.5 × 10⁹ yr) = 1.54 × 10⁻¹⁰ yr⁻¹. Only about one atom in six billion decays in any given year.

6. A deuterium atom weighs 2.014102 u. Hydrogen-1 weighs 1.007825 u and a neutron weighs 1.008665 u. Find the binding energy per nucleon, using 1 u = 931.5 MeV.

1.112 MeV per nucleon. Parts: 1.007825 u + 1.008665 u = 2.016490 u. Mass defect: 2.016490 − 2.014102 = 0.002388 u. Energy: 0.002388 u × 931.5 MeV/u = 2.224 MeV. Per nucleon: 2.224 ÷ 2 = 1.112 MeV, far below the 8.79 MeV of iron-56.

7. A technician absorbs 0.50 mGy of neutron radiation, quality factor 10. What is the dose equivalent, and how does it compare with 0.50 mGy of gamma?

5.0 mSv. 0.50 mGy × 10 = 5.0 mSv. The same 0.50 mGy of gamma has a quality factor of 1, so it counts as only 0.50 mSv. Equal energy, ten times the expected harm.

8. A meter reads 90 µSv/h at 2.0 m from a gamma source. What is the reading at 6.0 m?

10 µSv/h. The distance triples, since 6.0 ÷ 2.0 = 3.0. Intensity falls as 1 ÷ d², so divide by 3.0² = 9. Then 90 µSv/h ÷ 9 = 10 µSv/h.

9. Why does a stable heavy nucleus need far more neutrons than protons, while carbon-12 gets by with an even split?

Because electrical repulsion grows faster than the strong force reaches. The strong force only pulls on close neighbours, but every proton repels every other proton across the whole nucleus. Extra neutrons add pull without adding repulsion, so the band of stability bends above the N = Z line.

10. Positron emission and electron capture both lower Z by 1 and leave A alone. How can you tell them apart?

By what leaves the atom. Positron emission throws a positron out, which soon annihilates with an electron and makes two 0.511 MeV photons. Electron capture pulls an inner electron in and gives off a characteristic X-ray as an outer electron fills the hole. Heavier nuclides favour capture.

11. Fission of uranium-235 releases about 200 MeV and fusion of deuterium with tritium releases 17.6 MeV. Why is fusion still called the richer reaction?

Because you have to divide by the nucleons involved. Fission: 200 MeV ÷ 236 nucleons = 0.85 MeV each. Fusion: 17.6 MeV ÷ 5 nucleons = 3.5 MeV each. Per unit of mass, fusion pays about four times as much.

12. An alpha emitter is far less dangerous than a gamma emitter on a bench, yet far more dangerous once inhaled. Explain both halves of that.

It comes down to where the energy lands. Alpha particles cannot get through dead skin, so an outside source does nothing. Inhaled, the same particles dump all their energy into a tiny volume of living tissue, and their quality factor of 20 reflects that. Gamma rays pass through the body either way, which is why they need lead or thick concrete.

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