Nuclear Equations, Decay Series and the Stability of Nuclei – Radioactivity (Senior 6 Chemistry) | Ganzaa

Radioactivity

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Senior 6 · Chemistry

Radioactivity

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Lesson 2 of 5

Nuclear Equations, Decay Series and the Stability of Nuclei

25 min

From uranium to lead, one step at a time

A piece of uraninite, the black ore from which uranium is mined, is a slow chemical factory. Each uranium-238 nucleus in it will, eventually, become a nucleus of lead-206, but not in one jump. It passes through thirteen radioactive nuclides on the way, including radium, which the Curies extracted from tonnes of such ore, and radon, a gas that seeps out of uranium-bearing rocks. To follow such changes you need to write nuclear equations, and to understand why some nuclei are stable and others are not.

Definition

Nuclear equation: an equation showing the nuclides and particles before and after a nuclear reaction, in which the total mass number and the total atomic number (charge) are the same on both sides. Example: ${}^{226}_{88}\mathrm{Ra} \rightarrow {}^{222}_{86}\mathrm{Rn} + {}^{4}_{2}\mathrm{He}$ .

Decay series: a chain of successive decays by which a heavy radioactive nuclide changes, through a series of radioactive daughters, into a stable nuclide. Example: the uranium-238 series ends at lead-206.

Artificial transmutation: changing one element into another by bombarding its nuclei with particles. Example: ${}^{14}_{7}\mathrm{N} + {}^{4}_{2}\mathrm{He} \rightarrow {}^{17}_{8}\mathrm{O} + {}^{1}_{1}\mathrm{H}$ (Rutherford, 1919).

Band (belt) of stability: the narrow region of a plot of neutron number against proton number in which the stable nuclides lie.

Writing and balancing nuclear equations

Two totals must balance: the mass numbers (top) and the atomic numbers or charges (bottom). Electrons in the shells are ignored. Use these symbols:

Particle	proton	neutron	electron (β⁻)	positron (β⁺)	alpha	deuteron	gamma
Symbol	${}^{1}_{1}\mathrm{p}$ or ${}^{1}_{1}\mathrm{H}$	${}^{1}_{0}\mathrm{n}$	${}^{0}_{-1}\mathrm{e}$	${}^{0}_{+1}\mathrm{e}$	${}^{4}_{2}\mathrm{He}$	${}^{2}_{1}\mathrm{H}$	${}^{0}_{0}\gamma$

Artificial transmutation is how most radioisotopes used in medicine and industry are made, usually by bombarding a stable nuclide with neutrons in a reactor:

${}^{59}_{27}\mathrm{Co} + {}^{1}_{0}\mathrm{n} \rightarrow {}^{60}_{27}\mathrm{Co}$ (cobalt-60, for radiotherapy and sterilising)
${}^{32}_{16}\mathrm{S} + {}^{1}_{0}\mathrm{n} \rightarrow {}^{32}_{15}\mathrm{P} + {}^{1}_{1}\mathrm{H}$ (phosphorus-32, a tracer)
${}^{14}_{7}\mathrm{N} + {}^{1}_{0}\mathrm{n} \rightarrow {}^{14}_{6}\mathrm{C} + {}^{1}_{1}\mathrm{H}$ (carbon-14, made naturally in the upper atmosphere by neutrons from cosmic rays)
${}^{9}_{4}\mathrm{Be} + {}^{4}_{2}\mathrm{He} \rightarrow {}^{12}_{6}\mathrm{C} + {}^{1}_{0}\mathrm{n}$ (the reaction in which James Chadwick identified the neutron in 1932)

Worked example 1: completing nuclear equations

Problem. Identify X in each equation. (a) ${}^{131}_{53}\mathrm{I} \rightarrow {}^{131}_{54}\mathrm{Xe} + \mathrm{X}$ ; (b) ${}^{239}_{94}\mathrm{Pu} \rightarrow \mathrm{X} + {}^{4}_{2}\mathrm{He}$ ; (c) ${}^{27}_{13}\mathrm{Al} + {}^{4}_{2}\mathrm{He} \rightarrow {}^{30}_{15}\mathrm{P} + \mathrm{X}$ ; (d) ${}^{18}_{9}\mathrm{F} \rightarrow {}^{18}_{8}\mathrm{O} + \mathrm{X}$ .

Solution. Balance top and bottom separately.

(a) $A$ : $131 = 131 + A$ , so $A = 0$ ; $Z$ : $53 = 54 + Z$ , so $Z = - 1$ . X is ${}^{0}_{-1}\mathrm{e}$ , a β⁻ particle.

(b) $A = 239 - 4 = 235$ ; $Z = 94 - 2 = 92$ : ${}^{235}_{92}\mathrm{U}$ .

(c) $A = 27 + 4 - 30 = 1$ ; $Z = 13 + 2 - 15 = 0$ : a neutron, ${}^{1}_{0}\mathrm{n}$ . (This is how Irène Joliot-Curie and Frédéric Joliot made the first artificial radioisotope in 1934.)

(d) $A = 0$ ; $Z = 9 - 8 = + 1$ : ${}^{0}_{+1}\mathrm{e}$ , a positron. Fluorine-18 is the positron emitter used in PET scans.

The uranium-238 decay series

Figure 2.1 plots each member of the series by its atomic number and mass number. Every α decay moves the nuclide 2 to the left and 4 down; every β⁻ decay moves it 1 to the right. The chain ends at lead-206, whose nucleus is stable.

Figure 2.1: The uranium-238 decay series (gamma emissions not shown).

Worked example 2: counting the steps in a series

Problem. Thorium-232 decays through a series to lead-208. How many α and how many β⁻ particles are emitted per nucleus?

Solution. Only α decay changes $A$ , by 4 each time: $\frac{232 - 208}{4} = 6$ α particles. Six α decays would lower $Z$ from 90 to $90 - 12 = 78$ , but lead has $Z = 82$ , so $82 - 78 = 4$ β⁻ decays must have raised it. Overall: ${}^{232}_{90}\mathrm{Th} \rightarrow {}^{208}_{82}\mathrm{Pb} + 6\,{}^{4}_{2}\mathrm{He} + 4\,{}^{0}_{-1}\mathrm{e}$ . Check: $A$ : $208 + 24 = 232$ ; $Z$ : $82 + 12 - 4 = 90$ . ✓

Why are some nuclei stable?

Protons repel each other electrically; the strong nuclear force, which acts between all nucleons but only over very short distances, holds the nucleus together. Neutrons add to the strong force without adding repulsion. In light nuclei about one neutron per proton is enough ( $\frac{N}{Z} \approx 1$ , e.g. ${}^{12}_{6}\mathrm{C}$ , ${}^{16}_{8}\mathrm{O}$ ). In heavier nuclei the repulsion grows faster, so more neutrons are needed: $\frac{N}{Z}$ rises to about 1.5 at lead-208 ( $\frac{126}{82} = 1.54$ ). Beyond bismuth ( $Z = 83$ ) no number of neutrons is enough, and every nuclide is radioactive.

Figure 2.2: The band of stability (sketch).

The position of an unstable nuclide relative to the band predicts how it decays, because each decay moves it towards the band:

Above the band (too many neutrons, $\frac{N}{Z}$ too high): β⁻ emission turns a neutron into a proton, lowering $\frac{N}{Z}$ .
Below the band (too few neutrons): positron emission or electron capture turns a proton into a neutron, raising $\frac{N}{Z}$ .
Very heavy ( $Z > 83$ ): α emission removes 2 protons and 2 neutrons, reducing the size of the nucleus.

Two further patterns: nuclides with even numbers of both protons and neutrons are the most common stable nuclides, while only a handful with odd Z and odd N are stable (such as ${}^{2}_{1}\mathrm{H}$ and ${}^{14}_{7}\mathrm{N}$ ); and nuclei with 2, 8, 20, 28, 50, 82 or 126 protons or neutrons (the magic numbers) are especially stable. Lead-208, with 82 protons and 126 neutrons, is "doubly magic".

Worked example 3: predicting the mode of decay

Problem. Predict the most likely mode of decay of (a) ${}^{14}_{6}\mathrm{C}$ , (b) ${}^{20}_{11}\mathrm{Na}$ , (c) ${}^{90}_{38}\mathrm{Sr}$ , (d) ${}^{241}_{95}\mathrm{Am}$ , and write the equation for (a).

Solution.

(a) $N = 8$ , $Z = 6$ : $\frac{N}{Z} = 1.33$ , above the value of about 1 for a light nucleus: β⁻. ${}^{14}_{6}\mathrm{C} \rightarrow {}^{14}_{7}\mathrm{N} + {}^{0}_{-1}\mathrm{e}$ .

(b) $N = 9$ , $Z = 11$ : $\frac{N}{Z} = 0.82$ , below the band: β⁺ or electron capture (it is in fact a positron emitter).

(c) $N = 52$ , $Z = 38$ : $\frac{N}{Z} = 1.37$ ; the stable strontium isotopes have 46 to 50 neutrons, so strontium-90 is neutron-rich: β⁻.

(d) $Z = 95 > 83$ : α emission.

Warning

Common mistakes

Balancing only the mass numbers. Check the bottom numbers (charges) as well: they decide which element the product is.
Writing a β⁻ particle as ${}^{0}_{+1}\mathrm{e}$ or a positron as ${}^{0}_{-1}\mathrm{e}$ . The subscript is the charge.
Using " $\frac{N}{Z} = 1$ means stable" for heavy nuclei. The stable ratio rises with $Z$ ; compare with the band, not with 1.
Counting β⁻ decays from the change in mass number. β⁻ decay does not change $A$ ; count α first, then fix $Z$ with β⁻.

Tip

For any series question: α = ΔA ÷ 4, then β⁻ = Z(final) − Z(start) + 2α. For Th-232 → Pb-208: α = 6, β⁻ = 82 − 90 + 12 = 4.

ChemistryUTAustin: Nuclear Stability, Band of Stability. Watch how the neutron-to-proton ratio of the stable nuclides rises with atomic number, and how a nuclide's position above or below the band predicts its decay.

✦

Practice

Complete: (a) ${}^{60}_{27}\mathrm{Co} \rightarrow {}^{60}_{28}\mathrm{Ni} + \_\_$ ; (b) ${}^{222}_{86}\mathrm{Rn} \rightarrow \_\_ + {}^{4}_{2}\mathrm{He}$ ; (c) ${}^{10}_{5}\mathrm{B} + {}^{1}_{0}\mathrm{n} \rightarrow {}^{7}_{3}\mathrm{Li} + \_\_$ .
Write the equation for the electron capture of ${}^{55}_{26}\mathrm{Fe}$ .
Uranium-235 decays through a series to lead-207. How many α and β⁻ particles are emitted per nucleus?
Using Figure 2.1, write the equations for the three steps from ${}^{238}_{92}\mathrm{U}$ to ${}^{234}_{92}\mathrm{U}$ .
Explain why the neutron-to-proton ratio of stable nuclei increases with atomic number.
Predict the decay mode of (a) ${}^{32}_{15}\mathrm{P}$ , (b) ${}^{18}_{9}\mathrm{F}$ , (c) ${}^{210}_{84}\mathrm{Po}$ , given that the stable isotopes are ${}^{31}_{15}\mathrm{P}$ and ${}^{19}_{9}\mathrm{F}$ .
A student claims that ${}^{238}_{92}\mathrm{U}$ changes to ${}^{206}_{82}\mathrm{Pb}$ by emitting 8 α particles only. Evaluate the claim using a balanced overall equation.

Show answersHide answers

(a) ${}^{0}_{-1}\mathrm{e}$ ; (b) ${}^{218}_{84}\mathrm{Po}$ ; (c) ${}^{4}_{2}\mathrm{He}$ .
${}^{55}_{26}\mathrm{Fe} + {}^{0}_{-1}\mathrm{e} \rightarrow {}^{55}_{25}\mathrm{Mn}$ .
α: $\frac{235 - 207}{4} = 7$ ; Z after 7 α: $92 - 14 = 78$ ; β⁻: $82 - 78 = 4$ .
${}^{238}_{92}\mathrm{U} \rightarrow {}^{234}_{90}\mathrm{Th} + {}^{4}_{2}\mathrm{He}$ ; ${}^{234}_{90}\mathrm{Th} \rightarrow {}^{234}_{91}\mathrm{Pa} + {}^{0}_{-1}\mathrm{e}$ ; ${}^{234}_{91}\mathrm{Pa} \rightarrow {}^{234}_{92}\mathrm{U} + {}^{0}_{-1}\mathrm{e}$ .
Proton–proton repulsion acts across the whole nucleus and grows roughly with $Z^{2}$ , while the strong force acts only between neighbouring nucleons; extra neutrons add attraction without repulsion, so heavier nuclei need proportionally more of them.
(a) One neutron more than the stable isotope: β⁻. (b) One neutron fewer than the stable isotope: β⁺ (or EC). (c) $Z > 83$ : α.
Wrong. 8 α would give $Z = 92 - 16 = 76$ , not 82. Six β⁻ decays are also needed: ${}^{238}_{92}\mathrm{U} \rightarrow {}^{206}_{82}\mathrm{Pb} + 8\,{}^{4}_{2}\mathrm{He} + 6\,{}^{0}_{-1}\mathrm{e}$ (A: $206 + 32 = 238$ ; Z: $82 + 16 - 6 = 92$ ).

Sources and credits

Image: Uraninite-3cm.jpg by 책읽는달팽, via Wikimedia Commons, licensed under CC BY-SA 3.0.
Video: Nuclear Stability, Band of Stability, ChemistryUTAustin, embedded from YouTube.
The text, tables, decay-series chart, band-of-stability sketch, worked examples and Practice items were written and drawn by Ganzaa.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1From uranium to lead, one step at a time

2Writing and balancing nuclear equations

3The uranium-238 decay series

4Why are some nuclei stable?

From uranium to lead, one step at a time

Writing and balancing nuclear equations

The uranium-238 decay series

Why are some nuclei stable?