PhysicsICSE Class 10 · Radioactivity

Balancing Nuclear Equations: Missing Particles, Decay Chains and Fission (ICSE Class 10)

Updated 16 October 2026 · Fact-checked

In every nuclear equation the total of the mass numbers (top) and the total of the atomic numbers (bottom) are the same on both sides. To find a missing nucleus or particle, add up each total on the left, subtract the known terms on the right, and read off A and Z, multiplying by any coefficient.

Add the top numbers (A) and the bottom numbers (Z) on each side; both totals must match, so a missing term is the difference.

Illustration drawn for QuestionBuddy.

Understand Balancing Nuclear Equations

Two rules. In any nuclear reaction or decay:
1. Mass numbers balance: the sum of the top numbers ( $A$ ) on the left equals the sum on the right (the number of nucleons is conserved).
2. Atomic numbers balance: the sum of the bottom numbers ( $Z$ ) on the left equals the sum on the right (charge is conserved).

The energy released is written as '+ energy' or shown by γ; it does not change either total.

Symbols you must know.

Particle	Symbol	A	Z
α particle (helium nucleus)	$_{2}^{4} He$	4	2
β particle (electron)	$_{- 1}^{0} e$	0	−1
γ ray	$γ$ (or $_{0}^{0} γ$ )	0	0
neutron	$_{0}^{1} n$	1	0
proton	$_{1}^{1} H$ or $_{1}^{1} p$	1	1
deuteron (heavy hydrogen nucleus)	$_{1}^{2} H$	2	1

Finding a missing term.

Unknown nucleus: $A = (total A on left) - (known A on right)$ ; the same for $Z$ . Then $Z$ tells you the element.
Unknown particle: work out its $A$ and $Z$ and match the table: (1, 0) neutron, (1, 1) proton, (4, 2) α, (0, −1) β, (0, 0) γ.
A coefficient multiplies both numbers: $3_{0}^{1} n$ adds 3 to the top total and 0 to the bottom.
β has $Z = - 1$ , so it lowers the right-hand total by 1; that is why the daughter's $Z$ goes up by 1.

Four kinds of equation you will meet.
1. α decay: $_{Z}^{A} X \to_{Z - 2}^{A - 4} Y +_{2}^{4} He$
2. β decay: $_{Z}^{A} X \to_{Z + 1}^{A} Y +_{- 1}^{0} e$
3. Fission: $_{92}^{235} U +_{0}^{1} n \to$ two medium nuclei $+ x_{0}^{1} n$ (left totals: 236 and 92)
4. Fusion and bombardment: light nuclei or particles combine. (The syllabus names α and β decay, fission and fusion; bombardment reactions are not named, but the 2027 specimen paper asks one, so they are practised here with the same rules.) An example is the reaction that makes carbon-14 in the air when cosmic-ray neutrons hit nitrogen:
$_{7}^{14} N +_{0}^{1} n \to_{6}^{14} C +_{1}^{1} H$

Decay chains. Treat each emission as a separate equation, writing the daughter in full before the next step. The emitted particles are not reactants for the next step.

Counting α and β in a series. Only α changes $A$ :

$n_{α} = \frac{A _{1} - A _{2}}{4}$
Then $n_{β} = 2 n_{α} - (Z_{1} - Z_{2})$ , because each α lowers $Z$ by 2 and each β raises it by 1.

Final check. Always add up both totals on each side once more before writing the answer.

Balancing Nuclear Equations: formulas and key facts

Balancing rules: $Σ A_{left} = Σ A_{right} ∣ Σ Z_{left} = Σ Z_{right}$; Energy and γ change neither.
Particle symbols: $α =_{2}^{4} He ∣ β =_{- 1}^{0} e ∣ n =_{0}^{1} n ∣ p =_{1}^{1} H ∣ γ =_{0}^{0} γ$; Read off a missing particle from its A and Z.
Counting emissions: $n_{α} = \frac{A _{1} - A _{2}}{4} ∣ n_{β} = 2 n_{α} - (Z_{1} - Z_{2})$; Only α changes A.
Fission left side: $_{92}^{235} U +_{0}^{1} n : A = 236, Z = 92$; Never forget the absorbed neutron.

How to solve Balancing Nuclear Equations questions

Use the same four steps for every nuclear equation, whether it is a decay, a fission or a bombardment.

1Write every term with both numbers, including particles: $_{0}^{1} n$ , $_{2}^{4} He$ , $_{- 1}^{0} e$ .
2Add the top numbers on the left; add the bottom numbers on the left.
3Subtract the known terms on the right (multiplied by their coefficients) to get the unknown A and Z.
4Name the unknown: by Z for a nucleus (look up the element), or by (A, Z) for a particle.
5For a chain, repeat for each step, writing every daughter nucleus.
6Check: re-add both totals on each side.

Balancing Nuclear Equations in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

How carbon-14 is made

Suppose you read that the carbon-14 used in dating is made all the time high in the atmosphere, where neutrons from cosmic rays strike nitrogen-14 nuclei.

Complete $_{7}^{14} N +_{0}^{1} n \to_{6}^{14} C + X$ and identify X.

Try it first, then show the solutionSolution

Left: $A = 14 + 1 = 15$ , $Z = 7 + 0 = 7$ .
X: $A = 15 - 14 = 1$ , $Z = 7 - 6 = 1$ .
(1, 1) is a proton, $_{1}^{1} H$ .

Answer: X is a proton, ¹₁H.

Memory hook: Neutron in, proton out: nitrogen becomes carbon.

The discovery of the neutron

Imagine reading that in 1932 James Chadwick fired α particles at beryllium-9 and found a new, uncharged particle coming out, while carbon-12 was formed.

Use balancing to show that the new particle was a neutron.

Try it first, then show the solutionSolution

$_{4}^{9} Be +_{2}^{4} He \to_{6}^{12} C + X$ .
Left: $A = 13$ , $Z = 6$ . X: $A = 13 - 12 = 1$ , $Z = 6 - 6 = 0$ .
A = 1, Z = 0: a neutron, $_{0}^{1} n$ .

Answer: X = ¹₀n, the neutron.

Memory hook: Z = 0 means no charge: neutron.

Tricks to remember Balancing Nuclear Equations

Short memory aids. Say each one aloud once, then use it on a question the same day.

Top with top, bottom with bottom
Never mix A and Z: two separate sums, two separate answers.
Use it when: Every equation.
Read the pair
(1,0) n, (1,1) p, (4,2) α, (0,−1) β, (0,0) γ.
Use it when: Naming a missing particle.
Coefficient counts twice
A number in front multiplies both A and Z.
Use it when: Fission neutrons and fusion products.
One step at a time
In a chain, write each daughter before the next arrow.
Use it when: Decay series.

Balancing Nuclear Equations: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Identify the particle · 1 mark · MCQBoard level

In the reaction $_{13}^{27} Al +_{0}^{1} n \to_{11}^{24} Na + X$ , X is:
(a) a proton
(b) a neutron
(c) an α particle
(d) a β particle

Try it first, then show the solutionSolution

X: $A = 28 - 24 = 4$ , $Z = 13 - 11 = 2$ : $_{2}^{4} He$ .

Answer: (c) an α particle

Example 2: α decay of neptunium · 2 marks · Short answerBoard level

Neptunium-237 ( $_{93}^{237} Np$ ) is an α emitter. Write the balanced equation, naming the product as Pa, and find the number of neutrons in the product.

Try it first, then show the solutionSolution

$_{93}^{237} Np \to_{91}^{233} Pa +_{2}^{4} He$ (A: 237 = 233 + 4; Z: 93 = 91 + 2). (1)
Neutrons: $233 - 91 = 142$ . (1)

Answer: ²³⁷₉₃Np → ²³³₉₁Pa + ⁴₂He; 142 neutrons

Example 3: Rutherford's transmutation · 2 marks · Short answerHarder board level

When α particles strike nitrogen-14, oxygen-17 and another particle are produced: $_{7}^{14} N +_{2}^{4} He \to_{8}^{17} O + X$ . Identify X and show that the equation balances.

Try it first, then show the solutionSolution

X: $A = 18 - 17 = 1$ , $Z = 9 - 8 = 1$ : a proton, $_{1}^{1} H$ . (1)
Check: $14 + 4 = 17 + 1 = 18$ ; $7 + 2 = 8 + 1 = 9$ . (1)

Answer: X = ¹₁H (proton)

Example 4: Two routes to radium · 2 marks · Short answerHarder board level

Actinium-227 ( $_{89}^{227} Ac$ ) changes into radium-223 ( $_{88}^{223} Ra$ ) by emitting one α particle and one β particle. The emissions can happen in either order. Name the intermediate nucleus for each order (Z = 87 is francium, Fr; Z = 90 is thorium, Th), and write both equations for the order in which the β particle comes first.

Try it first, then show the solutionSolution

β first: $_{90}^{227} Th$ is formed; α first: $_{87}^{223} Fr$ is formed. (1)
$_{89}^{227} Ac \to_{90}^{227} Th +_{- 1}^{0} e$ , then $_{90}^{227} Th \to_{88}^{223} Ra +_{2}^{4} He$ . (1)

Answer: Th-227 if β comes first; Fr-223 if α comes first. ²²⁷₈₉Ac → ²²⁷₉₀Th + ⁰₋₁e; ²²⁷₉₀Th → ²²³₈₈Ra + ⁴₂He

Example 5: Three α and two β · 3 marks · Short answerHarder board level

A nucleus $_{Z}^{A} X$ emits three α particles and two β particles in some order. Write the mass number and atomic number of the final nucleus, and state whether the order of the emissions matters for the final answer.

Try it first, then show the solutionSolution

Three α: A falls by $3 \times 4 = 12$ ; Z falls by $3 \times 2 = 6$ . (1)
Two β: A unchanged; Z rises by 2. Final: $_{Z - 4}^{A - 12} Y$ . (1)
The order changes the intermediate nuclei but not the final A and Z, because the totals simply add. (1)

Answer: A − 12, Z − 4; the order does not change the final nucleus.

Example 6: Counting a long series · 3 marks · Short answerHOTS / competency

After a long chain of α and β decays, neptunium-237 ( $_{93}^{237} Np$ ) ends up as bismuth-209 ( $_{83}^{209} Bi$ ). Work out (a) the α count and (b) the β count. (c) A classmate claims the chain could contain exactly 6 α particles if more β particles were emitted. Use the mass numbers to show that this is impossible.

Try it first, then show the solutionSolution

(a) $n_{α} = \frac{237 - 209}{4} = 7$ . (1)
(b) Seven α lower Z by 14, to 79; Z must rise to 83, so $n_{β} = 4$ . (1)
(c) Six α would lower A only to $237 - 24 = 213$ ; β particles do not change A, so the chain could never reach A = 209. (1)

Answer: (a) 7 (b) 4 (c) 6 α leave A = 213, and β cannot change A

Example 7: Case study: a student's fission equation · 4 marks · Case-basedHOTS / competency

A student balances a fission of uranium-235 that gives tin-131 ( $_{50}^{131} Sn$ ) and molybdenum-103 ( $_{42}^{103} Mo$ ).
(a) Find the correct number of neutrons released.
(b) The student forgot the neutron absorbed on the left. What number of neutrons would the student get?
(c) Check the atomic numbers.
(d) Write the correct equation.

Try it first, then show the solutionSolution

(a) Left mass total $235 + 1 = 236$ ; $x = 236 - 131 - 103 = 2$ . (1)
(b) Using 235 instead: $235 - 131 - 103 = 1$ , a wrong answer. (1)
(c) $92 = 50 + 42$ ✓. (1)
(d) $_{92}^{235} U +_{0}^{1} n \to_{50}^{131} Sn +_{42}^{103} Mo + 2_{0}^{1} n + energy$ . (1)

Answer: (a) 2 (b) 1 (c) 92 = 50 + 42 (d) As above

Where students lose marks in Balancing Nuclear Equations

The slips examiners see most often, and what to write instead.

Mistake: Ignoring the coefficient of neutrons
CISCE examiners' comments, 2024 Q3(v): completing a fission equation, candidates wrote the mass number of barium as 143 instead of 141, and ignored the 3 neutrons, which gave the wrong atomic number for krypton.
Wrong working
Counting 3 ¹₀n as 1 (or leaving it out) in the mass total.
Right working
3 ¹₀n adds 3 to the top total and 0 to the bottom total.
Do this: Multiply each particle by its coefficient.
Usually costs: 1–2 marks.
Mistake: Working a chain in one go
CISCE examiners' comments, 2024 Q7(ii): candidates were confused by a sequential nuclear equation and could not work backwards through more than one emission.
Wrong working
Undoing the α and the β together and getting the wrong parent.
Right working
Write each step as a separate equation; emitted particles are not reactants for the next step.
Do this: One arrow, one equation.
Usually costs: 1–2 marks.
Mistake: Forgetting the absorbed neutron; wrong neutron notation
CISCE examiners' comments, 2025 Q8(ii): candidates wrote the mass number as 235 instead of adding the absorbed neutron, and wrote the neutron in the wrong notation.
Wrong working
²³⁵₉₂U + n → ²³⁵₉₂U
Right working
²³⁵₉₂U + ¹₀n → ²³⁶₉₂U
Do this: Write the neutron as ¹₀n and add its 1.
Usually costs: 1 mark.
Mistake: Changing A in β decay
CISCE examiners' comments, 2024 Q1(iv): candidates added 1 to the number of nucleons after β emission because the atomic number rises.
Wrong working
'After β emission the number of nucleons goes up by 1.'
Right working
A is unchanged (β has A = 0).
Do this: β is ⁰₋₁e: it changes only the bottom total.
Usually costs: 1 mark.

Exam tips

Show both sums (top and bottom) in your working; examiners award a mark for the balancing.
Write particles in full symbol form, not just 'n' or 'e'.
Name the element from Z when the question gives the symbol choices.
In fission, start every time with 236 and 92 on the left.

Practice questions on Balancing Nuclear Equations

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The correct symbol for a β particle in a nuclear equation is:
(a) $_{1}^{0} e$
(b) $_{- 1}^{0} e$
(c) $_{- 1}^{1} e$
(d) $_{2}^{4} He$
Hint
An electron has almost no mass and charge −1.
Answer
(b) $_{- 1}^{0} e$
Q2 · 1 mark · Very short answerBoard level
Potassium-40 decays by β emission. Complete: $_{19}^{40} K \to_{?}^{40} Ca +_{- 1}^{0} e$
Hint
Bottom: 19 = Z − 1.
Answer
Z = 20: $_{20}^{40} Ca$ .
Q3 · 1 mark · Very short answerBoard level
Identify X: $_{5}^{10} B +_{0}^{1} n \to_{3}^{7} Li + X$
Hint
Left totals: 11 and 5.
Answer
A = 4, Z = 2: an α particle, $_{2}^{4} He$ .
Q4 · 2 marks · Short answerBoard level
Write the A and Z of a neutron and of a proton, and explain why a neutron adds nothing to the bottom total.
Hint
Charge.
Answer
Neutron $_{0}^{1} n$ (A = 1, Z = 0); proton $_{1}^{1} H$ (A = 1, Z = 1). The bottom number is the charge (number of protons); a neutron has no charge, so it adds 0.
Q5 · 2 marks · Short answerHarder board level
Complete and name X: $_{13}^{27} Al +_{2}^{4} He \to_{15}^{30} P + X$
Hint
Left: 31 and 15.
Answer
A = 31 − 30 = 1, Z = 15 − 15 = 0: X is a neutron, $_{0}^{1} n$ .
Q6 · 2 marks · Short answerHarder board level
Polonium-216 ( $_{84}^{216} Po$ ) is formed when its parent nucleus emits one α particle. Find A and Z of the parent and name it (Z = 86 is radon, Rn).
Hint
Add the α particle back: A + 4, Z + 2.
Answer
Parent: A = 216 + 4 = 220, Z = 84 + 2 = 86: radon-220, $_{86}^{220} Rn \to_{84}^{216} Po +_{2}^{4} He$ .
Q7 · 2 marks · Short answerHarder board level
Identify X: $_{11}^{23} Na +_{1}^{2} H \to_{11}^{24} Na + X$ . How is the product related to the target nucleus?
Hint
Same Z?
Answer
X: A = 25 − 24 = 1, Z = 12 − 11 = 1: a proton. Na-24 and Na-23 have the same Z: they are isotopes.
Q8 · 2 marks · Short answerHarder board level
In a fission, $_{92}^{235} U$ absorbs a neutron and splits into $_{57}^{147} La$ , $_{35}^{87} Br$ and some neutrons. A student writes '3 neutrons'. Is the student right? Give the correct number with working.
Hint
Add the absorbed neutron.
Answer
No. Left total $= 235 + 1 = 236$ ; $236 - 147 - 87 = 2$ neutrons. (With 3 neutrons the right side would total 237.) Atomic: $57 + 35 = 92$ ✓.
Q9 · 2 marks · Short answerHOTS / competency
A nucleus $_{Z}^{A} X$ changes into $_{Z - 2}^{A - 8} Y$ by α and β emissions. Find the number of each.
Hint
Use A first.
Answer
$n_{α} = \frac{8}{4} = 2$ , lowering Z by 4; Z actually falls by 2, so $n_{β} = 2$ .
Q10 · 2 marks · Short answerHOTS / competency
A student writes: $_{5}^{11} B +_{2}^{4} He \to_{7}^{14} N +_{- 1}^{0} e$ . Show that the equation does not balance and correct it.
Hint
Check both totals.
Answer
Left: A = 11 + 4 = 15, Z = 5 + 2 = 7. Right: A = 14 + 0 = 14, Z = 7 − 1 = 6. Neither balances. The missing particle needs A = 15 − 14 = 1 and Z = 7 − 7 = 0, a neutron: $_{5}^{11} B +_{2}^{4} He \to_{7}^{14} N +_{0}^{1} n$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.A and Z of an α particle?Show answer
4 and 2.
2.A and Z of a β particle?Show answer
0 and −1.
3.Particle with A = 1, Z = 0?Show answer
Neutron.
4.Particle with A = 1, Z = 1?Show answer
Proton.
5.Number of α in a series from A₁ to A₂?Show answer
(A₁ − A₂)/4.

Balancing Nuclear Equations: frequently asked questions

Why do both mass number and atomic number have to balance?

The number of nucleons (mass number) and the total charge (atomic number) are both conserved in nuclear reactions. Energy is released, but it does not change either total.

Do I need to know artificial transmutation reactions for ICSE?

The syllabus names α and β decay, fission and fusion. Bombardment reactions use exactly the same balancing rules, and the 2027 specimen paper includes one (lithium-6 struck by a neutron), so practising them helps.

How do I name the new element?

Its atomic number Z decides the element. Questions usually give the symbol or a short list; match Z to it.