IIT-JAM Nuclear Chemistry — Decay Kinetics, Binding Energy and Q-values

By Aniket Bhardwaj · 18 September 2026 · IIT-JAM Chemistry

Nuclear chemistry is one of the most self-contained units in the IIT-JAM syllabus. It has a small set of ideas — what makes a nucleus stable, how an unstable one decays, how fast, and how much energy is released — and each of them turns into a short calculation. If you already know first-order kinetics from chemical kinetics, half the unit is free: radioactive decay is first-order kinetics with different symbols. This article covers the whole unit with four fully computed examples, including the mass–energy calculations that students most often get wrong.

Nuclear stability and the n/p ratio

A nucleus is held together by the strong nuclear force acting between nucleons, working against the electrostatic repulsion of the protons. For light nuclei stability sits near a neutron-to-proton ratio of 1; as Z rises, more neutrons are needed to dilute the proton–proton repulsion, and the stability band curves upward to about 1.5 near lead. Beyond Z = 83 no nuclide is stable. Two useful patterns follow, and both are asked as reasoning questions:

Above the band (too many neutrons) a nucleus converts a neutron into a proton — β⁻ emission.
Below the band (too many protons) it converts a proton into a neutron — β⁺ emission or electron capture. Electron capture is favoured for heavier nuclei because β⁺ emission requires an energy release of at least about 1.02 MeV to create the positron–electron mass.
Magic numbers of protons or neutrons (2, 8, 20, 28, 50, 82, and 126 for neutrons) give extra stability, and even–even nuclides are far more common than odd–odd ones.

Decay modes and the group displacement law

Mode	Emitted	Change in A	Change in Z	Periodic-table shift
Alpha (α)	⁴₂He nucleus	−4	−2	Two groups left
Beta minus (β⁻)	electron + antineutrino	0	+1	One group right
Beta plus (β⁺)	positron + neutrino	0	−1	One group left
Electron capture	(absorbs an inner electron); X-rays emitted	0	−1	One group left
Gamma (γ)	photon	0	0	No change — de-excitation only

These two rules (Fajans and Soddy) are all you need to follow a decay series. The natural series are the 4n (thorium), 4n+2 (uranium) and 4n+3 (actinium) families; the 4n+1 (neptunium) series is not found in nature because its longest-lived member has decayed away over geological time.

The equations of the unit

Decay is first order: −dN/dt = λN ⇒ N = N₀ e^−λt
Half-life: t_½ = ln 2 / λ = 0.6931 / λ · mean life τ = 1/λ = t_½/ln 2
Activity: A = λN, and A = A₀ e^−λt (activity decays with the same λ)
Age from activity: t = (1/λ) ln (A₀/A)

Mass–energy: E = Δm c² · 1 u = 931.494 MeV
Binding energy: BE = [Z·m(¹H) + N·m_n − m(atom)] × 931.494 MeV
Q-value: Q = (mass of reactants − mass of products) × 931.494 MeV

Units to keep straight: λ carries the reciprocal of whatever time unit t_½ was given in. Activity is in becquerel (1 Bq = 1 disintegration per second) or curie (1 Ci = 3.7 × 10¹⁰ Bq exactly, by definition). N is a number of atoms, never a mass and never a number of moles.

Worked example 1 — decay constant, number of atoms and activity

Phosphorus-32 has t_½ = 14.3 days. Find λ in s⁻¹ and the activity of a 1.00 µg sample, in Bq and in Ci.

λ = ln 2 / t_½ = 0.69315 / 14.3 = 0.048472 day⁻¹

In seconds: 0.048472 / 86400 = 5.610 × 10⁻⁷ s⁻¹

Number of atoms: n = 1.00 × 10⁻⁶ g ÷ 31.97 g mol⁻¹ = 3.128 × 10⁻⁸ mol
N = 3.128 × 10⁻⁸ × 6.022 × 10²³ = 1.884 × 10¹⁶ atoms

A = λN = 5.610 × 10⁻⁷ × 1.884 × 10¹⁶ = 1.06 × 10¹⁰ Bq = 10.6 GBq

In curie: 1.06 × 10¹⁰ ÷ 3.7 × 10¹⁰ = 0.286 Ci

Notice how small a mass produces a very large activity — that is the whole reason radioactive tracers can be used at chemically negligible concentrations.

Worked example 2 — how much is left, checked two ways

What fraction of the same ³²P sample remains after 30.0 days?

Route 1 — exponential. λt = 0.048472 × 30.0 = 1.4542

N/N₀ = e^−1.4542 = 0.2336, i.e. 23.4%

Route 2 — counting half-lives. 30.0 / 14.3 = 2.098 half-lives, so N/N₀ = (½)^2.098 = 2^−2.098 = 0.2336. Same answer.

The second route is worth practising because it gives you an instant sanity check: two half-lives would leave 25%, and 2.098 half-lives is a little more than two, so the answer must be a little under 25%. Any answer far from that is arithmetic error.

Worked example 3 — radiocarbon dating

A wooden object gives 9.2 disintegrations per minute per gram of carbon; living material gives 15.3 dpm per gram. With t_½(¹⁴C) = 5730 years, find its age.

1/λ = t_½/ln 2 = 5730 / 0.69315 = 8266.6 years

A₀/A = 15.3 / 9.2 = 1.6630, and ln 1.6630 = 0.50865

t = 8266.6 × 0.50865 = 4.2 × 10³ years (about 4200 years)

Because both activities are per gram of carbon, the sample size cancels — you never need to know how much wood was tested. Dating assumes the atmospheric ¹⁴C level was the same when the tree died as it is in the modern reference; real laboratories apply a calibration for that, which is why a textbook answer and a published date can differ.

Worked example 4 — binding energy and a Q-value

(a) Binding energy per nucleon of ⁴He. Use atomic masses: m(¹H) = 1.007825 u, m_n = 1.008665 u, m(⁴He) = 4.002603 u.

2 × 1.007825 = 2.015650 · 2 × 1.008665 = 2.017330 · sum = 4.032980 u

Δm = 4.032980 − 4.002603 = 0.030377 u

BE = 0.030377 × 931.494 = 28.30 MeV, so BE per nucleon = 28.30 / 4 = 7.07 MeV

Why atomic masses work here: the two electrons carried in with the two ¹H atoms are exactly the two electrons in the neutral He atom, so they cancel. Mixing nuclear masses with atomic masses is the classic error.

(b) Q-value for ²³⁸U → ²³⁴Th + ⁴He. m(²³⁸U) = 238.050788 u, m(²³⁴Th) = 234.043601 u, m(⁴He) = 4.002603 u.

Products: 234.043601 + 4.002603 = 238.046204 u

Δm = 238.050788 − 238.046204 = 0.004584 u

Q = 0.004584 × 931.494 = 4.27 MeV, released. A positive Q means the decay is energetically allowed; a negative Q means it cannot happen spontaneously.

Fission, fusion and the binding-energy curve

Binding energy per nucleon rises steeply from hydrogen, peaks near iron and nickel at about 8.8 MeV, and falls slowly towards uranium. Everything about nuclear energy follows from that single shape: heavy nuclei release energy by splitting towards the peak (fission), light nuclei release energy by joining towards the peak (fusion), and iron-group nuclei release energy by neither. In fission of ²³⁵U, thermal neutrons are captured, the nucleus splits into two unequal fragments plus two or three neutrons, and those neutrons sustain a chain reaction once the assembly exceeds its critical mass. Moderators such as heavy water or graphite slow the neutrons, because ²³⁵U captures slow neutrons far more efficiently than fast ones.

Nuclear reactions are written compactly as X(a, b)Y — target, incoming particle, outgoing particle, product. For example ¹⁴N(α, p)¹⁷O is the historic first artificial transmutation. Check both mass number and charge balance on every equation you write.

Common mistakes that cost marks

Unit mismatch in λt. If t_½ is in days, λ is per day and t must be in days. Converting one but not the other is the single most common error in this unit.
Treating activity as a number of atoms. A = λN. They differ by a factor of λ, which can be many orders of magnitude.
Thinking half-life depends on how much you have. It does not, and it is also independent of temperature, pressure and chemical form to an excellent approximation — because decay is a nuclear process, not a chemical one.
Using 1 Ci = 3.7 × 10¹⁰ the wrong way. Curie is the bigger unit: divide becquerel by 3.7 × 10¹⁰ to get curie.
Forgetting electrons in mass-defect problems. Use atomic masses consistently on both sides, or nuclear masses consistently on both sides — never a mixture.
Getting the beta rule backwards. β⁻ emission increases Z by one; the nucleus loses a neutron and gains a proton even though a negative particle leaves.
Rounding 931.494 too early. A mass defect is a small difference between two large numbers, so keep all six decimal places until the subtraction is done.

How to prepare this unit

Sub-topic	What you must be able to do without hesitation
Stability	Predict the decay mode from the n/p ratio; recognise magic numbers
Decay equations	Balance A and Z; apply the group displacement law; follow a decay series
Kinetics	Convert between λ, t_½ and τ; find N, A or t from any two of the others
Dating	Radiocarbon and long-lived isotope dating from an activity ratio
Energetics	Mass defect, binding energy per nucleon, Q-value of a decay or reaction
Applications	Tracers, fission and moderators, fusion, radiation units

Use that as a revision checklist. For the syllabus itself and the current paper pattern, read the official IIT-JAM notification for your year — it is the only reliable source.

Check every decay calculation instantly. The half-life calculator converts between half-life, decay constant, elapsed time and the fraction remaining, so you can verify a whole problem set in the time one hand calculation takes.

Open the Half-Life Calculator →

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