Nuclear Chemistry Formulae

Introduction

Nuclear formulae describe statistical transformations of large atom populations and energy associated with mass differences. The same letter A may mean mass number in nuclide notation or activity in a decay equation; context and units distinguish them. A source's activity is not a person's radiation dose. This sheet keeps the exponential model, nuclear bookkeeping and energy calculation separate, with their assumptions visible.

Core explanation

For a single radionuclide with constant decay probability per unit time and no production, expected number of undecayed nuclei is N(t) = N₀e^(−λt) . The decay constant λ has units time⁻¹ and relates to half-life by t₁/₂ = ln 2/λ . An equivalent form is N(t) = N₀(1/2)^(t/t₁/₂) . These equations predict an expected population; individual nuclei decay randomly. After one half-life, half the initial expected count remains; after two, one quarter. The model assumes the radionuclide's nuclear decay constant is unchanged by ordinary chemical conditions and ignores production, separation or daughter buildup unless explicitly included.

Activity is expected decays per time: A act(t) = λN(t) . Its SI unit is becquerel, 1 Bq = 1 s⁻¹ in the decay-counting sense. Because λ is fixed for one isotope under ordinary conditions, activity follows the same exponential factor as N . If atoms are physically removed from a detector region, measured activity there can decline faster than the radioactive decay law without changing the nuclear half-life. A decay chain with radioactive daughters needs coupled equations; the total detector count rate is not necessarily one simple exponential.

For energy, Einstein's mass–energy relation gives rest energy E = mc² . A nuclear reaction energy release can be estimated from mass defect Δm between specified initial and final particles: Q = (m initial − m final)c² when masses include the appropriate particles and the sign convention defines positive release. Atomic masses can be used with care when electron counts cancel or are included consistently. Units must match: kilograms times m² s⁻² yield joules; atomic-mass-unit conversions to MeV use a defined factor. The formula does not imply all rest mass of a fuel sample is converted to radiation; the relevant quantity is the small mass difference for the particular reaction.

Absorbed dose is energy deposited in material per unit mass, D = E deposited/m tissue , with unit gray (Gy = J kg⁻¹). It cannot be calculated from source activity alone: radiation type and energy, geometry, shielding, time and absorption fraction matter. Effective or equivalent dose in sieverts adds weighting conventions and is not simply another spelling of gray. The IAEA radiation protection standards distinguish these quantities.

Step-by-step reasoning

1. Define whether the unknown is remaining atoms, activity, energy release or absorbed dose. 2. Use time units consistent with decay constant or half-life. 3. Apply repeated halving or exponential decay for one isolated radionuclide under stated assumptions. 4. For nuclear reaction energy, balance particles and use a consistent mass table and sign convention. 5. For dose, identify deposited energy and receiving mass; do not substitute Bq as though it were Gy.

Visual explanation

Plot N(t) and A act(t) as parallel decaying curves, each halving at the same labeled intervals but with different vertical units. A nuclear reaction balance has masses on both sides and a small difference arrow to energy. A separate source-to-tissue path includes distance and shielding before a deposited-energy box, showing why activity does not directly become dose.

Real-world analogy

A large batch of unstable timers, each with random trigger time, can show predictable average decline even though one timer is unpredictable. This resembles radioactive population decay. The analogy omits nuclear transformations and emitted radiation, so it cannot explain dose or energy spectra.

Real-world example

A radiotracer's signal in a body region may drop because radioactive nuclei decay and because biological processes move the tracer elsewhere. The observed disappearance half-time can therefore be shorter than the radionuclide's physical half-life. A clinician or researcher must separate physical decay from distribution and excretion before interpreting images or dose. The chemical form of the tracer influences biological fate even though the nuclide's ordinary nuclear half-life is unchanged.

Why?

Why calculate activity as λN rather than as mass alone? Equal masses of two radionuclides can contain different numbers of nuclei and have very different decay constants. The activity reflects both population and per-nucleus decay probability. The relation also explains why activity and atom count halve together for one isolated nuclide.

Common misconception

“Half-life is when all activity disappears.” It halves each interval. “A single atom decays exactly at its half-life.” Decay is stochastic. “Bq is dose.” It counts transformations per time. “All mass becomes energy in a nuclear reaction.” The reaction energy follows the mass difference between specified initial and final states. “Observed count rate always follows one exponential.” Geometry, daughters and removal can alter it.

Worked example

A radionuclide has half-life 8.0 h and starts at 1600 Bq. After 24.0 h, three half-lives pass, so expected activity is 1600 × (1/2)³ = 200 Bq . The same factor applies to expected remaining parent nuclei if none are produced or removed. The decay constant is λ = ln 2/8.0 h ≈ 0.0866 h⁻¹ , or about 2.41 × 10⁻⁵ s⁻¹. If a detector sees only 100 Bq in a body region, that does not prove the isotope's half-life changed; tracer transport or detection geometry may have changed. For a separate mass defect of 1.00 × 10⁻¹² kg, Q ≈ Δmc² ≈ 9.00 × 10⁴ J using c ≈ 3.00 × 10⁸ m s⁻¹ .

Quick check

1. What fraction remains after three half-lives? Answer: One eighth of the initial expected parent population or activity. 2. Does 100 Bq mean 100 J kg⁻¹ dose? Answer: No. Bq is decays per second; Gy is deposited energy per mass.

Exam focus

State the radionuclide and time basis. Use λ = ln 2/t₁/₂ with coherent units. Distinguish expected nuclear decay from physical transport or biological elimination. Balance nuclear reactions before mass-defect arithmetic. Separate source activity from deposited dose and state what information is missing for a dose estimate.

Advanced insight

In a decay chain, daughter activity may first rise as the daughter is produced and then fall, so the total activity can be non-monotonic. Branching ratios split one parent among several products. Mass–energy bookkeeping must include kinetic energy, emitted particles and sometimes recoil when relating a tabulated Q value to what a detector measures. These refinements show why a simple formula should be paired with a clearly defined system.

Summary

One isolated radionuclide follows exponential expected decay, with activity proportional to remaining nuclei and half-life set by its decay constant. Nuclear reaction energy comes from mass difference, while absorbed dose concerns energy deposited per receiving mass. The quantities have different units and cannot be interchanged.

Practice questions

1. A sample starts at 800 Bq with half-life 5 h. What is expected activity after 10 h? Answer: 200 Bq after two half-lives. 2. Find λ in h⁻¹ for a half-life of 10 h. Answer: λ = ln 2/10 ≈ 0.0693 h⁻¹ . 3. What energy corresponds to 2.0 × 10⁻¹² kg mass difference? Use c = 3.0 × 10⁸ m s⁻¹ . Answer: E = mc² = 1.8 × 10⁵ J . 4. Why is a detector's falling local count rate not necessarily a changed nuclear half-life? Answer: Transport, excretion, shielding or geometry can change local detection independently of nuclear decay.