Mass-energy equivalence, mass defect and binding energy, the binding energy per nucleon curve behind fission and fusion, and the exponential law of radioactive decay.
12 definitions to learn word for word, 10 equations in their own column, 6 diagrams and graphs. Read it, cover it, test yourself.
23.1aMass-energy equivalence, mass defect and binding energy
Definitions
Mass-energy equivalence
mass and energy are equivalent; a change in energy ΔE corresponds to a change in mass Δm given by ΔE = c2Δm.
Mass defect
the difference between the total mass of the separate nucleons and the mass of the nucleus.
Binding energy
the minimum energy needed to separate a nucleus completely into its individual protons and neutrons (the energy equivalent of the mass defect).
Unified atomic mass unit (u)
one twelfth of the mass of a carbon-12 atom (1 u = 1.66 × 10-27 kg).
Key points
A nucleus has less mass than its separate nucleons: the mass defect.
Energy is released when a nucleus forms; that energy must be supplied to pull it apart.
Mass defect = Z mp + (A - Z) mn - mnucleus.
1 u ≈ 931.5 MeV of energy; work in kg and J if unsure.
Energy released in a reaction = (mass of reactants - mass of products) × c2.
Released energy appears as kinetic energy of products and as γ-ray photons.
Separate nucleons have more energy (and mass) than the bound nucleus; the gap is the binding energy = c2 × mass defect (blue = protons, white = neutrons).
Watch out!
The nucleus is lighter than its parts, not heavier.
Binding energy is energy needed to separate a nucleus, not energy stored inside it.
Use nuclear masses, or consistent atomic masses (electrons then cancel); do not mix them.
23.1bBinding energy per nucleon, fission and fusion
Definitions
Binding energy per nucleon
the binding energy of a nucleus divided by the number of nucleons in it.
Nuclear fission
the splitting of a large (heavy) nucleus into two smaller nuclei of similar mass, with release of energy (and neutrons).
Nuclear fusion
the joining of two light nuclei to form a heavier nucleus, with release of energy.
Key points
Higher binding energy per nucleon = more stable nucleus.
The curve peaks near iron-56 (about 8.8 MeV per nucleon).
Energy is released when binding energy per nucleon increases: products are more tightly bound.
Fusion of light nuclei (left of the peak) and fission of heavy nuclei (right of the peak) both release energy.
Energy released = total BE of products - total BE of reactants.
Fusion needs very high temperature so nuclei overcome electrostatic repulsion; fission is triggered by neutron absorption.
Binding energy per nucleon against A: fusion and fission both move towards the iron peak and release energy.
Fission: a slow neutron is absorbed by U-235, which splits into two daughter nuclei plus 2 or 3 neutrons, releasing energy.
Watch out!
Fusing nuclei heavier than iron absorbs energy; energy comes out only when BE per nucleon rises.
Products have greater binding energy but less total mass than the reactants.
23.2aRandom decay, activity and the decay constant
Definitions
Random (decay)
it cannot be predicted which nucleus will decay next, or when a given nucleus will decay.
Spontaneous (decay)
decay is not affected by external factors such as temperature, pressure or chemical state.
Activity (A)
the number of nuclei that decay per unit time (1 Bq = 1 decay per second).
Decay constant (λ)
the probability that an individual nucleus will decay per unit time.
Key points
Evidence for randomness: the count rate fluctuates about a mean value.
Each nucleus has the same probability of decaying per second, so activity ∝ number of undecayed nuclei.
Measured count rate is less than the activity: the detector catches only some emissions; subtract background count.
Large λ means rapid decay and a short half-life.
Number of nuclei N = (mass / molar mass) × NA.
Count rate from a long-lived source over time: random fluctuations about a steady mean.
Watch out!
Random does not mean unpredictable in bulk: for large N the decay rate is predictable.
Correct count rates for background radiation before using them.
23.2bExponential decay and half-life
Definitions
Half-life (t½)
the mean time taken for half of the active nuclei in a sample to decay.
Key points
N, A and count rate all fall exponentially with the same half-life.
After n half-lives the fraction remaining is (½)n.
Half-life is constant for an isotope, independent of the starting amount.
Straight-line test: ln A against t is linear with gradient -λ and intercept ln A0.
Read t½ from a decay curve at several points and average.
Undecayed nuclei fall as N; daughter nuclei grow as N0 - N.
Exponential decay: every half-life halves what remains, N0 to N0/2 to N0/4.
ln A against t: a straight line with gradient -λ, so t½ = ln 2 / λ.
Watch out!
λ = 0.693/t½, not 1/t½. Units must match: λ in s-1 needs t in s.
Half-life is not half the time for the sample to decay completely; it never fully decays.
Gradient of ln A against t is -λ; drop the minus sign only when quoting λ.