Reaction energies, activity and decay balances
Introduced| English |
|---|
| activity/ækˈtɪvɪti/ |
| Q value/kjuː ˈvæljuː/ |
A decision before an answer
- One atomic mass unit is 931.5 MeV, so a reaction energy is a small difference between large masses.
- Your goal: Compute reaction energy from complete atomic or nuclear mass bookkeeping.
Bookkeep the masses
- The reaction energy is Q=(m_initial−m_final)c²; Q>0 releases kinetic energy. Track every species, and keep the mass convention consistent: tabulated atomic masses include electrons.
- For β− decay and for α decay with a neutral helium product, atomic electron counts balance directly; β+ or electron capture needs explicit electron-mass terms. Use 1 u=931.5 MeV/c².
Using atomic masses 238.0508 u (²³⁸U), 234.0436 u (²³⁴Th) and 4.0026 u (⁴He), the alpha-decay Q value is about:
Δm=0.0046 u; 0.0046×931.5≈4.28 MeV.
Compute Q and check balance
- Alpha decay of ²³⁸U with atomic masses 238.0508 u → 234.0436 u + 4.0026 u gives Δm=0.0046 u and Q≈4.28 MeV. Fusion D+T with 2.0141+3.0161−4.0026−1.0087=0.0189 u gives Q≈17.6 MeV.
- A reaction also requires charge and nucleon-number balance: ¹⁴N+α→¹⁷O+p balances 7+2=9 and 14+4=18.
Two samples contain equal numbers of radioactive nuclei; sample X has half the half-life of sample Y. The activity of X is:
A=λN and λ=ln2/T½, so halving T½ doubles A at fixed N.
Convert half-life to activity
- Activity is A=λN with decay constant λ=ln2/T½; one becquerel is one decay per second. For T½=2 h and N=6×10¹², A=6×10¹²×ln2/7200≈5.8×10⁸ Bq. After three half-lives both N and A fall by 1/8.
- Equal activities do not imply equal atom counts: shorter half-life at fixed N gives larger A.
U-238 alpha decay: Q=(238.0508−234.0436−4.0026)×931.5≈4.28 MeV, carried mostly by the alpha by momentum conservation. D+T fusion releases ≈17.6 MeV. A sample with T½=2 h and 6×10¹² atoms has A≈5.8×10⁸ Bq. Fission of a heavy nucleus releases of order (8.5−7.6)×240≈200 MeV.
A sample starts with activity 8×10⁸ Bq and half-life 3 h. After 9 h its activity is ____ ×10⁸ Bq.
Nine hours is three half-lives: 8×10⁸/8=1×10⁸ Bq.
Compare binding changes
- In a chain A→B→C with a long-lived parent (λ_A≪λ_B), B accumulates until its activity approaches the parent activity (secular equilibrium); write the balance dN_B/dt=λ_A N_A−λ_B N_B.
- The binding-energy-per-nucleon curve peaks near iron, so heavy fission (about 7.6 to 8.5 MeV per nucleon over roughly 240 nucleons, of order 200 MeV) and light fusion both release energy; compare total binding, not nucleon counts.
Mixing atomic and nuclear masses without tracking electrons, or reporting a positive Q for a reaction that needs energy input. Balance charge and nucleon number first, then check the sign of Q.
Which answer fits this case?
Compute reaction energy from complete atomic or nuclear mass bookkeeping
For β− decay, tabulated atomic masses of parent and daughter require an extra electron-mass term in Q.
Atomic electron counts already balance for β− (and for α with neutral He); β+ needs the explicit 2m_e.
Keep the distinctions
- Q value Q值 — The rest-mass energy converted to kinetic energy in a reaction, (m_initial−m_final)c².
- activity 活度 — The decay rate λN of a radioactive sample, measured in becquerel.
- Compute reaction energy from complete atomic or nuclear mass bookkeeping.
- Calculate activity from half-life and a simple decay-chain balance.
- Compare fission and fusion binding changes with charge and nucleon conservation.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.