Energías de reacción, actividad y balances de desintegración
Introducido| English | Significado |
|---|---|
| activity/ækˈtɪvɪti/ | actividad |
| Q value/kjuː ˈvæljuː/ | valor Q |
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:
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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:
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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.
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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.
¿Qué respuesta se ajusta a este caso?
Calcular la energía de reacción mediante el balance completo de masas atómicas o nucleares
For β− decay, tabulated atomic masses of parent and daughter require an extra electron-mass term in Q.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.