What you should know about . . .

1. Nuclei and [Q12,Q13]

A 1/3 • nuclear notation: Z and A total =⇒ Z X; size: r = r0A , with r0 ≈ 1.2 fm • Strong interaction between color neutral objects (such as nucleons) is short range (< 2 fm) P • Binding energy: Eb = ( parts Eparts) − Esys P 2 • Mass defect: ∆m = ( parts mparts) − msys = Eb/c = ZmH + Nmn − matom 2/3 2 1/3 2 • binding formula (A > 20): Eb(A, Z) = aI A − aS A − aC Z /A − aA(A − 2Z) /A

with aI = 15.56 MeV, aS = 17.23 MeV, aC = 0.697 MeV, aA = 23.285 MeV 2 • per , eb = Eb/A and binding energy vs. mass (E = mc )

2. (beta, alpha, gamma) [Q13,Q14]

• Predicting stability and type of decay: energy level diagrams, binding energy formula, atomic mass

− + − + + + + − • beta: β =⇒ n −→ p + e + νe β =⇒ p −→ n + e + νe EC =⇒ p + e −→ n + νe • alpha: nuclei (2 protons, 2 ) • gamma: photons from decays of excited nuclear states

−λt • activity ≡ −dN/dt = λ N =⇒ solution: N(t) = N0 e =⇒ half-life t1/2 = ln 2/λ

 1.0084  • Zstable(A) for stability against β decay: Zstable = A 2+0.015 A2/3 . This formula gives the stable for even-odd nuclei (A is odd) or even-even nuclei (both Zstable and

N are even). For odd-odd nuclei (A even, N,Zstable odd) the neighboring even-even nuclei may have

lower atomic mass, in which case the stable nuclei have Z = Zstable ± 1. • arises from weak interactions; it leaves A unchanged. β− decay converts a to a , decreasing N by 1 and increasing Z by one. capture (EC) and β+ decay convert a proton in a neutron, increasing N by 1 and decreasing Z by one.

− • β decay and EC from i to f are energetically possible if the atomic masses satisfy the relation mi > mf . + β decay is energetically possible if mi > mf + 2me. 4 • is due to strong interactions. It arises from emission of a 2He nucleus, by tunneling through the Coulomb barrier. It changes A by 4. Parent and daughter nuclei of decay chains involving both α and β decays differ in by multiples of 4.

• Alpha decay is energetically allowed if dEb/dA ≤ 7.07 MeV where 7.07 MeV is the binding energy per 4 nucleon in a 2He nucleus. For β-stable nuclei, this happens for A larger than about 150. • Fission describes the decay of a large nucleus into two roughly equal-sized fragments. It additionally produces neutrons, since the fragment nuclei have smaller A and prefer a lower N/Z ratio for stability. The extra neutrons can trigger further fission events, leading to a chain reaction.

• Decay rate λ = dPdecay/dt = (1/N) dNdecay/dt

The activity dNdecay/dt = λN of a radioactive sample is measured in becquerel (1 Bq = 1 decay per second) or (1 Ci = 3.7 · 1010 Bq).

• Half-life t1/2 = (ln 2)/λ • Exponential decay law: N(t) = N(0) e−λt What you should know about . . . Laws of Thermodynamics

1. Zeroth Law of Thermodynamics: objects A and B are in thermal equilibrium if and only if TA = TB [T1]

• A thermometer is a device that quantifies temperature, calibrated to a standard temperature scale. • T (in K) = T (in ◦C) + 273.15 =⇒ Use T in kelvin in all formulas • dU = mc dT , where m is the mass of the substance and c is its specific heat.

2. First Law of Thermodynamics: ∆U = Q + W [T3]

• Q > 0 if heat energy entering system; W > 0 if work energy entering system

3. Second Law of Thermodynamics: ∆S ≥ 0 [T5]

• be able to explain in your own words why entropy increases

What you should know about . . . Ideal Gases [T2, T3]

−23 −5 1. Ideal gas law: PV = NkBT , where N is the number of molecules and kB = 1.38 × 10 J/K = 8.62 eV/K.

23 • n = number of moles = N/NA, where NA = 6.02 × 10 .

2. “Equipartition theorem” [T2]

1 1 2 3 • each degree of freedom gets 2 kBT on average =⇒ translational K = 2 m[v ]avg = 2 kBT 2 1/2 p • vrms ≡ ([v ]avg) = 3kBT/m 3 5 • monatomic gas: U = 2 NkBT ; diatomic gas: U = 2 NkBT

−∆E/kB T • mode is fully unfrozen when kBT > 2∆E (c.f. Boltzmann factor e )

3. Gas processes [T3]

• interpreting a PV diagram =⇒ideal gas law applies at each point on curve – work from W = − R P dV =⇒area under curve – net work from closed cycle is enclosed area – find T or N from ideal gas law and P , V – find ∆U from ∆T , then Q from First Law • isochoric: V = constant =⇒W = 0

• isobaric: P = constant =⇒W = −P (Vf − Vi) R R • isothermal: T = constant =⇒W = − P dV = −(NkBT ) dV/V = −NkBT ln(Vf /Vi) • adiabatic: Q = 0 =⇒PV γ = constant – γ = 5/3 = 1.67 for monatomic, 7/5 = 1.4 for diatomic, 1.33 for polyatomic gas What you should know about . . . Macrostates, Microstates, and Entropy

1. Definitions: [T4]

• macrostate specified by macroscopic variables (e.g., 3 of P , V , N, or T for ideal gas) • microstate specified by quantum state of every molecule • multiplicity Ω is number of microstates consistent with a macrostate (e.g., same U, N) • macropartition table: – know how to construct one given Ω(U, N)

– using U = UA + UB = constant; ΩAB = ΩA × ΩB – predicting the most probable macropartition • fundamental assumption: each accessible microstate is equally probable =⇒ relative probabilities of macropartitions equals ratio of (total) multiplicities

2. Einstein Solid with oscillator energy ε =hω ¯ [T4]

(3N + U/ε − 1)! 3N • Ω(N,U) = U = P n ε U = 3Nk T (3N − 1)!(U/ε)! i i B

3. Temperature and Entropy S = kb ln Ω [T5,T6]

• SAB = SA + SB

• given S, find multiplicity Ω = eS/kb • dS/dU = 1/T defines temperature

1 −E/kB T −E/kB T P Ei/kB T • Boltzmann factor: Pr(E) = Z e = e / all states e • Movie test for reversible vs. irreversible processes • explain why heat spontaneously flows from hot to cold bodies

4. Maxwell-Boltzmann Distribution and Average Energy [T7]

2   2 4 v −(v/vP ) 1/2 • Pr(speed within dv centered on v) = 1/2 e where vP ≡ (2kBT/m) π vP

  P −En/kB T P e−En/kB T Ene • Average energy: Eavg = En = Z P e−En/kB T 5. Calculating Entropy Changes [T8]

• ∆S = R dQ/T (N constant, quasistatic volume change) • ∆S = 0 for (quasistatic) adiabatic volume changes • isothermal =⇒ ∆S = Q/T (Q given or find from Q = −W ) • phase change: ∆S = ±mL/T with L the latent heat

• If no work, dQ = dU = mc dT =⇒ ∆S = mc ln(Tf /Ti) • Non-quasistatic processes =⇒ find ∆S from replacement process