Physics | Energy » Fundamental Equations

Source: http://www.doksinet Physics 341: Fundamental Equations Energy First Law: ∆U = Q + W Equipartitian: U = N · f · Ideal Gas Law: P V = nRT = q N kT Kinetic Theory: vrms = 3kT m 1 2 kT Processes: Isothermal compression: W = N kT ln (Vi /Vf ), Adiabatic compression: P V γ = constant; )  γ = (1 + 2/f  

Q Q ∂U ∂H Heat Capacity: CV = ∆T = ∂T V ; CP = ∆T = ∂T P V P Latent Heat: L = Q/m (Heat needed for phase transition per unit mass) Enthalpy: H = U + P V ~ Magnetic dipole: U = −~ µ·B Entropy Definition: S = k ln Ω Second Law: In any process the entropy increases or stays the same: ∆S ≥ 0 Approximate model multiplicities: √ 2 Approximations: N ! = 2πN N N e−N ; ln (1 + x) x − x2 q N↑ −N↑ N ! N Two-state system: Ω2−state (N, N↑ ) = N↑ !(NN−N 2πN↑ (N (1 − N↑ )−(N −N↑ ) −N↑ ) ( N ) ↑ )! q −1)! q N N N q Einstein solid: ΩES (N, q) = (q+N q!(N −1)! 2πq(N +q) (1 + q ) (1 + N ) iN h N 5 [ V (2πmU

)3/2 ] 3h2 N 32 Ideal Gas: ΩIdeal (N, V, U ) = h3 N !(3N/2)! v0VN e 2 , where v0 = ( 4πmU ) . Approximate model entropies: Two-state system: S2−state (N, N↑ ) −kN h N↑ N ln N↑ N + (1 − N↑ N ) ln (1 − i N↑ N ) Einstein solid: SES (N, q) k [q ln (1 + N/q) + N ln h (1 + q/N )] i 3h2 N 32 ) . Ideal gas (Sackur-Tetrode): SIdeal (N, V, U ) kN ln ( v0VN ) + 52 , where v0 = ( 4πmU Engines and Refrigerators Engine efficiency: e = Net work Heat in ≤1− Tc Th Refrigerator coefficient of performance: cop = Heat out Work in ≤ Tc Th −Tc H4 −H1 H3 −H1 H1 −H3 cop = H 2 −H1 Steam engine efficiency: e 1 − Throttling process refrigerator: Paramagnetic magnetization: M = N µ tanh ( µB kT ) Thermodynamic Properties Thermodynamic potentials: U (S, V, N ), H(S, P, N ), F (T, V, N ), G(T, P, N ) Thermodynamic relations: dU = T dS − P dV + µdN , dH = T dS + V dP + µdN , dF = −SdT − P dV + µdN , dG = −SdT + V dP + µdN Partial

derivative relations:


∂U ∂U ∂U = T ∂S V,N ∂V S,N = −P ∂N S,V = µ


∂H ∂H ∂H ∂S P,N = T ∂P S,N = V ∂N S,P = µ


∂F ∂F ∂F ∂T V,N = −S ∂V T ,N = −P ∂N T ,V = µ


∂G ∂G ∂G = −S, =V, =µ ∂T ∂P ∂N P,N T ,N T ,P 1 Source: http://www.doksinet Maxwell Relations:
=− dU: ∂P ∂S V,N ∂T ∂V
S,N ∂S dF: ( ∂P ∂T )V,N = ( ∂V )T ,N ; Claussius-Clapeyron Relation: ; ∂T ∂P S,N ∂S dG: ( ∂V ) = −( ∂P )T ,N
∂T P,N S2 −S1 ∂P L = = ∂T G V2 −V1 T (V2 −V1 ) dH: ∂V ∂S
P,N
= Boltzmann Factor and the Partition Function −βEs Boltzmann Factor and Probability: P (s) = e Z P −βEs Partition Function: Z = s e ; β = (kT )−1 P Average Values: Q = s Qs P (s); E = − Z1 dZ dβ  3/2 2 2 mβ e−βmv /2 Velocity distribution: D(v) = 4πv 2π 2-state paramagnet: Z2−state = 2 cosh (βµB0 ) Harmonic Oscillator: ZHO = (1 − e−βh̄ω )−1 Diatomic Molecule

rotations: Zrot = fs kT rot where fs = 1 (distiguishable) and fs = 12 (indistinguishable)
V mkT 3/2 ( 2πh̄2 ) Zrot ] + 1 Ideal gas: ln Zideal = N ln [ N Relation to Helmholtz Free Energy:
F = −kT lnZV mkT 3/2  ∂F Chemical potential: µideal = ∂N = −kT ln N ( 2πh̄2 ) Zrot T,V Gibbs Factor and Quantum Statistics −βNs (Es −µ) Gibbs factor: P (s, Ns ) = e Z P , where Es is the single particle energy. Grand Partition Function: Z = s,Ns e−βNs (Es −µ) 2 dk Density of States: dn = 2 V 4πk (2π)3 θ(kF − k); N = 3 V kF 3π 2 . DOS in terms of energy (nonrelativistic): dn = g0 1/2 d, where g0 = Fermi-Dirac Statistics: nF D (s) = 3N 3/2 . 2EF h̄2 k2 1 ; eβ(Es −µ) +1 EF = 2mF (nonrelativistic). R∞ kT <<EF 3 3/2 Average energy (nonrelativistic): UF D = g0 0 d eβ(−µ) − 5 N EF + +1 1 Bose-Einstein Statistics: nBE (s) = eβ(Es −µ) −1 ; µ = 0 for photons and phonons. 8π 3 dI c du 4 (hc)3 eβ −1 ; d = 4 d ; I = σT ; 1/3 s 6N

1/3 Debye Theory of Solids: TD = hc = h̄cks (6π 2 N ; 2k ( πV ) V ) 4 R TD /T 4 3 T <<TD 9N kT 3π N k 4 x T . Uphonon = T 3 − 3 ex −1 dx 0 5TD D Blackbody Radiation: du d = π 2 k2 2 4EF N T . λM T = 2.989 × 10−3 m/K Additional Information Constants: R = 8.315 J/mol·K; k = 1381 × 10−23 J/K = 8617 × 10−5 eV/K; NA = 6022 × 1023 ; h = 2πh̄ = 6.626 × 10−34 J·s; 1 atm = 1013×105 Pa; TST P = 298 K; 1 eV = 16×10−19 J; σ = 5.67 × 10−8 W/(m2 K4 ); hc = 1240 eV nm; h̄c = 1973 eV nm; u = 166 × 10−27 kg √
R∞ R∞ R∞ R∞ 2 2 2 d n Integrals: 0 dx xn e−x = n!; 0 dx e−x = 2π ; 0 dx x2n e−x = − dα dx e−αx ; 0 α=1 R∞ R∞ R ∞ x3 R ∞ x3 2 π4 7 π4 dx x2n+1 e−x = 12 0 ds sn e−s = n! 2 ; 0 ex −1 dx = 15 ; 0 ex +1 dx = 8 15 0 2