Physics 341: Fundamental Equations - Thermodynamics and Statistical Mechanics - Prof. Jame, Study notes of Thermal Physics

Download Physics 341: Fundamental Equations - Thermodynamics and Statistical Mechanics - Prof. Jame and more Study notes Thermal Physics in PDF only on Docsity! Physics 341: Fundamental Equations Energy First Law: ∆U = Q + W Ideal Gas Law: PV = nRT = NkT Equipartitian: U = N · f · 12kT Kinetic Theory: vrms = √ 3kT m Processes: Isothermal compression: W = NkT ln (Vi/Vf ), Adiabatic compression: PV γ = constant; γ = (1 + 2/f) Heat Capacity: CV = ( Q ∆T ) V = ( ∂U ∂T ) V ; CP = ( Q ∆T ) P = ( ∂H ∂T ) P Latent Heat: L = Q/m (Heat needed for phase transition per unit mass) Enthalpy: H = U + PV Magnetic dipole: U = −~µ · ~B Entropy Definition: S = k lnΩ Second Law: In any process the entropy increases or stays the same: ∆S ≥ 0 Approximate model multiplicities: Approximations: N ! = √ 2πNNNe−N ; ln (1 + x) ' x− x 2 2 Two-state system: Ω2−state(N,N↑) = N !N↑!(N−N↑)! ' √ N 2πN↑(N−N↑) ( N↑ N ) −N↑(1− N↑N ) −(N−N↑) Einstein solid: ΩES(N, q) = (q+N−1)! q!(N−1)! ' √ N 2πq(N+q) (1 + N q ) q(1 + qN ) N Ideal Gas: ΩIdeal(N,V, U) = [ V h3 (2πmU)3/2]N N !(3N/2)! ' [ V v0N e 5 2 ]N , where v0 = ( 3h 2N 4πmU ) 3 2 . Approximate model entropies: Two-state system: S2−state(N,N↑) ' −kN [ N↑ N ln N↑ N + (1− N↑ N ) ln (1− N↑ N ) ] Einstein solid: SES(N, q) ' k [q ln (1 + N/q) + N ln (1 + q/N)] Ideal gas (Sackur-Tetrode): SIdeal(N,V, U) ' kN [ ln ( Vv0N ) + 5 2 ] , where v0 = ( 3h 2N 4πmU ) 3 2 . Engines and Refrigerators Engine efficiency: e = Net workHeat in ≤ 1− Tc Th Refrigerator coefficient of performance: cop = Heat outWork in ≤ Tc Th−Tc Steam engine efficiency: e ' 1− H4−H1H3−H1 Throttling process refrigerator: cop = H1−H3H2−H1 Paramagnetic magnetization: M = Nµ tanh (µBkT ) Thermodynamic Properties Thermodynamic potentials: U(S, V,N), H(S, P, N), F (T, V,N), G(T, P,N) Thermodynamic relations: dU = TdS − PdV + µdN , dH = TdS + V dP + µdN , dF = −SdT − PdV + µdN , dG = −SdT + V dP + µdN Partial derivative relations:( ∂U ∂S ) V,N = T ( ∂U ∂V ) S,N = −P ( ∂U ∂N ) S,V = µ( ∂H ∂S ) P,N = T ( ∂H ∂P ) S,N = V ( ∂H ∂N ) S,P = µ( ∂F ∂T ) V,N = −S ( ∂F ∂V ) T,N = −P ( ∂F ∂N ) T,V = µ( ∂G ∂T ) P,N = −S, ( ∂G ∂P ) T,N = V , ( ∂G ∂N ) T,P = µ 1