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finalformulasheet - PHYS4125 Spring 2007 Final Exam...

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Unformatted text preview: PHYS4125 Spring 2007 Final Exam (Schroeder Ch. 1-7) Formulas and Constants First and Second Law Equipartition Theorem: Uthermal = N f kBT 1st Law of Thermodynamics: U = Q + W V Compressive Work: W Adiabatic Processes: PV = constant; VTf/2 = constant; = (f+2) / f Ideal Gas Law: PV = nRT = NkBT N 3 2 Transport Mean Free Path: Scattering time: A RMS velocity: vrms = (3kBT/m)1/2 Viscosity: Fx 0 !) Diffusion, Fick's Law: & % t v rms du x dz \$ # l 1 r 2 4 V N l Jx D dn dx " ! Heat Conduction: Q t kt A S Nk B ln 3Nh dT dx kt 1 CV lv 2 V Sackur-Tetrode Equation: V N 4 mU 2 ' (" Ideal Gas: 3N 2 1 V 2mU N ! h 3 N 3N 2 ! N 3N 2 Entropy of mixing: Entropy: S k B ln S k B ln n1 n2 ! n1 ! n 2 ! N ,V P Heat Capacity: C = Q/T or f P V dV Vo C Q T U T N ,V P 5 2 Statistics N ,n N! n! N n ! U ,N U ,V Engines and Refrigerators Definition of Coefficient of Performance: Joule-Thompson Throttling Process: H f Hi g f e d h COP benefit cost Qc W Qh Definition of Efficiency: e Qh Th Qc Qc Th Tc Tc benefit cost W Qh Qh Qc Th Definition of Chemical Potential: T S N y Thermodynamic Identity: dU T dS v T u x vt rs q Definition of Pressure: P T S V U V S ,N P dV dN U N Tc p i h w g processes. Third Law of Thermodynamics: T CV e fd c b a Entropy and heat: dS dQ T C V dT `Y X Definition of Temperature: W degrees of freedom and V TU or more generally 2 U for fN n pN systems with only quadratic Interactions T U S N ,V ; S CV T dT for quasi-static 0 0 0 as T 0 ;S T S TR F GE Q P I H C B q N 1 ! q! N 1 ! Stirling's Approximation: ln N ! Nln N 1 e n Gaussian Distribution: P N n 2 A D 8 7 Einstein Solid: N ,q 2 @ 1 Binomial Distribution: PN n N N n 2 2 2 3 N! pn r n! N n ! n and 9 6 5 4 where Npr S ,V Free Energy Enthalpy: H U PV Helmholtz Free Energy: F U TS Gibbs Free Energy: G U TS PV Thermodynamic Identities: dH TdS dF SdT PdV dN , dG Boltzmann Statistics Es kB T Boltzmann Factor: Partition Function: Average Values: e e Es Probability function: Z X ave P s s 1 e Z Es E s e s s Ideal Gas: quantum volume: Z total Z total Z 1 Z 2 Z 3 ... Z N (non-interacting distinguishable particles) 3 1 N Z (noninteracting indistinguishable particles) N! 1 vQ 2 h mkBT ! Free Energy and the Partition Function: F Partition Function of Composite Systems: k B T ln Z Maxwell Distribution Function: D v 4 v2 e m 2 kBT Equipartition (Again): if E q cq 2 then E ave 1 2kBT 3 2 X s P s 1 Z X s e Es E ave 1 Z mv 2 2 van der Waals Equation of State: P aN V2 2 V Nb Nk B T Clausius Clapeyron Relation for a Phase T T For fixed V, N: dF T 1 For fixed P, N: dS total T dG Chemical Potential: G N ; dS total 1 T ,P o T k B T ln P P dP Boundary: dT ~ T} | SdT s vr { z q p l o y !x k n m w j u vt i VdP dN , dN VdP o L T V Z N Chemical Potential: k B T ln Quantum Statistics Grand Partition FunctionGibbs Sum: s Bose-Einstein distribution: Fermi gases: Fermi energy: n BE F F 2h 2 2 12 e Energy Spectrum: u hc 8 3 3 e 1 Blackbody Radiation: Plank Distribution: n Pl "#! 1 F Chemical Potential: 2 kBT ... 1 hf 3 g V 3 2 1 Density of States: 8m 3 2 ! Heat Capacity: 2 CV Nk B T F 2 2 Degeneracy Pressure: P F 5V 2U 3V F Total energy: U 3 N 5 Fermi temperature: TF F kB 2 N 4 2N h2 8m 3N V Boltzmann occupation number: Fermi Dirac Distribution: n FD 1 e 1 1 e 1 n Boltzmann e 2 3 kBT 2 ... 3N ! Z Ţ entropy: S Nk B ln V Nv Q 5 2 partition function: Z 1 N! V Z internal vQ F internal T V Z internal Nv Q exp E s N s 0 2k B g e d c b N0 N N excited 1 T Tc f ` Y Bose-Einstein Condensation: kB T c 0.527 h 2 2 m 3 2 N V N for T a W CV X U low temperature: U V T high temperature: U 3Nk B T 3 4 Nk B T T3 D 4 4 5 3 12 5 Nk B 2 3 T TD Tc R Q T e P G Debye Model: U 9Nk B T 3 D D IH 4 T T x x 3 1 dx TD hc s 6N V S FD #EC 6 Phonons: Einstein Model CV 3Nk B e 1 2 A B@ kBT 2 15h c e 4 53 1 8 97 0 Stefan-Boltzmann: Power area 2 ( Entropy: S T 15 hc 32 k 4 V 3 B T 3 45 hc 5 ) & Heat Capacity: CV ' \$ Total Energy: 15 hc 32 5 k 4 V B 3 % U V 8 5 kBT 3 4 T 3 2 5 kBT 3 2 4 T4 1 3 Reference Data: kB = 1.3807 x 10-23 J/K = 8.617 x 10-5 eV / K NA = 6.022 x 1023 R = 8.315 J/mol K h = 6.626 x 10-34 J s = 4.136 x 10-15 eV s c = 2.998 x 108 m/s G = 6.673 x 10-11 Nm2/kg2 e = 1.602 x 10-19 C me = 9.109 x 10-31 kg mp = 1.673 x 10-27 kg B = 9.2741x10-24 J/T Unit Conversions: 1atm = 1.013 bar = 1.013 x 105 Pa = 14.7 lb / in2 = 760 mm Hg (T in oC) = (T in K) 273.15 (T in oF) = 9/5 (T in oC) + 32 1 oR = 5/9 K 1 cal = 4.186 J 1 Btu = 1054 J 1 eV = 1.602x 10-19 J 1 u = 1.661 x 10-27 kg ...
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This note was uploaded on 05/20/2011 for the course PHYS 4125 taught by Professor Staff during the Spring '07 term at LSU.

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