Thermodynamics and Statistical Mechanics: Principles and Equations, Summaries of Thermodynamics

Download Thermodynamics and Statistical Mechanics: Principles and Equations and more Summaries Thermodynamics in PDF only on Docsity! Thermodynamics and Statistical Mechanics 1 General Definitions and Equations • First Law of Thermodynamics: Heat Q is a form of energy, and energy is conserved. – dU = dQ− dW • Second Law of Thermodynamics: The entropy S of a system must increase. – ∆S ≥ 0 • Third Law of Thermodynamics: For a system with a nondegenerate ground state S → 0 as T → 0. – lim T→0 S = kB ln Ω0, where Ω0 is the degeneracy of the ground state • A system is equally likely to be in any of the quantum states accessible to it. • Equipartition theorem: each “degree of freedom” of a particle contributes 1 2 kBT to its thermal average energy – Any quadratic term in the energy counts as a “degree of freedom” – E.g. monoatomic: 3 translational, 0 rotational, 0 vibrational → 〈E〉 = 3 2 kT – E.g. diatomic: 3 translational, 2 rotational, 0 vibrational (2 at high T) → 〈E〉 = 5 2 kT ( 7 2 kT at high T ) • Binomial Distribution: P (n,N − n) = ( N n ) pn(1− p)N−n – ( N n ) = N ! n!(N − n)! – Normalization: N∑ n=0 P (n,N − n) = 1 – Mean: 〈n〉 = N∑ n=0 nP (n,N − n) = Np – Variance: 〈 n2 〉 − 〈n〉2 = N∑ n=0 n2P (n,N − n)− (Np)2 = Np(1− p) 1 • Gaussian Distribution: P (x) = 1√ 2πσ2 exp ( −(x− µ)2 2σ2 ) – Normalization: ∫ ∞ −∞ P (x) dx = 1 – Mean: 〈x〉 = ∫ ∞ −∞ xP (x) dx = µ – Variance 〈 (x− µ)2〉 = ∫ ∞ −∞ (x− µ)2 P (x) dx = σ2 – P (a < x < b) = ∫ b a P (x) dx • Heat capacity, c: Q = C∆T , C = mc, where c is the specific heat capacity • Molar heat capacity, C: Q = nC∆T , where n is the number of moles • Latent heat of fusion (Q to make melt): Q = mLf • Latent heat of vaporization (Q to vaporize): Q = mLv • Ideal Gas Law: PV = nRT = NkBT • For an ideal gas, Cp = CV +R, where CV = ( ∂Q ∂T ) V , CP = ( ∂Q ∂T ) P • Adiomatic index: γ = CP CV – γ = 5 3 for monoatomic gases, γ = 7 5 for diatomic gases • Stirling’s Approximation: ln(n!) ≈ n ln(n)− n, for n >> 1 2 Energy equations • Thermodynamic Identity: dE = TdS − PdV + µdN • Helmholtz Free Energy: A = E − TS ⇒ dA = −PdV − SdT + µdN • Enthalpy: H = E + PV → dH = TdS + µdN + V dP • Gibbs free energy: G = A+ PV = E + PV − TS → dG = V dP − SdT + µdN • Maxwell relations: ∂ ∂xj ( ∂Φ ∂xi ) = ∂ ∂xi ( ∂Φ ∂xj ) 2