Fundamental Thermodynamic Relations and Equations
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Equations of State and Real Gases
Critical Point Conditions: (∂p/∂v)Tc = (∂²p/∂v²)Tc = 0
- Real Gas Law: pv = ZRT
- Van der Waals Equation: (p + a/v²)(v - b) = RT
- Reduced Variables: vr = v/vc; Tr = T/Tc; pr = p/pc
- Compressibility Factor: Z = Z(pr, Tr)
- Semi-Reduced Volume: vr,sem = v/vc
Thermodynamic Differentials and State Functions
Functions of Temperature and Pressure (T, p)
- Enthalpy Differential: dh = CpdT + [v - T(∂v/∂T)p]dp
- Internal Energy Differential: dU = [Cp - p(∂v/∂T)p]dT - [T(∂v/∂T)p + p(∂v/∂p)T]dp
Functions of Temperature and Volume (T, v)
- Entropy Differential: dS = (Cv/T)dT + (∂p/∂T)vdv
- Enthalpy Differential: dh = [Cv + v(∂p/∂T)v]dT + [T(∂p/∂T)v + v(∂p/∂v)T]dv
- Internal Energy Differential: dU = CvdT + [T(∂p/∂T)v - p]dv
Functions of Pressure and Volume (p, v)
- Entropy Differential: dS = [(Cv/T)(∂T/∂p)v]dp + [(Cp/T)(∂T/∂v)p]dv
- Enthalpy Differential: dh = [Cv(∂T/∂p)v + v]dp + [Cp(∂T/∂v)p]dv
- Internal Energy Differential: dU = [Cv(∂T/∂p)v]dp + [Cp(∂T/∂v)p - p]dv
Heat Capacity and Thermal Coefficients
- Mayer's Relation: Cp - Cv = T(∂v/∂T)p(∂p/∂T)v
- Alternative Mayer's Relation: Cp - Cv = -T(∂v/∂T)p²(∂p/∂U)T = Tv(α²/κt)
- Expansion Coefficient (α): α = (1/v)(∂v/∂T)p
- Pressure Coefficient (β): β = (1/p)(∂p/∂T)v
- Isothermal Compressibility (κt): κt = (-1/v)(∂v/∂p)T > 0
- Adiabatic Compressibility (κs): κs = (-1/v)(∂v/∂p)S
- Ratio of Compressibilities: κt/κs = γ
- Speed of Sound: c = √((∂p/∂ρ)S)
Thermodynamic Potentials
- Internal Energy: U(S, v); dU = TdS - pdv; (∂U/∂S)v = T; (∂U/∂v)S = -p
- Helmholtz Free Energy: A(T, v) = U - TS; dA = -SdT - pdv; -S = (∂A/∂T)v; -p = (∂A/∂v)T
- Enthalpy: H(S, p) = U + pV; dH = TdS + vdp
- Gibbs Free Energy: G(T, p) = U - TS + pV; dG = -SdT + vdp
Joule-Thomson Effect
Coefficient: μJT = (∂T/∂p)h = [T(∂v/∂T)p - v] / Cp
- Cooling (Left of Inversion Curve): μJT > 0; at constant h, if p₂ < p₁, then T₂ < T₁.
- Heating (Right of Inversion Curve): μJT < 0; at constant h, if p₂ < p₁, then T₂ > T₁.
Maxwell Relations and Clausius Equations
- (∂T/∂v)S = -(∂p/∂S)v
- (∂S/∂v)T = (∂p/∂T)v
- (∂T/∂p)S = (∂v/∂S)p
- -(∂S/∂p)T = (∂v/∂T)p
- Clausius Relations: (∂Cp/∂p)T = -T(∂²v/∂T²)p; (∂Cv/∂v)T = T(∂²p/∂T²)v
Chemical Potential and Phase Equilibrium
- Chemical Potential (μi): μi = (∂U/∂Ni)S,v,Nj = (∂A/∂Ni)T,v,Nj = (∂H/∂Ni)S,p,Nj = (∂G/∂Ni)T,p,Nj
- Fundamental Equation: dU = TdS - pdV + ΣμidNi
- Euler Equation: U = TS - pV + ΣμiNi
- Gibbs-Duhem Equation: SdT - Vdp + ΣNidμi = 0
- Gibbs Phase Rule: L = C - F + 2
- Clapeyron Equation: (dp/dT)α,β = (sα - sβ) / (vα - vβ) = (1/T)(hα - hβ) / (vα - vβ)