Pure Substances

Concept

Pure Substances — Phases, Property Relations, and Equations of State (Continuum + Statistical Foundations)

Scope: rigorous, step-by-step derivations connecting macroscopic thermodynamics to microscopic statistical mechanics for pure substances. Covers phase behavior, Clapeyron relations, Maxwell identities, cubic EoS, virial expansions, departure functions, fugacity, and property evaluation workflows. Includes stability, spinodals, and speed of sound.

1. Definitions and Phase Concepts

Pure substance: a system with uniform chemical identity throughout. May exist in multiple phases (solid, liquid, vapor) but all phases have the same chemical composition.

Saturation states: pairs of equilibrium states where two phases coexist at the same T,PT, P. Along the saturation curve, two intensive variables are not independent; specifying one fixes the other.

Quality xx in liquid–vapor mixtures: x=mvml+mv,v=(1−x)vf+xvg,  h=(1−x)hf+xhg,  s=(1−x)sf+xsg.x = \frac{m_v}{m_l+m_v}, \qquad v = (1-x) v_f + x v_g,\; h = (1-x) h_f + x h_g,\; s = (1-x) s_f + x s_g.

Critical point: (Tc,Pc,vc)(T_c, P_c, v_c) at the end of the liquid–vapor coexistence curve, where latent heat vanishes and properties of the phases become identical. Near TcT_c many properties show non-analytic behavior.

Gibbs phase rule (non-reactive): F=C−P+2=1−P+2=3−P.F = C - P + 2 = 1 - P + 2 = 3 - P. For a single component: single phase F=2F=2, two-phase F=1F=1, three-phase F=0F=0 at the triple point.

2. Fundamental Relation and Exact Differentials

For a simple compressible system: dU=T dS−P dV.dU = T\,dS - P\,dV. Legendre transforms give dH=T dS+V dP,dA=−S dT−P dV,dG=−S dT+V dP.dH = T\,dS + V\,dP,\quad dA = -S\,dT - P\,dV,\quad dG = -S\,dT + V\,dP. Because U,H,A,GU, H, A, G are state functions with continuous second derivatives in single-phase regions, mixed partials commute. This yields Maxwell relations:

(∂T∂V)S=−(∂P∂S)V,(∂T∂P)S=(∂V∂S)P,(∂S∂V)T=(∂P∂T)V,(∂S∂P)T=−(∂V∂T)P.\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V,\quad \left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P,\quad \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V,\quad \left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P.

Heat capacities: cv=(∂u∂T)v,cp=(∂h∂T)p.c_v = \left(\frac{\partial u}{\partial T}\right)_v,\qquad c_p = \left(\frac{\partial h}{\partial T}\right)_p. From dHdH and dUdU plus Maxwell relations:

cp−cv=T(∂P∂T)v(∂v∂T)p. c_p - c_v = T\left(\frac{\partial P}{\partial T}\right)_v\left(\frac{\partial v}{\partial T}\right)_p.

Define thermal expansion α\alpha and isothermal compressibility κT\kappa_T: α=1v(∂v∂T)p,κT=−1v(∂v∂p)T.\alpha = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_p, \qquad \kappa_T = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_T. Then cp−cv=Tvα2κT.c_p - c_v = \frac{T v \alpha^2}{\kappa_T}.

Speed of sound aa: a2=(∂p∂ρ)s=−v2(∂p∂v)s=cpcv 1ρκT.a^2 = \left(\frac{\partial p}{\partial \rho}\right)_s = -v^2\left(\frac{\partial p}{\partial v}\right)_s = \frac{c_p}{c_v}\,\frac{1}{\rho\kappa_T}. Derivation uses identities linking isentropic and isothermal derivatives.

3. Phase Equilibrium and Clapeyron Equation

For two phases α,β\alpha,\beta of a pure substance at equilibrium at T,PT, P: equality of chemical potentials μα(T,P)=μβ(T,P).\mu^{\alpha}(T,P) = \mu^{\beta}(T,P). Differentiating: dμ=−s dT+v dPd\mu = -s\,dT + v\,dP gives −(sα−sβ) dT+(vα−vβ) dP=0.-(s^{\alpha}-s^{\beta})\,dT + (v^{\alpha}-v^{\beta})\,dP = 0. Thus the slope of the coexistence curve is dPdT=ΔsΔv=hβ−hαT (vβ−vα)=Δhα→βT Δvα→β.\boxed{\frac{dP}{dT} = \frac{\Delta s}{\Delta v} = \frac{h^{\beta}-h^{\alpha}}{T\,(v^{\beta}-v^{\alpha})} = \frac{\Delta h_{\alpha\to\beta}}{T\,\Delta v_{\alpha\to\beta}}}. This is the Clapeyron equation.

Clausius–Clapeyron approximation for liquid–vapor when vg≫vlv_g \gg v_l and vapor is ideal, Δv≈vg=RT/P\Delta v \approx v_g = RT/P: dln⁡PsatdT=ΔhlvRT2⇒ln⁡Psat=−ΔhlvR 1T+C.\frac{d\ln P_{sat}}{dT} = \frac{\Delta h_{lv}}{R T^2} \quad\Rightarrow\quad \ln P_{sat} = -\frac{\Delta h_{lv}}{R}\,\frac{1}{T} + C. This motivates Antoine-like correlations for Psat(T)P_{sat}(T).

Maxwell construction in the subcritical isotherm of a cubic EoS enforces equal areas to ensure equal Gibbs free energy of coexisting liquid and vapor.

4. Stability, Spinodals, and Criticality

Mechanical stability requires (∂P∂V)T<0,κT>0.\left(\frac{\partial P}{\partial V}\right)_T < 0, \qquad \kappa_T > 0. The spinodal is where (∂P∂V)T=0,\left(\frac{\partial P}{\partial V}\right)_T = 0, separating metastable and unstable states. The critical point satisfies (∂P∂V)Tc=0,(∂2P∂V2)Tc=0.\left(\frac{\partial P}{\partial V}\right)_{T_c} = 0, \quad \left(\frac{\partial^2 P}{\partial V^2}\right)_{T_c} = 0. For cubic EoS this gives closed-form (Tc,Pc,vc)(T_c, P_c, v_c) relations.

5. Equations of State (EoS)

5.1 Ideal Gas from Statistical Mechanics

Canonical partition function for a monatomic ideal gas (distinguishable states corrected by 1/N!1/N!): ZN(T,V)=1N![VΛ3]N,Λ=h22πmkBT.Z_N(T,V) = \frac{1}{N!}\left[\frac{V}{\Lambda^3}\right]^N, \qquad \Lambda = \sqrt{\frac{h^2}{2\pi m k_B T}}. Helmholtz free energy A=−kBTln⁡ZNA = -k_B T \ln Z_N gives P=−(∂A∂V)T,N=NkBTV⇒Pv=RT.P = -\left(\frac{\partial A}{\partial V}\right)_{T,N} = \frac{N k_B T}{V} \Rightarrow P v = R T. Internal energy: U=32NkBTU=\tfrac{3}{2} N k_B T, hence cv=32Rc_v=\tfrac{3}{2}R for monatomic.

5.2 Virial Expansion (Real-Gas Corrections)

Define compressibility factor Z=PvRTZ = \tfrac{P v}{R T}. For not-too-dense gases: Z=1+B(T)v+C(T)v2+⋯=1+B′(T)P+C′(T)P2+⋯ .Z = 1 + \frac{B(T)}{v} + \frac{C(T)}{v^2} + \cdots = 1 + B'(T)P + C'(T)P^2 + \cdots. Coefficients derive from cluster integrals of the intermolecular potential ϕ(r)\phi(r) (e.g., Lennard–Jones). Second virial coefficient: B(T)=−12∫(e−βϕ(r)−1) dr.B(T) = -\tfrac{1}{2}\int (e^{-\beta \phi(r)} - 1)\, d\mathbf r. This links macroscopic non-ideality to microscopic forces.

5.3 Cubic EoS (Van der Waals, Redlich–Kwong, Soave–RK, Peng–Robinson)

A generic cubic form in vv: P=RTv−b−a(T)v(v+b)+b(v−b).P = \frac{R T}{v - b} - \frac{a(T)}{v(v+b) + b(v-b)}. Van der Waals: P=RTv−b−av2P = \frac{R T}{v-b} - \frac{a}{v^2}. Critical constraints yield a=27R2Tc264Pc,b=RTc8Pc.a = 27\frac{R^2 T_c^2}{64 P_c}, \qquad b = \frac{R T_c}{8 P_c}. RK/SRK/PR introduce temperature-dependent a(T)=acα(T)a(T)=a_c \alpha(T) via acentric factor ω\omega to fit vapor pressure. For Peng–Robinson: P=RTv−b−a(T)v(v+b)+b(v−b),ac=0.45724R2Tc2Pc,  b=0.07780RTcPc.P = \frac{R T}{v-b} - \frac{a(T)}{v(v+b)+b(v-b)},\quad a_c=0.45724\frac{R^2 T_c^2}{P_c},\; b=0.07780\frac{R T_c}{P_c}. The α(T)\alpha(T) function uses κ(ω)\kappa(\omega) to match saturation data.

Fugacity for a pure fluid from cubic EoS follows from residual chemical potential μR\mu^R (see §6). Vapor–liquid equilibrium satisfies f(l)=f(v)f^{(l)}=f^{(v)} or ϕ(l)P=ϕ(v)P\phi^{(l)} P = \phi^{(v)} P, where ϕ\phi is the fugacity coefficient.

5.4 Fundamental-Equation EoS in a(τ,δ)a(\tau,\delta)

Modern formulations tabulate non-ideal Helmholtz energy a=A/(RT)a = A/(R T) as a function of reduced variables τ=Tc/T\tau=T_c/T, δ=ρ/ρc\delta=\rho/\rho_c. Properties follow by differentiation:

P=ρRT(1+δ aδ),u=RT2aτ,s=R(τaτ−a),h=u+P/ρ. P = \rho R T\left(1 + \delta\, a_\delta\right),\quad u = R T^2 a_\tau,\quad s = R\left(\tau a_\tau - a\right),\quad h = u + P/\rho.

This route underpins high-accuracy standards for water and refrigerants.

6. Residual/Departure Functions and Fugacity

Define residual (departure from ideal gas at same T,PT,P) for molar property MM: MR=M−Mig(T,P)M^R = M - M^{ig}(T,P).

From any EoS in P(T,v)P(T,v) form and the fundamental relations: (∂sR∂P)T=−(∂v∂T)P+RP,(∂hR∂P)T=v−T(∂v∂T)P−RTP.\left(\frac{\partial s^R}{\partial P}\right)_T = -\left(\frac{\partial v}{\partial T}\right)_P + \frac{R}{P}, \qquad \left(\frac{\partial h^R}{\partial P}\right)_T = v - T\left(\frac{\partial v}{\partial T}\right)_P - \frac{R T}{P}. Integrate from zero pressure where residuals vanish. Common working formulas with compressibility factor ZZ:

gR=RTln⁡ϕ,hR=RT[Z−1−T(∂Z∂T)P],sR=R[ln⁡Z−∫0P(∂Z∂T)P′dP′]. g^R = R T\ln \phi,\quad h^R = R T\left[ Z - 1 - T\left(\frac{\partial Z}{\partial T}\right)_P \right],\quad s^R = R\left[ \ln Z - \int^P_0 \left(\frac{\partial Z}{\partial T}\right)_{P'} dP' \right].

For cubic EoS, closed forms exist for ln⁡ϕ\ln \phi in terms of ZZ, A=aP/(R2T2)A=aP/(R^2T^2), and B=bP/(RT)B=bP/(RT).

Fugacity ff of a pure substance: μ(T,P)=μig(T,P)+RTln⁡ϕ,f=ϕP.\mu(T,P) = \mu^{ig}(T,P) + R T\ln \phi, \quad f = \phi P. Phase equilibrium at fixed TT: f(l)(T,P)=f(v)(T,P)f^{(l)}(T,P) = f^{(v)}(T,P).

7. Property Evaluation Workflows

7.1 Saturation properties from correlations or EoS

  1. Given TT, compute Psat(T)P_{sat}(T) from correlation or solve two-phase equality f(l)=f(v)f^{(l)}=f^{(v)} with EoS.
  2. Compute phase molar volumes from EoS roots or Helmholtz formulation.
  3. Evaluate h,s,uh,s,u via residuals plus ideal-gas references.

7.2 Single-phase properties at (T,P)(T,P)

  1. Solve EoS for vv given T,PT,P (unique root outside two-phase).
  2. Compute ZZ, then residuals hR,sR,gRh^R,s^R,g^R.
  3. Add ideal-gas contributions at TT using cpig(T)c_p^{ig}(T) correlations.

7.3 Two-phase mixture properties

Given (T,P)(T,P) in the dome, obtain xx from lever rule using vv or ss. Interpolate extensive properties linearly in xx.

8. Non-analytic Critical Behavior (Brief)

Close to TcT_c, real fluids deviate from mean-field cubic predictions. Empirically, order parameter Δρ∼∣T−Tc∣β\Delta\rho \sim |T-T_c|^{\beta} with critical exponent β≈0.326\beta\approx 0.326. While not derived here, be aware that cubic EoS embed mean-field exponents and need crossover corrections for accuracy near TcT_c.

9.1 Intermolecular Potentials and Second Virial

Given pair potential ϕ(r)\phi(r), the configurational integral yields the second virial coefficient (as in §5.2). Attractive wells make B(T)B(T) more negative; repulsions raise B(T)B(T).

9.2 Corresponding States and Acentric Factor

For many nonpolar fluids, reduced variables produce approximate collapse: Z=Z(Tr,Pr,ω)Z = Z(T_r, P_r, \omega). The acentric factor ω\omega encodes shape/polarity corrections and feeds α(T)\alpha(T) in cubic EoS.

9.3 Lattice and Cell Models (Outline)

Simple lattice-gas or cell models map to Ising-like systems and produce phase separation and criticality. They justify qualitatively the van der Waals attraction–repulsion decomposition.

10. Metastability and Nucleation (Engineering Relevance)

Metastable states persist between binodal and spinodal lines. Actual boiling often initiates via heterogeneous nucleation at imperfections. Superheat limits are near the spinodal. Design must allow for nucleation kinetics, not only equilibrium curves.

11. Diagrams and Practical Use

  • PP–vv–TT surface: visualize isotherms and the two-phase dome.
  • TT–ss and hh–ss: useful for cycle analysis; isentropes are vertical in TT–ss.
  • Compressibility charts: generalized Z(Tr,Pr)Z(T_r,P_r) for gases when specific EoS data are absent.

Workflow to trace an isentropic path with an EoS: integrate ds(T,p)=(cpT)dT−(αvκT)dpds(T,p)=\left(\frac{c_p}{T}\right)dT - \left(\frac{\alpha v}{\kappa_T}\right) dp using cp,α,κTc_p,\alpha,\kappa_T computed from the EoS.

12. Worked Derivations

12.1 Derive $c_p-c_v = T v

alpha^2/\kappa_TStartfromidentities:(1)Start from identities: (1)c_p - c_v = T\left(\frac{\partial s}{\partial T}\right)_p - T\left(\frac{\partial s}{\partial T}\right)_v.(2)Using. (2) Using ds = \left(\frac{\partial s}{\partial T}\right)_v dT + \left(\frac{\partial s}{\partial v}\right)_T dvandMaxwelland Maxwell (\partial s/\partial v)_T = (\partial p/\partial T)_v .(3)Evaluateatconstant. (3) Evaluate at constant pusingJacobianstorelateusing Jacobians to relatedvtotodT:: (\partial v/\partial T)_p = v\alpha.(4)Combinetoobtainthestatedresultusing. (4) Combine to obtain the stated result using \kappa_T = -\tfrac{1}{v}(\partial v/\partial p)_T$ and cyclic relations.

12.2 Clapeyron from Gibbs Potentials

For pure phases, dμ=−s dT+v dPd\mu = -s\,dT + v\,dP; at coexistence, dμα=dμβd\mu^\alpha = d\mu^\beta. Rearranging yields dP/dT=Δs/ΔvdP/dT = \Delta s/\Delta v. Substituting Δs=Δh/T\Delta s = \Delta h/T gives the heat-of-transition form.

12.3 Fugacity Coefficient from Cubic EoS (Sketch)

Express residual Gibbs gR=∫(v−RT/P) dPg^R = \int (v - R T/P)\, dP at constant TT, substitute cubic Z(P)Z(P), integrate analytically to obtain ln⁡ϕ\ln \phi in terms of ZZ, A=aP/(R2T2)A=aP/(R^2T^2), and B=bP/(RT)B=bP/(RT). Set equality of ϕP\phi P across phases to compute saturation and phase splits.

13. Summary of Key Equations

  • Fundamental: dU=TdS−PdVdU=T dS - P dV; dH=TdS+VdPdH=T dS + V dP; dA=−SdT−PdVdA=-S dT - P dV; dG=−SdT+VdPdG=-S dT + V dP.
  • Maxwell set; heat capacities relation cp−cv=Tvα2/κTc_p-c_v = T v \alpha^2/\kappa_T.
  • Clapeyron: dP/dT=Δh/(T Δv)dP/dT = \Delta h/(T\,\Delta v); Clausius–Clapeyron for liquid–vapor.
  • Virial: Z=1+B/v+C/v2+⋯Z=1+B/v+C/v^2+\cdots, B(T)B(T) from ϕ(r)\phi(r).
  • Cubic EoS: PR/SRK forms with a(T),ba(T), b; critical parameter relations.
  • Residuals: gR=RTln⁡ϕg^R=RT\ln\phi, hR,sRh^R, s^R from ZZ.
  • Helmholtz fundamental a(τ,δ)a(\tau,\delta) derivatives for high-accuracy properties.
  • Speed of sound: a2=(∂p/∂ρ)s=γ/(ρκT)a^2 = (\partial p/\partial \rho)_s = \gamma/(\rho\kappa_T) with γ=cp/cv\gamma=c_p/c_v.
  • See second-law.md for entropy, exergy, and stability inequalities.
  • See Appendix C for Jacobian identities and derivative transforms.
  • See Fluid_Dynamics/07_Compressible_Flow.md for use of real-gas a(T,p)a(T,p) in nozzle and wave calculations.