Evolution of Equations of State
Module EOS · 2105603 Advanced Chemical Engineering Thermodynamics
8 August 2026
The argument of this course
Introduction
State the organising thesis and the six questions applied to every model.
Six questions asked of every model
- What problem existed?
- Why did the previous equation of state fail?
- What new physics was introduced?
- How was the mathematics modified?
- What limitations remain?
- Why was another equation of state developed?
The hierarchy of physics contained in an equation of state.
One hundred and fifty years in one line
Introduction
Place every model of the course on a common historical and conceptual axis.
Physical meaning
Classical models (navy) add mean-field physics. Molecular models (amber) add structure: chains, hydrogen bonds, charges. Data-driven models (teal) learn the residual free energy directly.
Engineering implication
The models in industrial use today span a century of physics. A simulator flowsheet routinely mixes a 1976 cubic for the hydrocarbons with a 2001 molecular model for the solvent.
What an equation of state is for
Chapter 1 · Why do we need an equation of state?
Identify the EOS as the closure relation that converts measurable P, T and composition into every other thermodynamic property.
Physical meaning
The EOS supplies the missing relation among P, V, T and composition. Everything else follows from classical thermodynamics without further assumption.
Engineering implication
In a process simulator the EOS is evaluated more often than any other model. Its error propagates into compressor duty, vessel volume, relief sizing and custody metering.
Where the ideal gas law is good enough
Chapter 2 · The ideal gas law
Delimit the region of the reduced state plane in which PV = RT is accurate to one and to five per cent.
Physical meaning
Ideality is a low-density statement. The gas behaves ideally when the mean separation is large compared with the molecular diameter and the thermal energy is large compared with the well depth.
Engineering implication
For light gases above about twice the critical temperature and below a few bar, PV = RT is adequate for line sizing and vent calculations. Outside that window it is not defensible.
Two distinct failure modes
Chapter 2 · The ideal gas law
Distinguish failure by density from failure by proximity to saturation, and read both from the Z curve.
Physical meaning
The initial fall below unity is attraction. The eventual rise above unity is the finite size of the molecules. Both are absent from PV = RT by construction.
Engineering implication
Near saturation the error is not a smooth few per cent: it is a discontinuity, because a real fluid condenses and the ideal gas law does not know that phases exist.
van der Waals 1873: the molecular picture
Chapter 3 · The van der Waals equation of state
Derive the two corrections from a molecular argument rather than accept them as empirical terms.
Excluded volume
Two hard spheres of diameter sigma cannot approach closer than sigma. The excluded volume per molecule is b = 2 pi N sigma cubed over 3, four times the hard-sphere volume.
Mean-field attraction
Each molecule feels an average attractive field proportional to the local density. The energy density therefore scales as density squared, giving a pressure reduction of a over V squared.
Isotherms, the dome and the critical point
Chapter 3 · The van der Waals equation of state
Read phase behaviour directly from the shape of the predicted isotherms.
Physical meaning
One analytic expression produces a liquid branch, a vapour branch and a critical point. Before 1873 no theory did this.
Engineering implication
Cubic equations remain the only models fast and robust enough to be called millions of times inside a distillation column solver.
The Maxwell equal-area construction
Chapter 3 · The van der Waals equation of state
Replace the mechanically unstable loop by the equilibrium saturation pressure using equality of Gibbs energy.
\int_{V^{L}}^{V^{V}}\left[P^{\mathrm{sat}} - P(V)\right]\mathrm{d}V = 0
What the rule enforces
Equality of molar Gibbs energy between the saturated liquid and the saturated vapour, which is the phase-equilibrium criterion applied along one isotherm.
The unstable branch
Where dP/dV is positive the fluid has negative compressibility and cannot exist. It is excluded by stability, not by the equation.
The verdict on van der Waals
Chapter 3 · The van der Waals equation of state
Identify the specific quantitative failure that motivates every later cubic equation.
Motivates Chapter 8
Redlich and Kwong changed the volume dependence of the attraction and made it decay with temperature.
The compressibility factor as a diagnostic
Chapter 4 · The compressibility factor
Interpret the sign of Z minus one, the location of the minimum, and the crossover, in molecular terms.
Z<1: \text{ attraction dominates} \qquad Z>1: \text{ repulsion dominates}
Z < 1
Attraction dominates: the fluid occupies less volume than an ideal gas at the same T and P.
Z > 1
Repulsion dominates: the finite molecular core resists compression and the fluid is stiffer than ideal.
Z = 1
Coincidence, not ideality. The two effects merely cancel.
The Boyle temperature
Chapter 4 · The compressibility factor
Connect the initial slope of Z against P to the second virial coefficient and define the Boyle temperature.
Lennard-Jones fluid: the second virial coefficient computed by numerical integration of the Mayer function gives a reduced Boyle temperature of 3.42.
\lim_{P\to 0}\left(\frac{\partial Z}{\partial P}\right)_{\!T} = \frac{B_{2}(T)}{RT}
B_{2}(T_{\mathrm{B}}) = 0
Reading the slope
Below the Boyle temperature the initial slope is negative: attraction wins first. Above it the slope is positive: the repulsive core is felt immediately.
The generalized compressibility chart
Chapter 4 · The compressibility factor
Read Z from reduced coordinates and recognise the chart as the engineering form of corresponding states.
Left: the classical chart form. Right: the same information as a contour map of Z over the reduced state plane, with the saturation line and the Z = 1 contour marked.
From the pair potential to the second virial coefficient
Chapter 5 · The virial equation of state
Connect a molecular-level potential to a macroscopic, measurable coefficient through the Mayer function.
B_{2}(T) = -2\pi N_{\mathrm{A}}\int_0^{\infty}\left[\mathrm{e}^{-u(r)/k_{\mathrm{B}}T}-1\right]r^{2}\,\mathrm{d}r
Left: the Lennard-Jones potential and its Mayer function at two temperatures. Right: the resulting second virial coefficient, negative at low temperature, crossing zero at the Boyle point.
Why truncation limits the virial equation
Chapter 5 · The virial equation of state
Quantify the density at which two-term and three-term truncations fail, and draw the practical boundary.
Physical meaning
Truncation is a statement about how many bodies interact simultaneously. At liquid density every molecule interacts with a dozen neighbours at once, so a two-body or three-body expansion cannot converge.
Engineering implication
For compressor and pipeline calculations in the gas phase the two-term virial equation is accurate and cheap. For anything involving a liquid it is not an option, which is why the cubic family survived.
Universality and its limits
Chapter 6 · The principle of corresponding states
Observe the collapse for simple fluids and the systematic departure for polar and associating fluids.
Physical meaning
Two parameters force a single universal critical compressibility. Real fluids have critical compressibilities from 0.23 to 0.29, so a two-parameter theory cannot be exact for all of them.
Engineering implication
Two-parameter charts are adequate for light hydrocarbons and permanent gases. For refrigerants, water and amines they are not, and a third parameter is mandatory.
Why the critical constants are not enough
Chapter 7 · The Pitzer acentric factor
Show that reduced vapour-pressure curves have different slopes, and that the slope carries the missing information.
Physical meaning
In reduced coordinates every vapour-pressure line is pinned at the critical point. What distinguishes fluids is how steeply the line falls, which reflects the strength and directionality of the intermolecular forces.
Engineering implication
This is a practical choice as well as a physical one: vapour pressure is the most widely measured property in chemical engineering, so a parameter defined from it is available for essentially every compound.
Pitzer’s acentric factor
Chapter 7 · The Pitzer acentric factor
Define the acentric factor from a single point on the reduced vapour-pressure curve and interpret it physically.
\omega \;\equiv\; -1 - \log_{10}\left(P_{\mathrm{r}}^{\mathrm{sat}}\right)_{T_{\mathrm{r}}=0.7}
Why reduced temperature 0.7
Close to the normal boiling point of many fluids, comfortably subcritical, easily measured, and far enough from the critical point to avoid critical anomalies.
Physical meaning
The acentric factor measures how far the intermolecular force field departs from spherical symmetry. It increases with chain length, with polarity and with hydrogen bonding.
The acentric factor across molecular families
Chapter 7 · The Pitzer acentric factor
Relate the numerical value of the acentric factor to molecular architecture, and state the three-parameter correlation.
Z = Z^{(0)}(T_{\mathrm{r}},P_{\mathrm{r}}) \;+\; \omega\,Z^{(1)}(T_{\mathrm{r}},P_{\mathrm{r}})
Three-parameter corresponding states
A simple-fluid reference surface plus a linear correction weighted by the acentric factor. The Lee-Kesler tables are the standard numerical implementation.
Where it still fails
Strongly associating fluids, whose reduced vapour-pressure curves are not straight, so a single slope parameter cannot represent them.
The temperature dependence of the attraction
Chapter 8 · Redlich-Kwong
Compare the alpha functions of the four classical cubics and see how each encodes a different physical assumption.
a(T) = a_{\mathrm{c}}\,\alpha(T_{\mathrm{r}},\omega), \qquad \alpha(1,\omega) = 1
What Redlich-Kwong bought, and what it did not
Chapter 8 · Redlich-Kwong
Separate the genuine improvement in the vapour phase from the persistent failures in the liquid phase.
Physical meaning
The vapour-phase compressibility improves substantially: errors of several per cent under van der Waals fall to about one per cent for simple fluids.
Engineering implication
Liquid densities remain poor and vapour pressures remain wrong for anything but simple fluids, because no molecular shape or polarity parameter enters the equation.
The effect on predicted vapour pressure
Chapter 9 · Soave-Redlich-Kwong
Quantify the improvement Soave’s alpha function delivers relative to the fixed Redlich-Kwong power law.
Physical meaning
The fixed power law of Redlich-Kwong is equivalent to assigning every substance the same acentric factor. Soave’s regression removes that constraint.
Engineering implication
Deviations exceeding thirty per cent at low reduced temperature translate directly into equilibrium-ratio errors, and therefore into stage counts and reflux ratios.
Liquid density: the reason Peng-Robinson won
Chapter 10 · Peng-Robinson
Compare saturated liquid volumes and coexistence envelopes from Soave and Peng-Robinson.
Left: saturated liquid molar volume of propane. Right: the coexistence envelope in density-temperature coordinates. The two models agree closely on the vapour branch and differ systematically on the liquid branch.
van der Waals, Redlich-Kwong, Soave and Peng-Robinson compared
Chapter 10 · Peng-Robinson
Assess the four classical cubics on the criteria that actually decide model selection in industry.
Physical meaning
Each row adds one piece of physics and each addition shows up in exactly one or two columns. No row adds physics that improves every column at once.
Engineering implication
The final column explains the persistence of the family: all four are solvable analytically or in a few Newton steps, which is what an inner-loop model must be.
Four terms, four pieces of physics
Chapter 11 · SAFT
Attribute each Helmholtz contribution to a specific molecular mechanism and to the cubic-equation defect it repairs.
Hard sphere
Excluded volume treated with the Carnahan-Starling result rather than a single constant b.
Chain
Connectivity of bonded segments. Replaces the acentric factor with a structural parameter.
Dispersion
Attraction between segments from perturbation theory rather than a mean field.
Association
Short-range directional bonding. No cubic equation contains this at all.
What the terms contribute, and what they buy
Chapter 11 · SAFT
Judge the relative magnitude of each contribution and the class of systems on which the investment pays.
Both panels are schematic illustrations of the qualitative behaviour, not output from a validated SAFT implementation.
Ions change the range of the forces
Chapter 12 · Electrolyte equations of state
Recognise that Coulomb interactions are long ranged and require terms with no counterpart in a neutral-fluid model.
The Debye screening length sets the range over which an ion feels the surrounding ionic atmosphere.
\kappa^{-1} = \left(\frac{\varepsilon_{\mathrm{r}}\varepsilon_{0}k_{\mathrm{B}}T}{e^{2}\sum_i \rho_i z_i^{2}}\right)^{\!1/2}
Concentrated solutions and the activity minimum
Chapter 12 · Electrolyte equations of state
Explain the minimum and upturn in the mean ionic activity coefficient in terms of competing physics.
Physical meaning
Screening lowers the activity coefficient. Solvent depletion and hydrated-ion repulsion raise it. The minimum is where the two mechanisms balance.
Engineering implication
Salt solubility limits, low-temperature precipitation in battery electrolytes and amine degradation in capture solvents all depend on the region beyond the minimum.
Application: battery electrolytes
Chapter 12 · Electrolyte equations of state
Map electrolyte thermodynamic outputs onto the design targets of an electrochemical cell.
Physical meaning
The solvation shell is the physical object that connects thermodynamics, transport and interfacial chemistry in a cell. An equation of state that resolves it describes all three consistently.
Engineering implication
Voltage window, solid-electrolyte-interphase chemistry, salt precipitation risk and low-temperature operating limit are all set by properties an electrolyte equation of state can compute.
Where the subject is going
Chapter 13 · Future directions
Identify the four active research directions and the specific limitation each one addresses.
Physical meaning
Each direction targets a specific gap: missing data, rigid functional form, unenforced thermodynamic consistency, or unmodelled electrochemical coupling.
Engineering implication
The practical near-term gain is parameter estimation for compounds that have never been measured, which is the binding constraint in specialty chemicals and in new battery chemistries.
Physics-informed machine learning equations of state
Chapter 13 · Future directions
Describe an architecture that learns a Helmholtz function under hard thermodynamic constraints.
A^{\mathrm{res}}_{\theta}(T,\rho,\mathbf{d}) \;\longrightarrow\; P,\;\mu_i,\;c_v \;\; \text{by automatic differentiation}
\lim_{\rho\to 0}A^{\mathrm{res}} = 0, \qquad c_v > 0, \qquad \oint V\,\mathrm{d}P = 0
The honest failure mode is extrapolation. Inside the training envelope such models are accurate; outside it they remain confident and are not. Any deployment must include an explicit domain-of-validity check.
Closing the argument
Conclusion
Restate the thesis of the course and convert it into a working rule for model selection.
The rule to take away
Do not ask whether an equation of state is accurate. Ask which physics it contains, and whether that physics is the physics of your problem. Every model in this course answers that question honestly if you know how to read it.