Tip: Drag T below T_c to see the unstable van der Waals loop. Maxwell construction replaces it with a flat coexistence line.
Section 02
The Idea, Step by Step
Take an ice cube out of the freezer and leave it on the counter. It doesn't slowly warm into room-temperature ice — it sits stubbornly at $0°$C while it melts, then the puddle heats up and finally boils away at $100°$C. The water molecules are the same the whole time; what changes is how they are arranged. Locked in a rigid lattice (solid), sliding over each other but still touching (liquid), or flying free and far apart (gas). Switching from one arrangement to another is a phase transition.
Two knobs decide which phase you get: the temperature $T$ (how hard the molecules jiggle) and the pressure $P$ (how hard they are squeezed together). The ideal gas law $PV = RT$ pretends molecules are points that completely ignore one another — fine for a thin gas, useless for a liquid. Johannes van der Waals fixed it in 1873 with two honest corrections. Molecules take up room, so the space they can move in shrinks from $V_m$ to $V_m - b$, where $b$ is the volume one mole of molecules occupies. And molecules gently pull on each other, an inward tug that lowers the measured pressure by an amount $a/V_m^2$. Put both together for one mole:
Feed in water's measured constants ($a = 0.554\ \text{Pa·m}^6/\text{mol}^2$, $b = 3.05\times10^{-5}\ \text{m}^3/\text{mol}$) and this single equation predicts water's critical temperature $T_c = 8a/27Rb \approx 647$ K — within a degree of the true value, from molecules alone.
That critical temperature is the whole point. Below $T_c$ you can squeeze gas until it condenses into liquid, and the van der Waals curve grows a wiggly "S"-shaped loop. Its middle branch — where squeezing harder would somehow expand the substance — is physically impossible, so nature snips it out and replaces it with a flat horizontal line (Maxwell's equal-area rule) along which liquid and vapour coexist side by side. Above $T_c$ the loop is gone entirely: you can travel from gas-thin to liquid-dense smoothly, never crossing a boundary, never seeing a meniscus. That borderless in-between state is a supercritical fluid. In the sim, the $T$ slider chooses which isotherm you see, while $a$ and $b$ reshape the entire landscape — the readouts for $T_c$, $P_c$ and $V_c$ update live as you drag.
Try this in the sim above. Drag $T$ down from $647$ K toward $\sim540$ K and watch the smooth curve sprout the unstable loop. Switch to Maxwell mode to see that loop replaced by the flat red liquid–gas coexistence line, with two equal areas balanced above and below it. Then open Molecules mode and cool below $T_c$ — the free-flying dots stop wandering and pull together into a dense, slow droplet, condensation happening right in front of you.
Section 03
Equation Derivation
The van der Waals equation generalises the ideal gas law by including molecular volume and intermolecular attraction. It produces the first correct theoretical description of liquid–gas phase transitions.
Governing Equation — van der Waals
$$\left(P + \frac{a n^2}{V^2}\right)(V - n b) = n R T$$
For one mole ($n=1$): $\;\left(P + a/V_m^2\right)(V_m - b) = RT$
Symbol
Meaning
SI Unit
$P$
Pressure
Pa
$V$
Volume
m³
$T$
Absolute temperature
K
$n$
Moles of gas
mol
$R$
Universal gas constant = 8.314
J·mol⁻¹·K⁻¹
$a$
Attractive parameter (cohesion)
Pa·m⁶·mol⁻²
$b$
Excluded volume per mole
m³·mol⁻¹
$T_c$
Critical temperature = $8a/(27Rb)$
K
$P_c$
Critical pressure = $a/(27b^2)$
Pa
$V_c$
Critical molar volume = $3b$
m³·mol⁻¹
Step 1 — Why correct the ideal gas law?
Ideal gas: $PV = nRT$ assumes (i) point particles, (ii) no intermolecular forces. Real molecules occupy space and attract each other.
Volume correction: molecules cannot overlap, so the available volume is reduced from $V$ to $V - nb$, where $b$ is the molar excluded volume.
Pressure correction: attractive forces near the wall reduce the impact pressure. Reduction is proportional to (density)² $\propto (n/V)^2$. So measured $P$ is less than ideal by $a(n/V)^2$.
Step 2 — Combining the corrections
$$P_{\text{ideal}} = P_{\text{measured}} + a\left(\frac{n}{V}\right)^2$$
$$V_{\text{ideal}} = V - nb$$
$$\Rightarrow\;\;\left(P + \frac{a n^2}{V^2}\right)(V - n b) = n R T$$
Step 3 — Critical point
At the critical point, the isotherm has an inflection: $\left(\partial P/\partial V\right)_T = 0$ and $\left(\partial^2 P/\partial V^2\right)_T = 0$.
Solving these two conditions on the van der Waals isotherm gives:
This is the law of corresponding states — all van der Waals gases obey the same equation in reduced variables.
Step 5 — Maxwell construction (coexistence)
Below $T_c$, the isotherm has an unphysical "S-shape" loop where $\partial P/\partial V > 0$ (mechanically unstable). Maxwell's equal-area rule replaces the loop with a horizontal line at $P_{\text{sat}}$ such that:
$$\int_{V_\ell}^{V_g} \left[P_{\text{sat}} - P_{\text{vdW}}(V,T)\right] dV = 0$$
This horizontal segment represents the liquid–gas coexistence: at $P_{\text{sat}}(T)$ both phases are present, with proportions set by the lever rule.
Mapping to the simulation
• Slider T selects the isotherm being plotted on the PV plane.
• Sliders a, b are the molecular parameters; T_c, P_c, V_c are derived live.
• In Molecules mode, particle density and clustering reflect the local phase: gas (sparse, fast), liquid (dense, slow). Crossing the binodal triggers droplet formation.
• The Maxwell mode shows the equal-area construction graphically — the unstable loop, the horizontal saturation line, and the two equal areas above/below it.
Reference: Reif — Fundamentals of Statistical and Thermal Physics, §5.8: "The van der Waals Equation"; Callen — Thermodynamics and an Introduction to Thermostatistics, 2nd Ed., Ch. 9: "First-Order Phase Transitions"; HRW 10th Ed., §19-9.
Section 04
Frequently Asked Questions
Below $T_c$, the cubic equation $(P + a/V^2)(V - b) = RT$ has three real roots for certain pressures. The middle root has $\partial P/\partial V > 0$, which is mechanically impossible (compressing a substance lowers its pressure?!). This unphysical region must be replaced by the horizontal Maxwell coexistence line. The two outer roots correspond to liquid (small $V$) and gas (large $V$) at the same pressure.
Key: The S-shape loop is the equation's "memory" of metastable states (superheated liquid, supercooled vapor) and one truly unstable region in the middle.
Everywhere — every pot of boiling water, melting ice cube, and frosted windowpane. Industrial applications include: refrigeration (vapor-compression cycle), supercritical CO₂ (decaffeination, dry cleaning), liquefied natural gas (LNG transport), steam turbines, and CO₂ extinguishers. Beyond ordinary fluids, the same mathematics describes ferromagnetic transitions (magnetisation appears below the Curie temperature), superconductivity, superfluidity in liquid helium, and Bose-Einstein condensation.
Key: The same critical exponents appear in liquid–gas, magnetic, and superfluid transitions — this is universality.
The PV mode plots the van der Waals isotherm $P(V)$ at the chosen temperature. As T crosses $T_c$, watch the curve change shape: above $T_c$ it's monotonic; at $T_c$ it has a flat inflection at the critical point; below $T_c$ it develops the characteristic loop. The Maxwell mode adds the equal-area construction so you can see the saturation pressure $P_{\text{sat}}(T)$ and the coexistence region. The Molecules mode runs an actual Lennard-Jones-like particle simulation that visually demonstrates gas → liquid clustering as you cool below $T_c$.
Key: Three views of the same physics — analytical (PV curve), graphical (Maxwell), and microscopic (molecules).
Because the van der Waals equation, when written in reduced variables $P_r, V_r, T_r$, contains no material-specific parameters at all. Every van der Waals gas obeys the same reduced equation, so $Z_c$ must be the same constant 3/8. Real gases come close (typical $Z_c \approx 0.27$–0.30) but not exact — the discrepancy reveals the limits of vdW theory and motivates better equations of state (Redlich-Kwong, Peng-Robinson) and the renormalisation-group treatment of true critical behaviour.
Key: Universality (same $Z_c$, same critical exponents) is a hallmark of phase transitions — a precursor of the deep RG insight that critical points lose memory of microscopic details.
Latent heat $L$ is the energy absorbed per mole when transforming between phases at fixed $T, P$ (e.g., melting ice or boiling water) — it goes entirely into changing molecular arrangement, not raising temperature. The Clausius-Clapeyron equation gives the slope of the coexistence curve in the P–T plane: $$\frac{dP}{dT} = \frac{L}{T \Delta V}$$ where $\Delta V$ is the volume change between phases. For water at 100°C, $L \approx 40.7$ kJ/mol, $\Delta V \approx$ 30 L/mol — yielding a slope $dP/dT \approx 3600$ Pa/K, exactly what experiment shows.
Key: Latent heat is not lost energy — it's stored in the structural difference between phases, recovered on the reverse transition.
Above $T_c$ and $P_c$, the substance is a supercritical fluid: it has gas-like diffusivity (great for penetration) and liquid-like density (great for dissolving). There is no longer any boundary between liquid and gas — you can travel continuously from one to the other without ever crossing a phase boundary. Supercritical CO₂ ($T_c=$ 304 K, $P_c=$ 7.4 MPa) is used industrially to extract caffeine from coffee beans without leaving toxic residue. Supercritical water dissolves organic compounds and oxygen simultaneously, useful for waste destruction.
Key: "Liquid" and "gas" are only meaningful below $T_c$. Above it, the distinction disappears entirely.
Near the critical point, microscopic details become irrelevant — only the symmetry of the order parameter and the dimensionality of space matter. Both the liquid–gas system and the uniaxial ferromagnet have a scalar order parameter ($\rho_\ell - \rho_g$ for fluids; magnetisation $M$ for magnets) in 3D, putting them in the same "universality class" (3D Ising). This is one of the great triumphs of 20th-century physics — Wilson's renormalisation group (Nobel Prize 1982) explained why such different systems share identical critical exponents.
Key: Universality classes are determined by symmetry and dimension, not by the specific substance.
Best Resource: HyperPhysics — "van der Waals Equation of State", hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/waal.html; MIT OCW 8.044 — Statistical Physics I, Lecture Notes on Phase Transitions; Khan Academy — "States of matter and intermolecular forces".
Section 05
Common Misconceptions
❌ Boiling and evaporation are the same thing.
✅ They aren't.
Evaporation occurs at any temperature, only at the surface, driven by the high-energy tail of the Maxwell-Boltzmann distribution. Boiling is a bulk phenomenon: it happens at one specific temperature where the saturation vapor pressure equals the ambient pressure, allowing bubbles of vapor to form and grow inside the liquid. This is why water boils at 100°C at sea level but only ~70°C on Mount Everest — lower ambient pressure means the liquid hits $P_{\text{sat}}$ at a lower $T$.
📖 HRW 10th Ed., §19-9: "The Phases of Matter"; Atkins — Physical Chemistry, Ch. 4.
❌ Adding heat always raises the temperature.
✅ Not during a phase transition.
During a first-order phase transition (melting, boiling), heat input goes entirely into changing the molecular arrangement — breaking bonds in melting, separating molecules in boiling — not into kinetic energy. A pot of boiling water stays at 100°C until every drop has evaporated, no matter how high the flame. The heat absorbed at constant temperature is the latent heat $L$. Forgetting this leads to wrong calculations in heating problems with phase changes (e.g., the famous "ice → water → steam" caloric balance).
📖 Serway/Jewett — Physics for Scientists and Engineers, 8th Ed., §20.3: "Latent Heat".
❌ The critical point is just where everything becomes gas.
✅ It's where the distinction between liquid and gas vanishes.
At the critical point, the densities of liquid and vapor become equal — there is no longer any meaningful boundary between them. Above $T_c$, you can compress a fluid to liquid-like density without it ever "becoming a liquid" because there is no condensation surface. This is why the question "is supercritical CO₂ a liquid or a gas?" has no meaningful answer — it's neither and both. Density fluctuations near $T_c$ become enormous, scattering all wavelengths of light, producing the famous "critical opalescence" — a substance that's normally transparent looks milky white.
❌ Pressure cookers cook faster because higher pressure means more force on the food.
✅ It's because higher pressure raises the boiling point.
In an open pot, water cannot exceed 100°C — extra heat just makes it evaporate faster. A pressure cooker traps the vapor, raising the internal pressure. By the Clausius-Clapeyron relation, the saturation temperature climbs: at 2 atm, water boils at ~120°C. Cooking happens via thermal reactions (denaturation, Maillard, hydrolysis), and reaction rates roughly double per 10 K rise (Arrhenius). So 120°C cooks roughly 4× faster than 100°C — purely a temperature effect, not a mechanical pressure effect.
📖 Atkins — Physical Chemistry, 11th Ed., §4.6: "Phase Diagrams of Pure Substances"; Clausius-Clapeyron section.
❌ The van der Waals equation gives an exact description of real gases.
✅ It's a major improvement over ideal gas, but still approximate.
The van der Waals equation captures the qualitative existence of phase transitions and the critical point, but it gets the critical exponents wrong: it predicts $\beta = 1/2$ for the order parameter near $T_c$, while experiment gives $\beta \approx 0.326$ (3D Ising universality). This is because vdW is a mean-field theory — it ignores fluctuations, which dominate near $T_c$. Modern equations of state (Peng-Robinson, SAFT) and the renormalisation group give much better agreement with experiment.
📖 Goldenfeld — Lectures on Phase Transitions and the Renormalization Group, Ch. 5.
❌ Ice has a higher density than water — that's why icebergs float upside-down.
✅ Ice has lower density than water; that's why icebergs float at all.
Water is one of very few substances where the solid is less dense than the liquid — at 0°C, ice density is 917 kg/m³, water is 1000 kg/m³. This anomaly is due to hydrogen bonding: ice's crystal structure is very open, leaving more empty space per molecule. The corresponding P–T phase diagram has a negative slope on the ice-water boundary ($dP/dT < 0$ via Clausius-Clapeyron with $\Delta V < 0$). Pressure melts ice, which is why ice skates work and why deep glacier ice can flow.
Misconception research: Loverude, Kautz & Heron — "Student understanding of the first law of thermodynamics: relating work to the adiabatic compression of an ideal gas", Am. J. Phys. 70, 137 (2002); Driver et al. — Making Sense of Secondary Science (Routledge); Meltzer — "Investigation of students' reasoning regarding heat, work, and the first law of thermodynamics", AJP 72, 1432 (2004).