Lesson 3 · 25 min

Phase diagrams: water and carbon dioxide

One pressure–temperature map tells you the state of a substance under any conditions. Water’s melting line leans left, dry ice can only sublime at 1 atm, and beyond the critical point liquid and gas become one.

In this lesson

Read a one-component phase diagram: what regions, boundaries, the triple point and the critical point mean
“Walk” along an isobar or isotherm and predict the phase changes a substance undergoes
Use the Clapeyron equation to explain why water’s melting line leans left
Explain dry-ice sublimation, freeze-drying and supercritical fluids

Dry ice on a stage shrinks away in a cloud of fog but never leaves a puddle. Instant coffee is dried while still frozen. The caffeine in decaf is washed out by carbon dioxide that is neither a liquid nor a gas. All three can be read off one chart — the phase diagram.

Predict first

A block of dry ice (solid CO₂) sits in air at 1 atm and is slowly warmed. What happens?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

It melts at −56.6 °C, then the liquid boils It sublimes straight to gas at −78.5 °C It melts at 0 °C It becomes a supercritical fluid

How to read a phase diagram

Temperature runs along the horizontal axis and pressure up the vertical one (on a log scale here).

Regions: in each region a single phase is stable — solid, liquid or gas.
Boundaries: on a line two phases coexist. The liquid–gas line is last lesson’s vapour-pressure curve; the solid–gas line is the sublimation curve and the solid–liquid line the melting curve.
Triple point: where the three lines meet and solid, liquid and gas coexist. Water: 0.01 °C and 611.657 Pa; CO₂: −56.6 °C and 5.18 bar.
Critical point: the end of the liquid–gas line. Water: 374 °C and 220.6 bar; CO₂: 31.0 °C and 73.8 bar. Beyond it there is no boundary between liquid and gas.

Walking along a horizontal line (constant pressure) from left to right means heating at constant pressure; walking up a vertical line (constant temperature) means compressing at constant temperature. Every line you cross is a phase change.

Phase boundaries computed from reference equations

One-component phase diagram

solid
liquid
gas (vapour)
supercritical fluid

Temperature · 25.0 °C Pressure · 1.01 bar

The sample in a closed vessel (schematic)

At this state

liquid

Liquid water.

Heating at this pressure from cold:

melts at 0.0 °C
boils at 100.0 °C

Water’s phase diagram. Try the freeze-drier: below the triple-point pressure, ice sublimes directly. Then try the pressure cooker. Look closely at the ice–water boundary: it leans up and to the left — squeezing ice makes it melt at a lower temperature, one of water’s anomalies.

The slope of a boundary: the Clapeyron equation

\frac{\mathrm{d}p}{\mathrm{d}T} = \frac{\Delta H_\mathrm{trs}}{T\,\Delta V_\mathrm{trs}}

The slope of any phase boundary is set by the enthalpy and the molar volume change of the transition.

$\Delta H_\mathrm{trs}$: molar enthalpy of the transition (positive for melting, boiling, subliming)
$\Delta V_\mathrm{trs}$: change in molar volume; negative when ice melts (−1.63 cm³/mol)
$T$: transition temperature (K)

Almost every substance expands on melting ( $\Delta V > 0$ ), so its melting line tilts up to the right: pressure raises the melting point, as for CO₂. Water is a rare exception: the hydrogen-bonded network of ice is open, making ice (0.917 g/cm³) less dense than liquid water (1.000 g/cm³). Melting shrinks the volume, $\Delta V < 0$ , and the melting line has a negative slope. The same fact makes ice float and lakes freeze from the top down.

Worked example

How steep is the ice melting line?

At 0 °C ice has $\Delta_\mathrm{fus}H = 6.01$ kJ/mol, and the densities of ice and water are 0.9167 and 0.9998 g/cm³. Find the slope of the melting line and estimate how much the pressure under a skate blade lowers the melting point.

Beyond the critical point: supercritical fluids

Climb the liquid–gas line: heating makes the liquid expand and thin out, while compression makes the gas denser, until their densities meet. At the critical point they become identical and the interface vanishes. Above both the critical temperature and pressure a substance is a supercritical fluid: it fills its container and diffuses like a gas yet dissolves things like a liquid, and its dissolving power can be tuned continuously with pressure.

CO₂’s critical point is only 31.0 °C and 73.8 bar, and CO₂ is non-toxic, cheap and simply evaporates away when the pressure is released. So supercritical CO₂ is used to decaffeinate coffee beans, extract hops and spices, and even to dry-clean clothes.

Concept check

A CO₂ fire extinguisher at 25 °C is at about 64 bar. What state is the CO₂ inside?

All gas Liquid and gas coexisting A supercritical fluid Dry ice

Try the calculation

Using the slope from the worked example (−135 bar/K, taken as constant), estimate the melting point of ice at 1000 bar (°C).

Key takeaways

Regions of a phase diagram are single phases; lines are two-phase coexistence; the triple point has three; the critical point ends the liquid–gas line.
The Clapeyron equation dp/dT = ΔH/(TΔV) sets each boundary’s slope; ice shrinks on melting, so water’s melting line has a negative slope.
CO₂’s triple-point pressure (5.18 bar) is above 1 atm, so dry ice can only sublime at ordinary pressure (−78.5 °C).
Above the critical point liquid and gas merge — supercritical CO₂ extracts and decaffeinates; below the triple-point pressure ice sublimes directly, the basis of freeze-drying.

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Data and references

IAPWS R14-08(2011), Revised Release on the Pressure along the Melting and Sublimation Curves of Ordinary Water Substance.
W. Wagner & A. Pruß, “The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance”, J. Phys. Chem. Ref. Data 31, 387 (2002).
R. Span & W. Wagner, “A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa”, J. Phys. Chem. Ref. Data 25, 1509 (1996).
P. Atkins, J. de Paula, J. Keeler, Atkins’ Physical Chemistry, 11th ed. (2018), Topic 4B (the Clapeyron equation).
R. Rosenberg, “Why is ice slippery?”, Physics Today 58(12), 50 (2005).