Phase Diagrams of Water and Carbon Dioxide

Introduction

Water and carbon dioxide make an instructive pair because their phase diagrams have the same basic ingredients but different practical paths. Water's ice–liquid boundary leans toward lower pressure as temperature rises, whereas the corresponding line for carbon dioxide has the more usual positive slope. At ordinary atmospheric pressure, ice may melt and water may boil; solid carbon dioxide instead sublimes. These observations follow from phase equilibrium and the location of each triple point, not from a mysterious property of the words “wet” and “dry.”

Core explanation

For a solid–liquid boundary, the Clapeyron equation is dP/dT = ΔH fus/(TΔV fus). Fusion absorbs heat, so ΔH fus is positive. Its slope therefore has the sign of ΔV fus = V liquid − V solid. Ordinary hexagonal ice has an open hydrogen-bonded structure and a greater molar volume than liquid water near melting. Melting collapses some of that open structure: ΔV fus is negative. Water's fusion boundary thus has a negative pressure–temperature slope. Raising pressure slightly favours the denser liquid and lowers the melting temperature of ordinary ice. This account applies to the familiar low-pressure phases; water has many high-pressure ice polymorphs, so one must not extrapolate a simple line indefinitely.

Most substances, including carbon dioxide in its familiar solid–liquid region, have a denser solid than liquid. Their fusion ΔV is positive, so the melting line slopes toward higher pressure with increasing temperature. The sign difference is a volume statement. It does not mean that adding pressure “provides heat”; pressure and heat are distinct routes through state space. Pressure changes the relative Gibbs energies through the P V contribution.

The triple point determines whether a liquid can occur along a fixed-pressure heating path. Carbon dioxide's triple-point pressure is above atmospheric pressure. At approximately one atmosphere, no stable liquid CO₂ region lies along an equilibrium heating path. A piece of dry ice warms, reaches the solid–vapour boundary and sublimes. To obtain liquid CO₂, one must work above the triple-point pressure and within the proper temperature window. Water's triple-point pressure is far below one atmosphere, so its familiar atmospheric-pressure path crosses solid–liquid and then liquid–vapour boundaries.

Water and CO₂ both have liquid–vapour coexistence curves ending at critical points. Above the critical temperature and pressure region, there is no distinct boiling transition separating liquid and vapour; the fluid's density can be varied continuously. Supercritical CO₂ is useful as an extraction solvent because its density and solvent behaviour can be tuned by pressure, then the CO₂ can be separated by changing conditions. Calling it a “gas under pressure” misses the absence of a liquid–vapour boundary in that region.

The diagram is an equilibrium map. Dry ice can make a visible white cloud in humid air, but the cloud is primarily condensed water droplets or ice particles formed as the surrounding air cools; pure CO₂ vapour is colourless. Likewise, atmospheric-pressure ice can be supercooled or water can persist metastably briefly beyond a boundary. Those kinetic delays do not redraw the equilibrium curves.

Step-by-step reasoning

First, identify the axes and the imposed path, such as heating at constant one atmosphere. Second, compare that pressure with the triple-point pressure. Third, trace intersections with equilibrium boundaries in temperature order. Finally, use the sign of ΔV fus in the Clapeyron equation to explain the fusion-line direction. This sequence separates phase availability from the details of transformation kinetics.

Visual explanation

Imagine pressure vertically and temperature horizontally. Water's familiar fusion line tilts up toward the left from its triple point; CO₂'s tilts up toward the right. Draw a horizontal line at one atmosphere. It crosses water's solid–liquid and liquid–vapour lines, but for CO₂ it lies below the triple point and crosses the sublimation line.

Real-world analogy

A map may show two roads with the same destination but different mountain passes. Your chosen altitude determines which road you can take. Similarly, the imposed pressure determines whether a heating path can reach a stable liquid region. The analogy helps with path reading, although phase boundaries are chemical-potential equalities rather than physical roads.

Real-world example

Dry ice is used to cool shipments because it turns directly into gaseous CO₂ under ordinary ambient pressure, leaving no pool of liquid CO₂. In contrast, melting ice leaves liquid water. Neither behaviour is explained merely by comparing their temperatures: one must compare ambient pressure with each substance's triple-point pressure and read the corresponding phase diagram.

Why?

Why can pressure help melt ordinary ice? The liquid occupies less volume than ordinary ice near the melting line. Increasing pressure lowers the Gibbs energy of that lower-volume phase relative to the solid, shifting coexistence toward a lower melting temperature. The effect is finite and phase-specific; frictional heating and surface effects also matter in real skating.

Common misconception

“Carbon dioxide cannot be liquid” is false. It cannot be a stable liquid at roughly one atmosphere, but sufficiently high pressure and a suitable temperature place it in a liquid region. Another error is to call any compressed fluid supercritical. Both critical temperature and the relevant pressure region matter, and the decisive feature is the lack of a distinct liquid–vapour coexistence boundary.

Worked example

Suppose a sample of solid CO₂ is heated reversibly at a pressure below its triple-point pressure. Predict its first equilibrium phase change. The constant-pressure line has no intersection with the solid–liquid boundary because the liquid region begins only above the triple-point pressure. It meets the solid–vapour boundary, so the solid sublimes. Repeating the thought experiment at a pressure above the triple point can instead permit melting before subsequent vaporisation, provided the selected path crosses those boundaries.

Quick check

1. Why does ordinary water have a negative fusion-line slope? Answer: Its fusion enthalpy is positive while its fusion volume change is negative: liquid water is denser than ordinary ice near melting, so Clapeyron gives dP/dT < 0.

Exam focus

State the phase and pressure assumptions before applying a slope rule. Use ΔV = V product phase − V initial phase, and do not confuse the slope of P versus T with that of T versus P. For an atmospheric-pressure path, compare the line with the triple point before claiming melting or boiling.

Advanced insight

The familiar negative fusion slope belongs to ice Ih near ordinary conditions. Water's many solid polymorphs make its full high-pressure diagram much richer. Each coexistence line obeys a local Clapeyron relation with the enthalpy and volume changes of the particular two phases. That is why “water always has a negative melting slope” is too broad.

Summary

Ice melts to a denser liquid near ordinary pressure, giving water's fusion line a negative dP/dT. Carbon dioxide's triple-point pressure exceeds atmospheric pressure, so dry ice sublimes along an ordinary-pressure heating path. Above a liquid–vapour critical point, the fluid changes continuously rather than boiling across a coexistence line.

Practice questions

1. A fusion transition has ΔH fus > 0 and ΔV fus > 0. What is the sign of dP/dT? Answer: Positive, since T is positive and the Clapeyron numerator and denominator are positive. 2. Why does the visible mist around dry ice not prove that CO₂ vapour is white? Answer: Gaseous CO₂ is colourless; cooling of moist surrounding air condenses water into visible droplets or ice particles. 3. What pressure comparison decides whether a stable liquid region can be crossed on a constant-pressure heating path? Answer: Compare the path's pressure with the substance's triple-point pressure, then locate the exact coexistence curves. 4. Does higher pressure always melt a solid? Answer: No. The shift depends on the volume difference between phases; for a solid denser than its liquid, pressure generally favours the solid along that fusion boundary.