Pressure-Volume isotherms of Carbon dioxide

Thomas Andrews produced the first complete experimental picture of how a real substance's pressure, volume and temperature relate to each other across both its gaseous and liquid states, by plotting a family of pressure-volume isotherms for carbon dioxide (Figure 6.12).

At a low fixed temperature, say $13\,^\circ\text{C}$, the isotherm starts as a smoothly falling curve as pressure rises and volume falls -- CO$_2$ is purely gaseous along this stretch. At some point, liquid CO$_2$ begins to separate out, and the isotherm flattens into a horizontal plateau: along this flat stretch, liquid and gas coexist side by side, and pressure stays constant even as volume keeps shrinking (more and more of the gas is simply converting to liquid at fixed pressure). Once the conversion is complete and only liquid remains, the isotherm turns sharply steep -- liquids are far less compressible than gases, so squeezing the now-fully-liquid CO$_2$ produces only a very small further decrease in volume even for a large increase in pressure.

Higher-temperature isotherms show exactly the same three-stage shape, but the flat coexistence plateau grows progressively shorter. At $31.1\,^\circ\text{C}$, the plateau has shrunk all the way down to a single point -- CO$_2$ passes directly from gas to liquid with no visible two-phase region at all. This special temperature is called the critical temperature (or liquefaction temperature) of CO$_2$, and the pressure at that point is 73 atm. Above this temperature, no amount of pressure -- however large -- can liquefy CO$_2$; it remains a gas at every pressure. This same qualitative picture, it turns out, holds for essentially every real gas, each with its own characteristic critical temperature, pressure and volume.

These are formalised as the critical constants:

• Critical temperature ($T_c$): the temperature above which a gas cannot be liquefied, no matter how high the pressure.
• Critical pressure ($P_c$): the minimum pressure needed to liquefy exactly 1 mole of the gas, at its critical temperature. …