Concept 04 / 07States of Matter
Liquefaction of Gases and Critical Constants
Andrews' isotherms for carbon dioxide, critical temperature, pressure and volume, the critical point, continuity of states, critical constants from the van der Waals equation, the relation Zc = 3/8 and methods of liquefying gases.
Andrews' experiments (1869). Thomas Andrews measured - isotherms of at several temperatures.
At high temperature () the isotherm resembles Boyle's hyperbola: stays a gas at any pressure.
At the isotherm has a point of inflection with a horizontal tangent: the critical point.
Below (for example ) compression first reduces the volume of the gas; then liquid appears and the volume falls at constant pressure (the flat portion, where liquid and vapour coexist at the vapour pressure); finally, when all is liquid, the curve rises steeply because liquids are hardly compressible.
Critical constants.
Critical temperature : the highest temperature at which a gas can be liquefied by pressure alone. Above , no pressure produces a separate liquid phase. ( , , , , , .)
Critical pressure : the pressure needed to liquefy the gas at .
Critical volume : the molar volume at and .
At the critical point the densities of liquid and vapour become equal and the meniscus disappears.
Continuity of states. A gas can be converted to a liquid without ever showing two phases, by going round the critical point: heat above , compress, then cool. So there is no sharp distinction between gas and liquid above ; the substance is a supercritical fluid. The word vapour is used for a gas below its critical temperature, which can be liquefied by pressure alone.
From the van der Waals equation. Requiring the isotherm at to have a horizontal inflection ( and ) gives
, , .
Hence for every van der Waals gas (real gases give about to ). Also , and , .
Below the van der Waals equation predicts an unphysical S-shaped loop; the real isotherm replaces it with a flat line placed so that the areas above and below are equal (Maxwell's construction; see figure).
Liquefaction methods. A gas must first be cooled below . Gases with high (, , , ) liquefy by pressure at or near room temperature. Permanent gases (, , , ) need cooling first, by adiabatic expansion or Joule-Thomson expansion (Linde's process). Joule-Thomson expansion cools a gas only below its inversion temperature ; and must be precooled because their is low.
Key Takeaways:
- is the maximum temperature for liquefaction by pressure.
- , , , .
- in units of .
- Higher means higher and easier liquefaction.
- Pitfall: a larger lowers , so liquefiability is judged by , not by alone.
Brain-Teaser 1: For , and . Estimate and .
Solution: , close to the measured . , close to the measured .
Brain-Teaser 2: The critical temperatures of gases A and B are and . Which has the larger , and which can be liquefied at room temperature by pressure alone?
Solution: , so A has the larger ratio, meaning stronger attractions relative to size. Room temperature () is below of A but above that of B, so only A can be liquefied by pressure alone.
Brain-Teaser 3: A cylinder of at shows a constant pressure of about as gas is withdrawn, until suddenly the pressure starts falling. Explain.
Solution: At (below ) the cylinder holds liquid in equilibrium with its vapour. The pressure equals the vapour pressure, which depends only on temperature. As gas is withdrawn, more liquid evaporates and the pressure stays constant. When the last liquid is gone, only gas remains and the pressure falls as Boyle's law predicts.
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