Chemistry · Ch 6 — Gaseous State
Compressibility factor Z
Compressibility factor Z
The extent to which a real gas departs from ideal behaviour is measured by the compressibility factor, Z, defined as the ratio of the real PV product to the ideal-gas value nRT:
For an ideal gas, PV always equals nRT exactly, so at every pressure and temperature -- a plot of Z against P for a perfectly ideal gas would be a horizontal straight line sitting right on . Any departure of Z away from 1 is a direct, quantitative measure of non-ideality.
Plotting Z against P for several real gases (Figure 6.8) shows two distinct patterns. H and He climb steadily above from the lowest pressures shown -- these are both very small, weakly-attracting molecules, so the "excluded volume" effect (their own finite size making PV a bit larger than nRT) dominates over any attraction essentially everywhere on the plot; this is the Z > 1 region. N, CH and especially CO instead dip below Z = 1 first -- here intermolecular attraction dominates at low-to-moderate pressure (pulling PV a bit below nRT, the Z < 1 region) before the excluded-volume effect takes over and drags Z back above 1 at high pressure. CO, with the strongest attractive forces of the gases shown, dips the deepest.
For every gas, at sufficiently low pressure and sufficiently high temperature, Z approaches 1 and the gas behaves essentially ideally. The physical reasoning: at low pressure the container is enormous compared to the total volume of the molecules themselves, so their individual volume is negligible, and the molecules are so far apart on average that attractive forces between them are negligible too. As pressure rises, gas density rises and molecules crowd closer together, so intermolecular forces become significant enough to measurably affect molecular motion, and the gas stops behaving ideally. Likewise, at high temperature the molecules' average kinetic energy is so large that intermolecular attraction becomes comparatively insignificant; as temperature falls, average kinetic energy falls too, and the same attractive forces become relatively more effective at influencing the gas's behaviour. …
What this figure shows. Z = PV/nRT (y-axis, 0 to 2.0) plotted against pressure P in atm (x-axis, 0 to 500) for H, He, N, CH and CO, with a horizontal reference line at Z = 1 labelled 'IDEAL GAS Z = 1'. H and He climb steadily above Z = 1 from the very lowest pressures (always in the 'Z > 1' region, since their molecules are so small and weakly attracting that the volume correction dominates at essentially all pressures). N and CH dip slightly below Z = 1 at low-to-moderate pressure (attraction dominates) before rising steeply above Z = 1 at high pressure (volume dominates); CO shows the same shape but dips the deepest, down to about Z = 0.25 near 100 atm, before rising above Z = 1 only past roughly 450 atm -- consistent with CO having th …
What this figure shows. Compressibility factor (y-axis, 0.8 to 1.2) plotted against pressure in bar (x-axis, 0 to 300) for nitrogen at four temperatures -- 220 K, 273 K, 327 K and 400 K -- alongside a dashed horizontal 'Ideal' line at Z = 1. All four curves start at Z = 1 at zero pressure. The 220 K curve dips the deepest, down to about Z = 0.9 near 100-150 bar, before climbing back through Z = 1 around 250 bar and continuing to rise; the 273 K and 327 K curves dip only slightly below 1 before rising; the 400 K curve barely dips at all and rises almost the whole way, showing that nitrogen's Boyle temperature -- above which no dip below Z = 1 occurs -- lies somewhere between 327 K and 400 K, and that raising the temperature steadily washes out the l …