User:Retired Pchem Prof/CompressibilityFactor

January 19, 2016

Article Compressibility factor

Physical reason for temperature and pressure dependence
Deviations of the compressibility factor, Z, from unity are due to attractive and repulsive Intermolecular forces. At a given temperature and pressure, repulsive forces tend to make the volume larger than for an ideal gas; when these forces dominate we get Z greater than unity. When attractive forces dominate, we get Z less than unity. The relative importance of attractive forces decreases as temperature increases (see effect on gases).

As seen above, the behavior of Z is qualitatively similar for all gases. Here we use molecular nitrogen, N$2$, to further describe and understand that behavior. All data used in this section were obtained from the NIST Chemistry WebBook. It is useful to note that for N$2$ the normal boiling point of the liquid is 77.4 K and the critical point is at 126.2 K and 34.0 bar.

First, we look at an overview covering a wide temperature range. At low temperature (100 K), the curve has a characteristic check-mark shape, the rising portion of the curve is very nearly directly proportional to pressure. At intermediate temperature (160 K), we have a smooth curve with a broad minimum; although the high pressure portion is again nearly linear, it is no longer directly proportional to pressure. Finally, at high temperature (400 K), Z is above unity at all pressures. For all curves, Z approaches the ideal gas value of unity at low pressure and exceeds that value at very high pressure.

To better understand these curves, we can take a closer look at the behavior for low temperature and pressure. All of the curves start out with Z equal to unity at zero pressure and Z decreases as pressure increases. N$2$ is a gas under these conditions, so the distance between molecules is large, but becomes smaller as pressure increases. This increases the attractive interactions between molecules, pulling the molecules closer together and causing the volume to be less than for an ideal gas at the same temperature and pressure. Higher temperature reduces the effect of the attractive interactions and the gas behaves in a more nearly ideal manner.

As the pressure increases, we eventually reach the gas-liquid coexistence curve, shown by the dashed line in the figure. When that happens, the attractive interactions have become strong enough to overcome the tendency of thermal motion to cause the molecules to spread out; so the gas condenses to form a liquid. On the coexistence curve, there are then two possible values for Z, a larger one corresponding to the gas and a smaller value corresponding to the liquid. Once all the gas has been converted to liquid, the volume decreases only slightly with further increases in pressure; then Z is very nearly proportional to pressure.

As we move along the coexistence curve to higher temperature and pressure, the gas becomes more like a liquid and the liquid becomes more like a gas. At the critical point, the two are the same. So for temperatures above the critical temperature (126.2 K), there is no phase transition; as pressure increases the gas gradually transforms into something more like a liquid. Just above the critical point there is a range of pressure for which Z drops quite rapidly (see the 130 K curve), but at higher temperatures the process is entirely gradual.

Finally, we consider the behavior at temperatures well above the critical temperatures. The repulsive interactions are essentially unaffected by temperature, but the attractive interaction have less and less influence. Thus, at sufficiently high temperature, the repulsive interactions dominate at all pressures.

We can see this in the graph showing the high temperature behavior. As temperature increases, the initial slope becomes less negative, the pressure at which Z is a minimum gets smaller, and the pressure at which repulsive interactions start to dominate, i.e. where Z goes from less than unity to greater than unity, gets smaller. At the Boyle temperature (327 K for N$2$), the attractive and repulsive effects cancel each other at low pressure. Then the compressibility remains at the ideal gas value of unity up to pressures of several tens of bar. Above the Boyle temperature, the compressibility factor is always greater than unity and increases slowly but steadily as pressure increases.