# Variation of compressiblity factor with temperature

While I certainly understand the order of temperatures, I can't find a reason for the curves to intersect at one common point.

Why do the curves intersect at one point? or do they really intersect at a common point or is it a coincidence that they are intersecting in this diagram?

• It is plausible that the isotherms intersect, it just means that the ratio of the real volume to the ideal volume are the same at those conditions; the ratios can be the same while the absolute values are different. Mar 7, 2017 at 20:18
• two lines intersecting is understandable but why do all the lines intersect at one point? I am trying to reason out why all the lines intersect at one common point. Mar 8, 2017 at 5:05
• Why all the isotherms intersect would require a more detailed analysis. Maybe for that system, that particular pressure could be the critical pressure so some fixed behavior of the system is expected. I happen think the intersections are coincidental given the information presented. Mar 8, 2017 at 12:59

Remember the mathematical definition of $Z$:

$$Z = \frac{pV}{nRT}$$

Given all the $Z$ graphs I've ever seen -- the single-point interaction of all these isotherms is just coincidence (and kind of misleading). There is nothing in thermodynamics that suggests for a given gas, there exists a $p$ for which $Z$ is constant, regardless of temperature. Even for constant $V$ and $n$, you have that $Z \propto \frac{p}{T}$, so changing $T$ can in no way leave $Z$ constant.

You can find graphs that look similar even for different gases at the same $T$:

Nonetheless, there is no reason for a given material to have the same $Z$ at a given $p$ regardless of temperature:

Generally speaking, if you zoom-out far enough, and have significantly different temperatures (like 100s of Kelvin apart), some $Z$ graphs may appear to have a uniform $(p,Z)$ point, but in reality the intersections are not uniform for any physical/mathematical reason. For this reason, graphs like that can be a little misleading.

• There is another graph in my textbook titled "Variation of Z with different pressure at different temperature" where at $P=0$, $Z$ is not equal to 1 for all the temperatures. Is that possible? Mar 9, 2017 at 7:47
• What's the gas? Mar 9, 2017 at 14:16
• The textbook does not mention any specific gas. Mar 9, 2017 at 14:19
• $Z$=0 at $p$=0, what you see near $p=0$ is really just very low pressures. I suppose for some gases at certain temperatures this is possible but not common, because generally, low pressure lowers potential energy to ~0, which is what an I.G. can be described as. Mar 9, 2017 at 19:36