Problem — Sci-fi physics: no molecular repulsion

Problem

Imagine a world in which molecules do not repel each other at high density (the Van der Waals parameter $b$ is zero, but $a$ is present). Discuss in five lines what would change for the liquid phase, compressibility, and the stability of the gas at low temperatures.

Solution

In the Van der Waals equation the term $b$ represents the molecules’ own volume (short-range repulsion), while $a$ describes attraction at a distance: $\left(p + \frac{a\,n^2}{V^2}\right)(V - nb) = nRT$ With $b = 0$ the excluded volume disappears: nothing stops the molecules from getting arbitrarily close. $\text{Compressibility} \to \infty : \quad V \to 0 \ \text{is allowed}$ Without repulsion there is no minimum volume, so the liquid phase (dense but incompressible) does not exist as a distinct state: nothing “holds up” the matter. $\text{At low } T: \frac{a\,n^2}{V^2} \ \text{dominates} \Rightarrow \text{collapse}$ $\ev{\text{no distinct liquid phase; unbounded compressibility; at low } T \text{ the gas collapses to a point under the attraction } a}$ It is the repulsion $b$ that gives condensed matter its incompressibility and stability.

Links

Topics: Kinetic theory of gases Concepts: Latent heat and changes of state Skills: Analysis of limiting cases and sci-fi physics