Zero volume at zero Kelvin

Question

Why does the volume of a gas become zero at 0 Kelvin? Can a Bose Einstein condensate be considered as matter? (I mean the volume becomes zero)

Answer 1

At constant pressure the volume of an ideal gas is given by Charles' law:

$$ V \propto T $$

and this law tells us that when the temperature $T$ falls to zero the volume $V$ also becomes zero.

But no gas is ideal and real gases show all sorts of non-ideal behaviour. For example real gases liquify then solidify as the temperatue falls. Real gases deviate from Charles law and their volume does not fall to zero at absolute zero.

Bose-Einstein condensates are indeed another form of matter and they don't have zero volume.

Answer 2

An ideal gas at 0 K doesn't necessarily have zero volume. It depends on the experimental setup; "experimental" in an idealized sense, since the ideal gas is a purely theoretical construct.

Let's consider it from a macroscopic point of view first, that is, within thermostatics. Its pressure $p$, volume $V$, temperature $T$, and mole number $n$ are related by the ideal gas law: $$pV = nRT,$$ with $R$ the ideal-gas constant.

If $T=0$ we have $pV=0$. This means that we can have non-zero volume, but the pressure will be zero. This equation also says that we could have non-zero pressure, but will have zero volume then.

Let's consider these two situations from a microscopic point of view now. The ideal gas is a collection of point particles: they have mass, momentum, kinetic energy but don't occupy any space; and they don't exert any kind of force on one another. The pressure of the gas on the walls of a container comes from the exchange of momentum between particles and walls. The temperature of the gas is the spatial average of the molecules' kinetic energies (I say more about statistical averages below).

At constant temperature we can control the pressure of an ideal gas by letting one wall be freely moving with a constant force exerted on the other side. For example a piston pressed by a weight, or pushed by a spring. In equilibrium the wall is still and the pressure exerted by the gas must equal the force/area on the piston.

We can control the volume by simply letting the walls be unmovable.

And how do we control the temperature of an ideal gas? One way is to let the walls of the container have local microscopic movements: when a particle of the gas is approaching a wall, we can let the impact region of the wall have an extra velocity – hence kinetic energy and momentum – towards the particle, or receding from it, for a very short time, then restore its position. This will give extra kinetic energy and momentum to the particle, or take some from it. Obviously this mental experiment is mimicking the effect of the molecules that constitute a real wall. Another way to control temperature is possible when the particles have some extra physical property that allow us to exert a force at a distance on them. Electrical or gravitational forces are excluded, however, otherwise the particles would exert forces on one another (this also tells us that the particles' masses must be enough small for gravitational attraction to be negligible).

At 0 K the particles of the ideal gas have no velocity: they stay still. If the gas is within an enclosure with unmovable walls, it will have that volume but will exert no pressure, as we found from the ideal-gas law. It's interesting to note that if we reach 0 K by modifying the particles' kinetic energies in the way described above, all the particles must eventually be in contact with the walls: the wall takes the last amount of kinetic energy from a particle, via a microscopic receding motion when the particles hits, and the particle thus stays still in contact with the wall.

Reaching 0 K at constant pressure, the moving wall eventually reaches the opposite wall of the container, so the final pressure is really just the contact, normal force exerted by the opposite wall, rather than the momentum released by the point particles.

This is just to give a pictorial intuition. The extreme situations implied by the processes envisaged above make clear that this is a purely mental construct. Already gravity alone introduces contradictions between the particles' having mass and yet be non-interacting within a volume shrinking to zero.

References for all this are, for example,

• Chapman, S., Cowling, T. G. (1995): The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases (3rd ed., Cambridge University Press).

• Truesdell, C. A., Muncaster, R. G. (1980): Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas: Treated as a Branch of Rational Mechanics (Academic Press).

Note on the definition of temperature

We can also define temperature as a statistical average if we introduce ensembles (Gibbs's statistical mechanics) or number-density functions (kinetic theory). The results are the same. More about all these kinds of temperatures and the interesting and troublesome relations between them can be found in

• Biró, T. S.: Is There a Temperature?: Conceptual Challenges at High Energy, Acceleration and Complexity (Springer 2001).