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From Boyle's Law, in very low pressure ($P \rightarrow 0$), it is a good approximation for real gases to behave as ideal gases. (refer to the graph)

Boyle's Law stated that $ P \propto \frac{1}{V} $

So I am thinking that when $P \rightarrow 0$, $V \rightarrow \infty$, then by $\mathrm{density} = m / V $, supposedly density of ideal gas molecules should be indefinitely small ($\mathrm{density} \rightarrow 0$)

However, it seems density is a measurable value and is involved in calculations of ideal gases. Then may I ask why it is not indefinitely small but a measurable value?

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2 Answers 2

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I think the problem may be that you are conflating the density of gas molecules vs the density of the substance.

The ideal gas molecules themself are considered as point particles with no dimensions, while if we take a volume element inside the ideal gas container, it has an average density. Essentially, the density we use with the ideal gas law is not the density of the actual molecule but the total mass of all these point particles in the container is divided by the total volume of the container.

It's sort of a weird idea to think about because they are point but at the same time, we say they are distributed in some way in space and also give a mass to the region they are spread out in (the whole container).

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  • $\begingroup$ Thanks! But if we 'divide by the total volume of container', which is infinite (to satisfy pressure of container --> 0 to eliminate intermolecular interactions for ideal gases), the density is still approaching 0? i.e. density = total mass of molecules / indefinitely large container? Sorry if I have mistaken your statement $\endgroup$
    – Questions
    Nov 12, 2020 at 6:10
  • $\begingroup$ The total mass will tend to infinite, since you are considering an infinitely large volume. So, the density becomes a limit of infinite/infinite. $\endgroup$
    – Manish S
    Nov 12, 2020 at 6:32
  • $\begingroup$ @Questions no you misunderstood limits, we are not taking it literally as infinite but just large enough that the approximations and all holds. Indeed, you can derive ideal gas law from some statistical mechanisms by imposing the conditions that you mentioned above. Actually there is also one extra condition for ideal gas law, that is high temperature. $\endgroup$ Nov 12, 2020 at 6:46
  • $\begingroup$ See here and here $\endgroup$ Nov 12, 2020 at 6:47
  • $\begingroup$ Read the comment in the answer of GiorgioP $\endgroup$ Nov 12, 2020 at 6:48
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Boyle’s law applies only if the temperature and the mass of the gas is constant. So clearly when pressure decreases and volume increases with the same mass (same number of particles) the density will decrease.

But as a practical matter when applying the gas laws one doesn’t deal with combinations of zero pressures and infinite volumes. Whether or not the density is measurable is a matter involving sensitivity of instrumentation and the test methodology.

Hope this helps.

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