# How can $1/V$ be equal to $0$ in Boyle's Law?

In relation to ideal gases, Boyle's Law states that pressure is inversely proportional to volume under constant temperature. In other words,

$$P \propto 1/V$$

Below is a graph that plots pressure, $$P$$, against inverse volume, $$1/V$$.

How can $$1/V$$ ever equal zero? How is this possible?

Formally we have $$\lim_{V\rightarrow \infty} \frac{1}{V}=0$$

I.e. the statement is true in the limit of infinite volume. For an ideal gas, this can be interpreted as saying that as the confining volume for the gas becomes infinite, the gas no longer exerts a normal pressure on the walls of the volume.

We meet this issue in many contexts in Mathematics and in Physics. I don't think you'll go far wrong by thinking of it like this...

We can consider larger and larger container volumes, that is values of $$1/V$$ that become smaller and smaller. So the points on the line on your graph can be plotted closer and closer to the origin. In fact as close as you care to demand, so that you can't see, even with huge magnification, that there isn't actually a point at the origin itself. The case of $$1/V=0$$ is the so-called 'limiting case'.

• Another example is a resonance curve (amplitude against forcing frequency for an oscillatory system). This time the graph doesn't go through the origin, but has a finite amplitude, $A_0$, say, at zero frequency. But surely we can't even talk about the amplitude at zero frequency, because the system won't be oscillating! Again, we can take the graph as close as we like to zero frequency and can calculate or measure the amplitudes at these very low frequencies. So we won't be able to tell by looking at the line, that it is meaningless at $f=0$. $A_0$ is the 'limiting value' of the amplitude. Jan 3, 2021 at 17:23

You've got to start with the ideal gas law

$$pV=mRT$$

Where, in this version, $$m$$ is the mass of the gas, $$R$$ is the specific gas constant and $$T$$ is the temperature. If $$m$$ and $$T$$ are constant, then $$mRT$$ = constant = $$C$$.

$$p=\frac{C}{V}$$

It is important to note that Boyle's law only applies to a closed system, i.e., a system where $$m$$ is constant. So even if the volume increases there are always the same number of gas molecules within that volume. Although the density of the gas (molecules per unit volume) keeps decreasing with increasing volume, it never becomes zero. And as long as there are gas molecules, there will be collisions between the molecules and any surfaces within or bounding that volume resulting in pressure proportional to the collision rate.

So if the volume were infinite, it would simply mean that an infinite amount of time would be required for a collision to occur. The rate of collisions approaches zero, but can never actually become zero as long as there are gas molecules in the volume.

Hope this helps.

How can $$1/V$$ ever equal zero?

Physically, it can't. This is just a point on a line - it does not have to correspond to an actual physical situation. It is like saying the average family has $$2.3$$ children - this does not mean any actual family can have $$2.3$$ children - it just means if you take a large number of families and divide the number of children by the number of families, this ratio approaches a limit of $$2.3$$.