Question

For a quantity of gas at a constant temperature, the pressure P is inversely proportional to the volume V. What is the limit of P as V approaches 0 from the right? Explain what this means in the context of the problem.

Answer 1

Answer:

$lim_{V \to 0^{+}} P = k\lim_{V \to 0^{+}} \frac{1}{V} =\infty$

This limit is not defined.

Explanation:

We need to remember first this law

The Boyle's law states that under a constant temperature when the pressure is inversely proportion to the volume.

So that means: $P \propto \frac{1}{V}$

And we can put an equal if we do this:

$P = \frac{k}{V}$ where k is the proportional constant.

For this case we want to find the following limit:

$lim_{V \to 0^{+}} P = \lim_{V \to 0^{+}} \frac{k}{V}$

And using properties of limits we have:

$lim_{V \to 0^{+}} P = k\lim_{V \to 0^{+}} \frac{1}{V}$

So for this case this limit tend to infinity since we are dividing a constant by a very low number positive near to 0.

So then we can conclude that:

$lim_{V \to 0^{+}} P = k\lim_{V \to 0^{+}} \frac{1}{V} =\infty$

This limit is not defined.

Interpretation: we are seeing that if the volume decrease considerable with the temperature constant by the inverse relation between P and V, the value of P increases to with no limit.