Problem

For a given quantity of gas, Boyle's Law states that the pressure exerted by the gas and the volume it takes up are inversely proportional to each other; that is,

or equivalently

Assume that the volume changes over time, so V is a function of t. Find the rate of change of the pressure with respect to time in two different ways: in terms of V, dV/dt only, and in terms of P, dV/dt only.

Solution

We want to find dP/dt. first, write

and remembering that V is also a function of t, we differentiate to get

Now, if we want to write this in terms of P, dV/dt only, we use the relationship

to derive

and we can write