Question

The pressure volume relationship is given by:

Answer 1

Hint: To determine the relationship between pressure and volume, think of a real life scenario and try to figure out what happens to the volume of the air inside the tyre when you push the handle of a cycle pump down.

Complete answer: At constant temperature, the volume of a given amount of gas changes inversely with pressure. This is called Boyle’s law.
Mathematically, boyle’s law is represented as:
\[V\]\[\propto \] \[\dfrac{1}{P}\]
\[V=\dfrac{K}{P}\] where K = Proportionality constant or Boyle’s Constant
$PV=K $
${{P}_{i}}{{V}_{i}}={{P}_{f}}{{V}_{f}}$
where \[{{P}_{i}}\] and \[{{V}_{i}}\] are the initial pressure and volume values, and\[{{P}_{f}}\]and \[{{V}_{f}}\]are the values of pressure and volume of the gas after change. So you can find any of the values, if three of these are given to you.
Boyle’s constant, K, does not depend on pressure and volume. But it depends on temperature and number of moles.
Let’s find out the unit of Boyle's constant, we know pressure is force per unit area. So let’s derive the units from the formula of pressure and volume. Volume has a unit of \[{{m}^{3}}\].
$P=\dfrac{F}{A}=\dfrac{F}{{{m}^{2}}} $
$V={{m}^{3}} $
$PV=K=\dfrac{F}{{{m}^{2}}}\times {{m}^{3}}=F\times m $
which is equal to work done (Force\[\times \] displacement) and work done has an unit of energy. So Boyle's constant has an energy unit. Therefore, it can have units like joules, kilojoules, erg.
So when you push the bicycle pump down, the pressure inside the pump will increase and as a result air inside the pump gets compressed and this air flows into the tyre which inflates the tyre.

Note:
Remember the relations discussed above hold true only when the number of moles and the temperature are both kept constant. If not, then you will have to use the ideal gas equation\[\left( PV=nRT \right)\] to arrive at the results.