Animated Solution for Chemistry - States of Matter: Which one of the following is the correct $pV$ vs $p$ plot at constant temperature for an ideal gas ? ($p$ and $V$ stand for pressure and volume of the gas respectively)

The problem asks us to identify the correct graph of $pV$ versus $p$ for an ideal gas at a constant temperature. To tackle any problem involving the macroscopic properties of an ideal gas, our first instinct should be to write down the ideal gas equation.

The ideal gas equation is given by:

Here, $p$ is the pressure, $V$ is the volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the absolute temperature.

Let's carefully analyze the conditions provided in the question. We are told that the temperature $T$ is constant. Furthermore, for a given sample of gas, the number of moles $n$ is also constant. The universal gas constant $R$ is, by definition, a constant.

Since $n$, $R$, and $T$ are all constants, their product must also be a constant. Therefore, we can write:

This relationship is the mathematical formulation of Boyle's Law, which states that at a constant temperature, the pressure of a fixed mass of gas is inversely proportional to its volume.

Now, we need to translate this mathematical relationship into a graphical form. We are asked to plot $pV$ on the y-axis and $p$ on the x-axis.

Let's substitute our variables into a standard coordinate geometry format. If we let $y=pV$ and $x=p$, our equation $pV=\text{constant}$ becomes: where $c$ is a constant.

In coordinate geometry, the equation $y=c$ represents a horizontal straight line parallel to the x-axis. This means that no matter how much we increase the pressure $p$ (moving along the x-axis), the value of $pV$ (the y-coordinate) remains exactly the same.

Looking at the given options, the graph in option (a) perfectly depicts a horizontal straight line. Therefore, it is the correct representation of the $pV$ vs $p$ plot at constant temperature.