According to the ideal gas law:
$pv = nRt $
My question is, why is the volume $v$ independent of the volume of individual components of the gas, i.e. molecules? As the sizes of individual molecules increase, should the volume of the gas itself not increase? Please explain why this is not the case. Thank you!
For an ideal gas at ambient pressure and temperature, the actual volume occupied by the molecules is so much smaller than the volume occupied by the gas molecules as they zoom about at high speed and bounce off the container walls and one another that it can be ignored, and the resulting approximations derived from the ideal gas law are satisfactory.
For high pressures and/or very low temperatures, there are other more complex equations of state which take the small but finite volume of the molecules into account and furnish better approximations.
The equation you mention stems from a model known as the “perfect gas” model.
In this model, the molecules of gas are approximated by material points, with a given mass, no size and a given (average) momentum (or speed or kinetic energy). In this model the molecules do not interact with each other. So basically, in this model, the molecules of gas are tiny, independent billard balls and all they do is hit the walls of the surrounding vessel with a given average energy.
It can be demonstrated that under these assumptions, there is a linear relationship between the average kinetic energy of the molecules and the temperature of the gas. In other words, a temperature t of the gas corresponds to a given kinetic energy of the molecules.
The pressure of the gaz on the surrounding vessel of volume V results from the sum of the particles’s kinetic energies (just like a swarm of billard balls hitting a wall).
As a consequence, in this simplistic model, the volume (resulting from the pressure) of the vessel containing the gas and the temperature of said gas are proportional.
In practice, under ordinary conditions, a real gas behaves almost exactly like the “perfect gas”.
As a consequence, since the actual size of the molecules was not taken into account at all in the “perfect gas” model and since experiments tell us that this model is in fact very accurate in practice, the volume of the gas has mostly to do with temperature (or with pressure). Basically the volume of the gas is given by the energy of the molecules colliding with the enclosure, not by their size.
To put it differently, in a gas, the average volume allocated to a molecule (the container’s volume divided by the number of molecules) is enormous compared to the volume of a molecule, therefore the volume of the container has all to do with the “push” exerted by the molecules in motion on the walls of the vessel and nothing to do with the very small volume they occupy in the gas containing vessel.
To emphasize it more, a gas is mostly empty space packed with kinetic energy and its volume is mostly linked with the energy of the molecules (the temperature of the gas), unlike a solid wich is mostly densely stacked molecules.
The ideal gas law treats gas molecules as point particles that interact with their containers but not each other, that answers it. Thanks! $\endgroup$