Pressure in Ideal Gas Law

In Boyle's law ($p$ inversely proportional to $V$ for constant $n$ and $T$), $p$ refers to the applied (external) pressure on the gas.

However, in the ideal gas law, my textbook says that $p$ refers to the pressure of the gas, which I think refers to the pressure applied by the gas. How is this possible, since the ideal gas law is derived from Boyle's law (and Charles' and Avogadro's laws)?

Shouldn't the $p$ in the ideal gas law then refer to the applied (external) pressure like in Boyle's law, not the pressure of (i.e. applied by) the gas?

• Everything should refer to the pressure exerted by the gas. The line on Boyle's law that you read was probably written with the assumption in mind that the internal and external pressures are equal. Oct 12, 2015 at 12:24
• But doesn't Boyle's law make more sense when P is the applied pressure? I thought the P in the ideal gas law needed to be changed, not the one in Boyle's law. Oct 12, 2015 at 12:39
• "Make more sense" is very vague... It makes more sense from an experimental point of view because the only way you can vary the volume is by varying the external pressure. However in that case the internal pressure will increase to match the external pressure, and Boyle's law will hold equally well regardless of which pressure you use. Which is why I said, it was probably written with the implicit assumption that the internal and external pressures were equal. Oct 12, 2015 at 13:02
• That is in the case of a container with movable walls, which will allow for equilibration of the internal and external pressure. In the case of fixed walls, you cannot experimentally verify Boyle's law anymore; however, the ideal gas law still holds true as long as you use the internal pressure. The ideal gas law relates properties of a system to each other, and in this case where the system cannot "see" anything outside since the walls are fixed, the external pressure cannot have any influence on the volume or temperature of the gas inside. Oct 12, 2015 at 13:09

1 Answer

At equilibrium (that is, if volume/temperature/mass is stable and not changing over time), the external pressure applied to the gas will equal the internal pressure of the gas.

This is (more or less) a consequence of Newton's third law. The external environment pushes on the gas with a certain force, and the gas pushes back with an equal and opposite force. At equilibrium, no work is being done on the gas or on the environment, so the forces have to balance. (This is very crude and hand-wavy - a more thorough description would require much more math.)

So (at equilibrium), there really isn't any difference between the two. In the Boyle's law case, the value is described as the external pressure, because the typical experimental setup in that case involves using something like a piston to control and vary the external pressure applied to the system. In the ideal gas case, you may not explicitly be controlling the external pressure, instead you're controlling temperature, volume, or particle numbers. In those case, you're normally measuring the internal pressure of the system, rather than (directly) controlling it or the external pressure. But (at equilibrium) the pressure of the gas pressing out is the same as the pressure of the container pushing in.

It's all just a matter of perspective, and how the system is set up.