# Gas Laws: Why is PV directly proportional to mT?

My book mentions that the three informal gas laws (Boyle's, Charles', and Gay-Lussac's) can be combined into a more general relation PV ∝ mT (the precursor to the Ideal Gas Law).

Where: P is pressure, V is volume, m is mass (taken as a measure related to quantity of gas molecules), and T is temperature on the absolute scale.

• From Boyle's Law: P ∝ 1/V
• From Charles' Law: V ∝ T
• From Gay-Lussac's Law: P ∝ T
• From simple observation of a balloon being inflated: m ∝ V

All these can be surmised from PV ∝ mT by letting some of these state variables be constant. But when we let V and T be constant we get P ∝ m and this is a relationship I don't understand. It's not any law that I was able to find in my textbook or online and I don't get it conceptually.

But for PV ∝ mT to be true, P ∝ m must also be true. So my question is what is the relationship between pressure and mass?

• I don't think m is mass. Isn't it number of particles in mols or something? The version I like is $pV = NkT$, which is $p=nkT$ using density $n=N/V$. Pressure is force per area, which is the same as energy per volume. $kT$ is the energy per particle, and $n$ is the particle density. So $nkT$ gives an energy density, as does $p$. Then, more particles at the same $T$ of course mean more total energy, and if volume is fixed, then that means higher energy density as well. – EL_DON Nov 1 '17 at 3:30

## 1 Answer

The relationship for physicists is: $$PV=NkT,$$ where $N$ is is the number of gas molecules/atoms, and $k$ is Boltzmann's constant. In chemistry they normally convert the $Nk$ to $nR$, where $n$ is measured in moles instead of being a simple count, and $R$ is the universal gas constant.

So, the proportionality from pressure to mass is incidental and depends on what the gass is made of, since, for a particular pure type of gas, you can convert from mass to number and back.

The reason that pressure is proportional to the number of atoms is because pressure is the average force per unit area exerted on the walls of the container. That force is cause by individual collisions of gas molecules with the walls, and the force is directly proportional to the collision rate. Naturally, if you double the number of molecules, all other things being equal, you'll double the collision rate, doubling the pressure. Thus pressure is proportional to particle number.

The interesting question is that when you have a collision the impulse, and thus the force, depends on both the mass and velocity of the colliding particle. So why is it proportional to number and not mass? Well, the reason is because temperature in gasses is a measure of the kinetic energy of the molecules. So, if you have two different gases at a given temperature, volume, and number of molecules, the one with more massive molecules will be moving slower in such a way that they have the same pressure.

• So P ∝ m is incidental and derived from the actual Ideal Gas Law, and not the other way around. My book made it sound like PV ∝ mT was the relationship that lead to the Ideal Gas Law (PV=NkT). The relationship between amount and pressure is much more evident. Thanks for clearing it up and the interesting insight! – V. Poghosyan Nov 1 '17 at 3:40
• @V.Poghosyan $PV\propto mT$ likely proceeds the ideal gas law, historically. It was, at the time, an observation. Now, we see it as logically derivable from the ideal gas law, which is also logically derivable from statistical mechanics. – Sean E. Lake Nov 1 '17 at 3:43