# 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

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.
• @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