Why P=nkT is better than PV=NRT

If you’ve ever taken a Chemistry course, you’ve run across PV=NRT. That is, of course, the ideal gas law. Real gasses approximate ideal gasses; the noble gasses (Helium, Neon, Argon, Krypton, Xenon) probably approximate it best. It tells you that the pressure times the volume of a gas is equal to the number of moles of that gas, times the ideal gas constant, times the temperature in Kelvins.
So, fine. It’s useful, and I’ve used it a lot. My problem is that as a physicist, I think that moles are an extremely gratuitous unit. Sure, I recognize that you’re more likely to be dealing with 32 grams of O2 than 32 individual molecules, but still, it’s yet one more concept that doesn’t do much for me. What’s more, the ideal gas constant is a constant that, at least as its name suggests, is of limited utility.
I much prefer this formulation:
P = n k T
All of the same information is there. However, instead of the ideal gas constant, we’ve got Boltzmann’s Constant, which is a much more fundamental constant. Yes, all the same information is there— except that it doesn’t come in units containing moles, so you don’t need to know the definition of moles to use it— and Boltzmann’s constant shows up as is in a lot of other equations.
On the left, we have pressure, the same as before. On the right, we have the number density of the gas. The variable n, instead of being just a number, is the number of particles per volume. OK, I will admit, that’s going to tend to be a huge number. If I did my calculations right, for a gas at room temperature it’s going to be something like 3×1025 m-3. So, I will admit that that is one advantage of the chemist’s way of formulating it: the numbers are easier to deal with.
The rest of the right is kT. What’s neat about that is that if you do physics (and probably chemistry as well, and probably many other natural sciences), you’re used to seeing kT all the time. Boltzmann’s constant times the temperature times a number of order 1 is the average kinetic energy of a particle in a gas that’s at temperature T. This (other than aethetically preferring k to R) is the primary reason I prefer this formulation of the ideal gas law. It’s got a piece in it that lets you directly connect this to other physics. “Aha”, you say, “this law is somehow related to the average energy of individual particles!” And, sure enough, if you realize that pressure is the rate at which particles are crossing an imaginary wall, times the amount of momentum that each particle carries with it across that imaginary wall, you realize that it should be related to the kinetic energy of that particle.
There’s another thing here. If you look at “nkT”, you’ll realize that that is just a number of order 1 times the kinetic energy density of the gas. kT is (close to) the kinetic energy of each particle, and n is the number of particles per cubic meter (or per cubic centimeter, if you like cgs units better). This leads immediately to the realization that the units of pressure are exactly the same as the units of energy density— something that seemed perverse to me the first time I came across the stress-energy tensor of relativity, as I’d been brainwashed into thinking they were entirely different things by the obscuration inherent in PV=NRT. To be sure, pressure and energy density aren’t the same thing, but they are related. (One could say that energy density is momentum flux in a temporal direction, and pressure is mometum flux in a spatial direction, but you need an appreciation of spacetime for that to be illuminating.)
It may be just me as a curmudgeonly physicist talking back to chemists who’ve figured out a more convenient way to deal with it. I’ve certainly come across curmudgeonly physicists who express disbelief and either horror or amused condescension that astronomers would use a unit so silly as the “Astronomical Unit”… and their reaction is simply the result of them not being used to it, and not realizing that that unit is extremely convenient for star systems, just like their fermi is extremely useful for atomic nuclei. However, I do really think that from a clarity of concept point of view, P=nkT is a much better way to state the ideal gas law than PV=NRT.