Why P=nkT is better than PV=NRT

Oct 08 2010 Published by under Astronomy & Physics, Science Education & Outreach

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.

9 responses so far

  • Bau Ur says:

    Speaking as a high school teacher: R has a more easily envisioned meaning than the Boltzmann's constant. I don't want students to just plug in the numbers and perform the calculations obediently as an abstract exercise. I want them to understand the meaning of each number in as direct a way as possible. They can use Boltzmann's constant later after they have the basic underpinnings .

    The concept of the mole is not just useful for ideal gas calculations. Ideal gas calculations are useful for practicing the concept of the mole. It is an oddly difficult thing for many students to wrap their heads around, and they need the practice, because the mole is important in other contexts too. PV=NRT is a nice way for them to start working with it. It's good for lower level students.

    So, nope, I will give up my R when you pry mah cold dead fingers from off'n it. :) I will send you my students whose imaginations are properly prepared, and at least already have practice thinking with moles, and you can explain your Boltzmann's constant to them after they get there. Not gonna do your work for you, man. :)

  • Bau Ur says:

    Damn, above, I just wrote R every time I meant N. Fix that for me if you post it, would you?

  • Bau Ur says:

    It was a late night, an early morning, no coffee, head full of polyurethane fumes from remodeling project. Sorry.

  • Kevin Fairchild says:

    I agree with you if you're teaching it at a college level, kT is a common term. On a high school level, I prefer PV/T = NR. But I haven't taught it in a chemistry class, only as review for AP Physics tests.

  • Alex Besogonov says:

    Heh. I can _still_ remember derivation for the Eavg=3/2kT law that I did on my high school exam. Derivation of PV=nkT is a piece of cake after that.

  • Deepto Chakrabarty says:

    Rob, an argument on AUs that may convince many (non-astro)physicists is to appeal to their love of recasting problems in terms of characteristic scales. In that context, the AU is just a reasonable scale length to make the problem dimensionless, not unlike the Bohr radius. In fact the Bohr radius comparison alone should stop the muttering.

  • Raskolnikov says:

    P=nT. Just put k=1.

  • The Giant says:

    Interesting to see you carrying on about gas!

  • Lab Lemming says:

    Gasses are generally measured (or reported) volumetrically, so removing V makes the equation a lot harder to use.

    Is a G size cylinder with 1.5 m^3 going to overpressurize if it heats up to 60C in the sun?