# Your Tires have Holes in them

McChoppinites!

So I’ve had bikes on the brain lately. I finally got my bike back from Iowa on Monday1, and while I was waiting for it to arrive, I purchased a bunch of new bike stuff. Nothing like RAGBRAI to make you want to pimp your bike out. Anyways, one of the things I purchased was a Garmin Edge 705.

Now, I never thought I’d purchase a dedicated GPS unit; I have an iPhone and that turns out to be more than enough GPS for me. But the Edge is a whole lot more than just GPS. It’s a bike computer! Or, as Petey would say, it’s a bikeputer. Which is to say it’s a cool sensor system with which to collect data whilst I ride my bike. As you should already know, I like the idea of collecting data automagically whilst doing activities. Indeed, I have plans a-brewing already about taking the Edge with me winter camping. Old Timey? No, but with the Lascar, I will have time, temperature, elevation, heart rate, and location data to do all sorts of things with! This of course will mean more graphs.

Back to the Edge, it clearly does a lot of stuff, and I want to learn about all of it so that I can fully exploit it’s sensing and reporting capabilities. So where to turn to learn more about it? The instruction manual? No, the intertubes of course! In short order I had found DC Rainmaker, who clearly has already learned almost everything there is to know about all of the Garmin sport computers.

DC Rainmaker has a lot of great information. Like how to get free maps for Garmin units. Tips on how organize your data screens. Even a piece about how to ski with GPS. But the thing that really got me going was in this post about using CO2 cartridges to inflate a bike tire after fixing a flat. At the end of it he tells you to that if you fill a tire with CO2 from a cartridge on the roadside, when you get home you should deflate the tire and refill it with normal atmospheric air. He then offers a link to a forum discussion in which someone explains the scientific reason why this is so.

First, I have almost no experience with this, having only ever filled a bike tire once with CO2 from a cartridge. So when I read that I was a little surprised. I think DC Rainmaker is correct. I think that a tire filled with pure CO2 will deflate faster than a normally filled tire. But I totally disagree with the “scientific” explanation. Let’s examine the original forum post.

First, note the comment from humble_biker. It’s hilarious. Magically expanding and shrinking carbon dioxide makes me laugh. It’s as if humble_biker has learned just enough chemistry to know the word “molecule” without actually knowing what a molecule is. If intended to be funny, a great joke and a tip of the hat to you, humble_biker.

All right, on to the given “scientific” explanation from khuon. First, it’s not much better of an explanation than the one proposed by the non scientist. However, this one is not funny because it uses math. Because khuon included multiple boldface equations I can only conclude he was being completely serious. This is further shown with the final statement:

Note – Writing equations with vBcodes sucks!

Pointing out how hard it is to markup equations is not a punchline my friend.

Okay, now that we’ve established that khuon is not attempting comedy, let us examine exactly why his explanation fails. First, khuon gratuitously throws in the ideal gas law and does a bunch of stuff with it. But this is, as far as I can tell, only used to explain why the cylinder and the valve get cold when using a CO2 cartridge. So, our first tip-off to faulty logic: using a well known equation to explain something that is unrelated to the problem. Also, his method was pretty messy. So, nothing really wrong yet, but as any college science professor will tell you, the inelegant use of unrelated equations in a solution is a huge red flag.

(Not a huge red flag – photo by BlueGoaॐ☮)

After the ideal gas law debacle, he gets to the “real” reason tire tubes filled with only carbon dioxide deflate faster than usual:

[It is] because of how the molecules in rubber attract CO2 better than Oxygen or Nitrogen. As a result, the CO2 permeates the rubber which then swells and thus allows more molecules to escape.

All right, I know I already established that muon2 or whatever is not trying to be funny, but that just makes me laugh. It’s hilarious. It’s even more hilarious than humble_biker‘s explanation. The idea that butyl rubber somehow attracts carbon dioxide better than diatomic oxygen is just great. It’s good stuff. But there’s more! Not only does it attract the carbon dioxide in a way that it doesn’t do with diatomic oxygen or nitrogen (which I humbly submit to kligon would be sufficient), but this in turn causes the rubber to swell, which causes more CO2 to escape. I think gluon runs into even more trouble here, but I won’t get into that (if CO2 causes the tube to swell and leak air faster, shouldn’t we be filling tires with CO2 depleted air??).

I have to admit that although I found fluon‘s explanation hysterical, I am, as a card caring physicist3, extremely disturbed by the overall blatant misuse of statistical physics that went on here. Being funny is one thing. Using the ideal gas law to make people think you know what you’re talking about when you clearly do not, is another. I know that most people don’t care, but when I see a physical equation being trotted out for show before a psuedo-scientific explanation of observed phenomena, it is akin to the pain that Lynne Truss must feel when she sees a sentence with a comma-free nonrestrictive clause.

Now that I’ve thrown stones I feel I must at the very least give some explanation as to what is going on with CO2-only inflated tires. But, to do that, I think I first must explain why tires deflate at all.

Yes, tires deflate. We all know it. And it’s not just bike tires. Car tires and balloons do it too. In physics this process, the air escaping, is known as effusion.4 Effusion is a special case of diffusion. There are a lot of examples of diffusive processes, the most commonly known being osmosis, but more on osmosis later. Anyways, it is now time for my gratuitous equation to show you all that I’m serious: $\frac{dN}{dt} = \frac{A}{2V}\sqrt{\frac{kT}{m}}N$. This equation describes the rate of change of the number of molecules ($N$) of a molecular mass ($m$) within a given volume ($V$) and at a certain temperature ($T$) that are effusing through an area ($A$) over a given time ($t$). If you do some calculus and algebra, you get the equation $N(t) = N_0 e^{-\frac{t}{\tau}}$. This equation will tell you the number of molecules left after a certain time given $N_0$ an initial number of molecules. $\tau$ is the characteristic time that it takes to get to a third of the initial number of molecules and $\tau = \frac{2V}{A}\sqrt{\frac{m}{kT}}$. If you assume that this process happens slowly, which it should or it’s not effusion, you can assume that the temperature does not change. Then, keeping the volume fixed you can use the ideal gas law, which I will not transcribe here because we already have it courtesy of moroun, and you can write the previous equation in terms of pressure: $P(t) = P_0 e^{-\frac{t}{\tau}}$. This is a useful equation for understanding why tires deflate. It basically says that if we were to leave a bike tire out that is inflated to some pressure $P_0$, after some time $t = \tau$, the tire would have a pressure equal to $\frac{P_0}{3}$. (See figure 1) All you need to do is measure $\tau$ and you can predict how long it will take how your bike tire to reach some lower pressure. But what else is $\tau$ telling us? Well, remember that $\tau = \frac{2V}{A}\sqrt{\frac{m}{kT}}$. So, knowing the volume, molecular mass of air, and temperature, we could calculate an effective area of the microscopic holes in our rubber bike tubes. Which I will leave as an exercise to you, the reader.

Figure 1

Figure 2 shows the predicted decay of air pressure in a tire that starts out at 120 psi. Suppose we know after three days the tire pressure is 80 psi. From that information we can predict the tire’s pressure at any time.

Figure 2

And there is the rub, that your seemingly impermeable rubber tube is actually a semi-permeable membrane through which gas can pass. The thing about semi-permeable membranes is that they are not one-way. As gas leaves, some actually enters. It is just that more leaves than enters. That is more or less what statistical mechanics is all about: elegant descriptions of statistical processes such as this. All right, so why does my purely CO2-filled tire deflate faster? Well, it’s sort of like osmosis. Yeah, back to osmosis. Osmosis is a little different because it involves water. Wikipedia describes osmotic pressure thusly: “the phenomenon of osmotic pressure arises from the tendency of a pure solvent to move through a semi-permeable membrane and into a solution containing a solute to which the membrane is impermeable.” It’s probably something you remember from chemistry, that two different solutions separated by a semi-permeable membrane “want”5 to equilibrate. This manifests itself as a force, or pressure, across the membrane. Overcoming that force is how you get reverse osmosis and some people get there fresh water, but I digress.

Our process does not involve water but gas molecules and to explain it you need a different physical quantity know as partial pressure. Partial pressure is the portion a particular molecule contributes to the total pressure. For instance, if you have a 50/50 mixture of O2 and CO2, than 50% of the pressure is provided by the O2 and 50% is provided by the CO2. The partial pressure of the CO2 is 50% of the total pressure. Easy right? The thing about partial pressure is that it’s a lot like normal pressure and it too wants to equilibrate. So let’s go back to our 120 psi bike tire sitting out in a 15 psi atmosphere. The partial pressure of the CO2 in the atmosphere is almost nothing. The partial pressure of the CO2 in our tire is 135 psia.6 So the difference is these pressures is also roughly 135 psi.

Now here is the tricky part and I’m going to be honest, I’m not exactly sure what to do with the partial pressure. I’m tempted to just use that in our equation. If you do that, then our equation says that given two of the same tires filled to the same pressure, but only one with pure CO2, after the characteristic time $\tau$ the CO2 tire has a 27% lower pressure. So let’s say we start with two 120 psi tires, one filled with a CO2 cartridge and one with ambient air, and that normally after 3 days our air-filled tire is at 80 psi. All that seems reasonable to me. This predicts that our CO2 only tire should be at 75 psi or about 7% lower pressure (figure 3). This, to me, seems like not a big deal. So, either I’m not using partial pressure correctly in the equation, or an entirely different equation is needed. Clearly an experiment is warranted.

Figure 3

The important thing to note is that the CO2-only tire deflates faster than an air filled tire because the partial pressure of CO2 in the bike tire is driving diffusion. This predicts that any single gas filled tire would deflate faster than normal, even an oxygen or nitrogen filled tire, albeit more slowly than the carbon dioxide only tire. I do not recommend it, but if you were to fill a bike tire with oxygen gas it would deflate to a given pressure sometime in between the time it would take an air filled tire and a carbon dioxide filled tire.

Not only does this show I’m serious, but that I’m way more serious because it’s a differential equation. Incidentally, there is another law, Graham’s law that relates how fast a gas effuses to its molecular weight. This is clearly not driving the CO2 diffusion in our tire because the CO2 has a higher molecular weight than air and therefore Graham’s law predicts a slower effusion.

1I wrote this six months ago, but I’m just posting it now…
2It was at this point I got tired of writing khoun . And then it became fun. I’m sorry, khoun , if you ever read this, for having some fun at your expense.
3I actually don’t have a card. But I do have a diploma and it says something about physics on it.
4In the case of car tires I’m not actually sure it’s effusion, due to the fact that car tires deflate so slowly. More likely, the driving phenomenon is straight up diffusion, with a little bit of effusion thrown in for good measure.
5Not that molecules ever “want” to do anything, they just “do stuff.” And they “do stuff” in a very predictable way, which is really fortunate for us, the users of physics.
6Psi stands for pounds-per-inch-square and it’s a relative measure of the pressure. In this case, it’s relative to the atmosphere. Psia is pounds-per-inch-square-absolute. It’s a measure of the absolute pressure. Atmospheric pressure is about 15 psia.