# How Fast is the Stuff in our Lungs? (Whiteboard Sketch #3)

Neither can we directly see it, nor taste, smell, hear, or touch it – still, we know that it persistently surrounds us, completely covers us, and that we probably cannot accomplish much more than, day after day, putting one foot in front of the other on the bed of its “ocean”, repeatedly trying to offer opposition to the natural force pointed towards the earth’s core in order not to fall, while its enormous weight is pushing on our shoulders from above and it impedes our advance, perceptible via a subtle sweep across our cheeks.

Untouchable, maybe even grim effects as this something might seem to have, it is crucial for our existence and any other life forms’ whatsoever on our planet.

Air is not the same everywhere. The concentrations of some of its constituents – like, e.g., the one of water or aerosols – vary from place to place. Its underlying gas mixture, though, is quite homogeneously distributed across the earth: For the most part, it is composed of nitrogen ($N_2$, 78%) and oxygen ($O_2$, 21%). Other substances like argon, $CO_2$ (etc.) make up the remaining percent.

So air is a mixture of different gases. All right, then. – But how can one imagine a gas to exactly look like?

What a stupid question, some of you might think. A gas is just a gas – what should be so exciting about it?! – However, it’s often already these little questions which provide fascinating insights into nature in the end. You will see what I am intending to get at with this. 😉

A simple model of a gas: Tiny balls which relentlessly bump into each other.

A gas consists of smallest particles – molecules or atoms – which can be individually described in a satisfying way by physical models, at least in principle. Alas, it is insufficient for well describing and modelling a gas to know the behavior of a single nitrogen molecule and subsequently “extrapolate” it to the entirety of the gas. A gas is just a too complex system to be described this way – too many particles, all interacting with each other, are involved. In a sugar cube’s volume of nitrogen gas there are about 30 quintillion (2.69·1019 per cm³) $N_2$ molecules. In order to successfully describe a gas, others ways have to be found.

Well, and we have found them: Instead of describing a single molecule and then projecting the result on the totality of the gas, we consider the whole gas and treat it in a statistical manner.

A very successful and simultaneously simple model has arisen from this approach – namely the model of the ideal gas, which treats the molecules, simply put, as tiny little balls flying freely and without the influence of any force through space as long as they don’t experience collisions with other “balls”. The collisions are “elastic”, which means that no energy is converted into other forms of energy than collision energy.

Brownian motion of small latex balls (diameter: 20 nm) in water. This motion stems from the countless and random collisions of the much smaller water molecules with the bigger beads.
(Credit: Jkrieger, Deutsches Krebsforschungszentrum, workgroup B040, via Wikimedia Commons)

It is not hard to see that this model doesn’t actually describe “reality”, especially when it comes to the microscopic level of nature (this is why it is called a model), however it is capable of describing our macroscopic world strikingly well. For example, we can interpret the pressure of a gas in a box as the sum of all the small momentum transfers from the individual gas particles to walls of the box. The strength of the ideal gas model and its statistical formalism is clearly the incredibly large number of involved particles.

But hey, how can I visualize this sea of endlessly clashing balls? – Is it like a smooth ball pit on a fair or rather like in the interior of a lotto drum?

As always when writing an article of the Whiteboard Sketch series, I won’t go into the details here, but only show you some whiteboard notes underpinning the statements which will finally be presented.
Still, here are some words on my whiteboard paintings: There is the well-known Maxwell-Boltzmann speed distribution (the red sketch in the image labelled “MB distribution”), which tells us how many gas particles (vertical axis) have a given speed (horizontal axis). The speed distribution, by the way, is strongly dependent on the gas temperature. Further, one can calculate things like the average speed etc. via this function.
By introducing another quantity, namely the “collision cross-section” $\sigma$ (sigma) – a mathematical area around a particle, within which other passing particles get deflected due to a occurring collision -, one can estimate the average distance (the “mean free path”) $\Lambda$ (lambda) a gas particle is able to cover before it bumps into the next particle, or the “mean free time” $\tau$ (tau) which is the average time passing between two collisions.

Whiteboard sketch on the calculation of some features of gas molecules.
(Blue: General outlines of the derivations of the formulae needed. Red: Concrete calculations using the example of nitrogen.)

Representatively of air under “normal” conditions (temperature: $T=273.15$ K, pressure: $p=10^5$ Pa), I have calculated some characteristics of nitrogen, thus of all the $N_2$ molecules (reminder: air consists of 78% nitrogen). The results we obtain are certainly surprising if heard for the first time!

# Now, how fast are the individual gas particles?

Nitrogen molecules dash around with an average speed of 455 meters per second (1638 kilometers per hour / 1018 miles per hour). But since there are some 30 quintillion particles in a cubic centimeter, as already mentioned above, little time remains for free flight: After a (mean) distance of only 83 nanometers and a time of 0.118 nanoseconds, the next collision with another nitrogen molecule occurs. This means that a nitrogen molecule bumps into other particles about 1010 times in just one second.

Such extraordinary numbers are more tangible when compared to others. Let’s try it:
Particles in air averagely move faster than a common bullet (~340 m/s) and nearly twice as fast as a Boeing 777 (~248 m/s).
The mean free path is comparable to the size of a HIV virus (~90 nm) or to about 80% the size of a chromosome (~100 nm).
However, the mean free time is such a short period of time that it is really hard to come up with a comparison which actually makes sense to us. Maybe I should just mention that light is able to travel only a distance of 3.5 centimeters in the mean free time of nitrogen. (This is not that much considering that light normally covers a distance of 30 billion centimeters in one second!)

Stunning numbers, right?

We can deem us glad for being such giants compared to air molecules and not noticing the frantic dashing and bumping of the microscopic world. Even without it, the world and our daily lives are often stressful enough in other ways, don’t you think? 😉