My young son is fascinated by large numbers (e.g., a google). One question he posed to me is if the human body over its lifetime collides with atoms/molecules a google times (I think not, but it's interesting to do the calculation and see what the number actually, to first order, is). The construct seems simple: $$ \mbox{Total Collision Count}\ = f\times A\times t, $$ where $f$ is the (average) surface collision frequency (collisions per unit area per time), $A$ is the surface area of a body (say 1.5 square meters), and $t$ is (being generous and rounding up) 100 years. Assume 1 atmosphere of pressure and 20 degrees Celsius. Also, again for simplicity, assume only surface contact is with air (!).

I'm quickly realizing that my stat mech textbook has far too much dust on it and I'm not sure where to start in determining $f$. I imagine Boltzmann statistics should play a role (ala Maxwell's distribution). I.e.: $$ f\propto\left(\begin{array}{c} \mbox{probability of particle} \\ \mbox{having velocity}\ \bar{v}\end{array}\right)\times\left(\begin{array}{c} \mbox{number of vectors}\ \bar{v} \\ \mbox{corresponding to speed}\ v_x\end{array}\right), $$ where the first factor I believe is $\propto e^{-mv^2/2kT}$ and the second factor needs to take into account we are only interested in surface collisions from one orientation. But at this point I'm not sure if I'm on the right track and, even if I am, where I go from here. Any help?


2 Answers 2


To evaluate the number of collisions in a lifetime does require some knowledge of Physics.

ne first needs an estimate of the speed of gas molecules at room temperature $(\rm 20^\circ C = 293\,K)$ using the average kinetic energy of a molecule $\frac 12 m v^2$, $m$ being the mass of a molecule, $\sim 5\times 10^{-26}\rm\,kg $, and $v$ an average relating to the speeds of the molecules, being equal to $\frac 12 k T$, where $k=\sim 10^{-23}\,\rm J/K$ is the Boltzmann constant and $T=300\,\rm K$ is the temperature.
This gives a speed for the molecules of approximately $400\,\rm m/s$.

When molecules hit a surface and rebound their momentum changes and this results in a force $F$ being exerted on the surface.
$F= \text{rate of change of momentum} =n( mv - (-mv)) = 2nmv$ where $n$ is the rate at which molecules hit the surface.
Atmospheric pressure $\frac FA$, where $A$ is the area of the surface is approximately $10^5$ pascals.
So for one square metre of surface $\frac {2nmv}{1}=10^5 \Rightarrow n \sim 10^{27}$ per second.
As this is an order of magnitude calculation and remembering that the surface area of a human changes with age, the surface area of a human can be taken to be one square metre.

If a human lives for $70$ years then the number of collisions with air molecules is

$365 \times 24\times 60\times 60 \times 10^{27} = \mathbf \sim 10^{34}$

are very large number but completely dwarfed by a google.

Even using the age of the universe, $1.38 \times 10^{10} \rm \, years$, produces only $\sim 10^{42}$ collisions!


You don't need thermodynamics for this. You need a sense of how big a googol is.

A googol, $10^{100}$, is really big. The universe is estimated to have about $10^{80}$ particles in it. So every particle would have to bounce off you 10 billion times to get you one $10$ billion$^{th}$ of the way there.

It might sound reasonable that all the air in the atmosphere might hit you. But how about if every atom in the Earth turned into air? It sounds a little improbable.

What about the Sun? The galaxy? The local cluster? There are several more layers to get to the observable universe.

So your son is right to be fascinated.


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