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My question is about estimating the radius of influence between two molecules; picture some mixture, comprised of water, oxygen gas (in small concentrations) and a molecule we denote $G$. In the mixture, oxygen gas can react with the excited state of the molecule $G$ but not with the ground state of $G$. After an excitation event, we can work out the proportion of $G$ moles which have become excited, but it's a little trickier to ascertain the next part about the chances of an oxygen molecule being sufficiency close to the excited $G$ to interact.

One approach is to calculate the fractional volume of oxygen per volume of mixture; this is relatively straight-forward to calculate and I've established a formula to do so for different oxygen concentrations; however, when I take the results from this and feed them into my model, they're too low by a factor of more than $10^5$; this could of course be due to the model being a poor choice, but my intuition tells me that what I really should be looking at is not the volume of oxygen but the "interaction volume" surrounding the O2 molecules; for example, if there is a field of radius of $R$ inside which electrical interaction is likely, then my volume calculation should not aim to find total volume of O2 gas but rather the total volume over which interaction is likely $V_{L}$, in a naive form assuming constant electrical attraction, something like;

$V_{L} = N_{o}\left(\frac{4}{3}\pi R^3 \right)$

where $N_{o}$ is the total number of oxygen molecules. This might be overly simplistic, as an interaction volume is likely to be a function of distance by Coulomb's law / electrodynamic effects but before I go and re-invent the wheel, does anyone know if there is a branch of molecular dynamics or atomic theory which deals with these kind of problems? I've messed around with Van Der Waals radii but they present the same problem in so much as they approximate the volume but not the sphere of interaction.

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In molecular dynamics, you want to think about "cross sections" and "mean free path" - two closely related concepts. You also have to consider the relative velocity of the molecules - is G of the same mass as the oxygen molecules? As you know, lighter molecules travel faster (same kinetic energy by equipartition theorem).

You might find this article helpful. It gives the mean free path as

$$\ell = \frac{k_BT}{\sqrt{2}\pi d^2p}$$

When you divide that by the velocity you get the mean time between collisions of similar molecules. When the masses and concentrations of the molecules are different, some scaling is required. You can find more information and diagrams at http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/menfre.html

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    $\begingroup$ Some very good suggestions there, thanks - I had been mulling over free path length but as it's not technically a gas I had been hesitant to use it; following on from your answer I had included my own below, if anyone is interested in the fine details! $\endgroup$ – DRG Jan 15 '15 at 13:33
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Following on from the excellent suggestion by Floris, I thought I'd add a quick work through lest anyone else encounters a similar question:

There is a way to link mean-free path $\ell$ with the diffusion constant; in the case of oxygen in water, the diffusion constant $D$ is roughly $2 \times 10^{-9}$m$^2$/s ; the relationship between the twain is roughly

$D = \frac{1}{3}\ell v_{T}$

We can then define the thermal velocity with the following relationship:

$v_{T} = \sqrt{\frac{8k_{B}T}{\pi m}}$

Where $m$ is the mass of an oxygen molecule and $T$ the temperature. At $T = 310.15$K , this approximately works out to a thermal velocity of 452.89 m/s, and thus $\ell =1.325 \times 10^{-11}$m , a minsicule distance.

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  • $\begingroup$ Yes that sounds about right. This link gives the mean free path of air molecules at ambient temperature as 68 nm, and that has to be roughly three orders of magnitude larger than in a liquid. $\endgroup$ – Floris Jan 15 '15 at 13:54

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