I come cross a question about calculate the MFP of a gas mixture, which contains several different kinds of molecules, each has different size, velocity and number density. My question is: How can I calculate the MFP of a mixture like this. Let's assume each component of the mixture follows the ideal gas law.
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2 Answers
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Below is educated guesswork. Someone more informed about statistical mechanics can perhaps do a better job.
The mean free path is usually estimated by employing following argument for a dilute gas. Suppose number density of gas molecules is $n$, and each has diameter $d$. Then collision cross-section of a molecule is $\pi d^2$. If a molecule moves a distance $l$ then the volume swept out by the collision cross section is $\pi d^2l$. For collision to occur there must be 1 molecule in this swept volume, i.e. $n\pi d^2l\sim 1$ which gives $l\sim (n\pi d^2)^{-1}$, where $l$ is now interpreted as the mean free path. All of this is standard stuff, and more rigorous expressions for mean free path can perhaps be derived. However let us build on this simple argument to get an estimate of mean free path in a mixture of gases.
Suppose there are $N$ species in the gas mixture, which we shall label using numbers $1$ through to $N$. Let $n_i$ be the number concentration of species $i$. Therefore the probability that a randomly picked molecule is of species $i$ is $P_i=n_i/(\sum_kn_k)$.
For a pair of molecules picked from species $i$ and species $j$ (not necessarily distinct) the collision cross-section would be $\pi (r_i+r_j)^2$, in which $r_\alpha$ is the radius of the molecule of species $\alpha$. Suppose you follow a molecule from species $i$. It may collide with molecule of species $j$ with probability $P_j$. We are assuming that there are several molecules of any given species. Given that it collides with molecule of species $j$, the corresponding mean free path is $l_{ij}\sim (n_j\pi (r_i+r_j)^2)^{-1}$. Therefore the average value of mean free path for molecule of species $i$ is $l_i=\sum_{j=1}^N P_jl_{ij}$. The average value of mean free path regardless of species is \begin{align} l& = \sum_{i=1}^N P_i l_{i}\\ &=\sum_{i,j=1}^N P_i P_jl_{ij}\sim \frac{1}{(\sum_kn_k)^2}\sum_{i,j=1}^N \frac{n_i}{\pi (r_i+r_j)^2} \end{align}
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The mean free path is $$\lambda\frac{N}{V}\pi r^2 \approx 1$$ where $r$ is the radius of a molecule. This gives $$\lambda = \frac{1}{(N/V)\pi r^2}$$
So we have $$\lambda_t = \frac{1}{\Sigma (x_i / \lambda_i)} \\ \Sigma x_i =1$$ where$\lambda_t$ is the mean value of the multi system and $x$ is the mole fraction.
