Often, we have a multi-component mixture of gases and do not want to deal with the diffusion coefficients for every pair of gases. We would like to define a pseudo-binary diffusion coefficient for each species relative to the mixture.

a. Show how using Fick's Law in the form:

\[\overrightarrow{\mathbf{N}}_{i}=-c_{t} D_{i m} \vec{abla} x_{i}+x_{i} \sum_{j=1}^{n} \overrightarrow{\mathbf{N}}_{j}\]

and the Stefan-Maxwell relations, equation (2.74), we can define such a coefficient based on individual binary diffusion coefficients for each pair of gases. $D_{i m}$ is the pseudo-binary diffusion coefficient.

b. Show in the limit where $x_{1} \approx x_{2} \approx 0$ that:

\[D_{3 m}=\frac{D_{31} D_{32}}{D_{31}+D_{32}} \quad D_{2 m}=D_{23} \quad D_{1 m}=D_{13}\]

c. Use your result to calculate the pseudo-binary diffusion coefficient for each species in the system $\mathrm{H}_{2} \mathrm{O}, \mathrm{He}, \mathrm{N}_{2}$.

\[\begin{array}{llll}\mathrm{H}_{2} \mathrm{O}-\mathrm{He} & 0.908 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s} & \mathrm{H}_{2} \mathrm{O}-\mathrm{N}_{2} & 0.256 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s} \\\mathrm{He}-\mathrm{N}_{2} & 0.687 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\end{array}\]

d. Can your results be extended to higher order mixtures?