$14-11.$ Ternary Gas with Two Nearly Equal Diffusivities: Diffusion in Air

The molecular weights and effective diameters of $N_2$ and $O_2$ are similar enough that their

binary diffusivities with a third gas are often nearly equal. Thus, for example, the diffusivities for

$\frac{N_2}{H_2}$ and $\frac{O_2}{H_2}$ are very similar, although much larger than for $\frac{N_2}{O_2}.$ For this reason, when cal$-$

culating the flux of something other than $N_2$ or $O_2,$ air is often approximated as a single component.

The objective is to determine how the Stefan$-$Maxwell equations can be simplified when two binary

diffusivities are nearly equal.

The third component is denoted as species $3.$ It is assumed that $D_{13}$~$=D_{23},$ but with $D_{23}{D}_{13}.$

Thus, species $2$ can be either $N_2$ or $O_2,$ whichever has the slightly higher diffusivity with species $3.$

A small parameter is defined as

$=\frac{D_{23}-D_{13}}{2D}=\frac{D_{23}-D_{13}}{D_{23}+D_{13}}$

where $D$ is the arithmetic mean of $D_{23}$ and $D_{13}.$ Of particular interest are the conditions under which

the flux of component $3$ is approximated well by Fick's law using $D,$ or

$N_3^{(0)}=C_3{v}^{(M)}-CDgrad{x}_3.$

$($a$)$ If $=0,$ show that $N_3=N_3^{(0)}$ exactly.

$($b$)$ If $0,$ show that

$N_3=N_3^{(0)}+[N_3^{(0)}(x_2^{(0)}-x_1^{(0)})+x_3^{(0)}(N_1^{(0)}-N_2^{(0)})]+O({}^2)$