## Related questions with answers

The general form of the Stefan-Maxwell equation for mass transfer of species i in an n-component ideal gas mixture along the z-direction for flux is given by

$\frac{d y_{i}}{d z}=\sum_{j=1, j \neq i}^{n} \frac{y_{i} y_{j}}{D_{i j}}\left(\nu_{j}-\nu_{i}\right)$

Show that for a gas-phase binary mixture of species A and B, this relationship is equivalent to Fick's rate equation:

$N_{A, z}=-c D_{A B} \frac{d y_{A}}{d z}+y_{A}\left(N_{A, z}+N_{B, z}\right)$

Solution

VerifiedIf we have gas-phase binary mixture of species A and B, we can substitute $i=A$, $j=B$ and $n=2$ into Stefan–Maxwell equation. Therefore, the Stefan–Maxwell equation for mass transfer of species A and B in an 2-component ideal gas mixture along the z-direction for flux, is:

$\begin{align*} \dfrac{d y_{i}}{d z}=\sum_{j=1, j \neq i}^{n} \dfrac{y_{i} y_{j}}{D_{i j}}\left(v_{j}-v_{i}\right)\\ \dfrac{d y_{\text{A}}}{d z}=\dfrac{y_{\text{A}}\cdot y_{\text{B}}}{D_{A B}}\left(v_{\text{B}_\text{z}}-v_{\text{A}_\text{z}}\right) \tag{1} \end{align*}$

According to the equation $\textbf{(24-9)}$, we can express mole fraction of component A and B, as:

$\begin{align*} y_{\text{A}}=\dfrac{c_{\text{A}}}{c} \tag{2}\\ y_{\text{B}}=\dfrac{c_{\text{B}}}{c} \tag{3} \end{align*}$

According to the $\textbf{Table 24.1}$ we can write:

$\begin{align*} y_{\text{A}}+y_{\text{B}}&=1\\ y_{\text{B}}&=1-y_{\text{A}}\tag{4} \end{align*}$

According to the book we can express $N_{\text{A}_{\text{z}}}$ and $N_{\text{B}_{\text{z}}}$, as:

$\begin{align*} N_{\text{A}_{\text{z}}}&=c_{\text{A}}\cdot v_{\text{A}_\text{z}} \tag{5}\\ N_{\text{B}_{\text{z}}}&=c_{\text{B}}\cdot v_{\text{B}_\text{z}} \tag{6} \end{align*}$

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