Understanding Equimolar Counter-Diffusion Fluxes
For a binary mixture involving components A and B, the total molar fluxes are denoted by $N_A$ and $N_B$, while the molar diffusive fluxes are $J_A$ and $J_B$. We need to determine the relationship between these fluxes under the condition of equimolar counter-diffusion.
Equimolar Counter-Diffusion Explained
- Equimolar means that the molar rates are equal.
- Counter-diffusion means the components move in opposite directions.
- Equimolar counter-diffusion specifically implies that component A moves in one direction and component B moves in the opposite direction, with their molar rates being identical in magnitude.
Analysis of Total Molar Flux ($N_A + N_B$)
Under equimolar counter-diffusion:
- The molar flux of A ($N_A$) is equal in magnitude but opposite in direction to the molar flux of B ($N_B$).
- This can be expressed as: $N_A = -N_B$.
- Summing these gives the total molar flux relationship: $N_A + N_B = N_A + (-N_A) = 0$.
Analysis of Diffusive Molar Flux ($J_A + J_B$)
The diffusive flux ($J_i$) is defined as the flux relative to the molar average velocity. For any binary mixture, the sum of the diffusive fluxes must always be zero:
$ J_A + J_B = 0 $
This fundamental relationship arises from the definition of diffusive flux relative to the molar average velocity. In the specific case of equimolar counter-diffusion, where $N_A + N_B = 0$, the molar average velocity is also zero. Consequently, $J_A$ equals $N_A$ and $J_B$ equals $N_B$. Since $N_A = -N_B$, it follows directly that $J_A = -J_B$, satisfying $J_A + J_B = 0$.
Conclusion
Based on the definitions and implications of equimolar counter-diffusion in a binary mixture:
- The sum of the total molar fluxes is zero: $N_A + N_B = 0$.
- The sum of the diffusive molar fluxes is also zero: $J_A + J_B = 0$.
Therefore, the condition $N_A+N_B = 0$ and $J_A + J_B = 0$ is true for equimolar counter-diffusion.
Important Questions from Molecular Diffusion
Consider a non-reactive binary mixture of two ideal gases A and B in an isothermal horizontal closed tube. Due to the concentration difference at the two ends of the tube, equimolal counterdiffusion occurs along the axial direction. No concentration gradients exist in the radial direction. The profile for partial pressure of component A ($p_A$) as a function of axial distance $z$ is shown in the figure. The partial pressure of component B ($p_B$) approaches zero at $z = 0.8 \text{ m}$. At a location $z = z_1$ the partial pressures of the components A and B are equal. The value of $z_1$ (in m) is _____ (rounded off to two decimal places).
- Consider the following two cases for a binary mixture of ideal gases A and B under steady state conditions. In Case 1, the diffusion of A occurs through non-diffusing B. In Case 2, equimolal counter diffusion of A and B occurs. In both the cases, the total pressure is 100 kPa and the partial pressures of A at two points separated by a distance of 10 mm are 10 kPa and 5 kPa. Assume that the Fick's first law of diffusion is applicable. What is the ratio of molar flux of A in Case 1 to that in Case 2?
- In a binary gas-liquid system, $N_{A,EMD}$ is the molar flux of a gas $A$ for equimolar counter diffusion with a liquid $B$. $N_{A,UMD}$ is the molar flux of $A$ for steady one- component diffusion through stagnant $B$. Using the mole fraction of $A$ in the bulk of the gas phase as $0.2$ and that at the gas-liquid interface as $0.1$ for both the modes of diffusion, the ratio of $N_{A,UMD}$ to $N_{A,EMD}$ is equal to ________ (rounded off to two decimal places).
- Consider two stationary spherical pure water droplets of diameters $d_1$ and $2d_1$. $CO_2$ diffuses into the droplets from the surroundings. If the rate of diffusion of $CO_2$ into the smaller droplet is $W_1$ mol $s^{-1}$, the rate of diffusion of $CO_2$ into the larger droplet is
- Consider steady-state diffusion in a binary A-B liquid at constant temperature and pressure. The mole-fraction of A at two different locations is $0.8$ and $0.1$. Let $N_{A1}$ be the diffusive flux of A calculated assuming B to be non-diffusing, and $N_{A2}$ be the diffusive flux of A calculated assuming equimolar counter-diffusion. The quantity $\frac{(N_{A1}-N_{A2})}{N_{A1}} \times 100$ is ________ (rounded off to one decimal place).
