Consider the steady, uni-directional diffusion of a binary mixture of A and B across a vertical slab of dimensions $0.2 \text{ m} \times 0.1 \text{ m} \times 0.02 \text{ m}$ as shown in the figure. The total molar concentration of A and B is constant at $100 \text{ mol m}^{-3}$. The mole fraction of A on the left and right faces of the slab are maintained at 0.8 and 0.2, respectively. If the binary diffusion coefficient $D_{AB} = 1 \times 10^{-5} \text{ m}^2 \text{ s}^{-1}$, the molar flow rate of A in $\text{mol s}^{-1}$, along the horizontal x direction is
Step 1: Given Data
Dimensions of slab:
Thickness in $x$-direction
$L = 0.02 \text{ m}$
Cross-sectional area perpendicular to diffusion:
$A = 0.20 \times 0.10 = 0.02 \text{ m}^2$
Total molar concentration:
$C = 100 \text{ mol m}^{-3}$
Mole fractions:
$y_{A1} = 0.8$
$y_{A2} = 0.2$
Diffusivity:
$D_{AB} = 1 \times 10^{-5} \text{ m}^2 \text{s}^{-1}$
Step 2: Use Fick’s law (steady equimolar counter diffusion)
Molar flux:
$N_A = -D_{AB} C \dfrac{dy_A}{dx}$
Since variation is linear:
$N_A = D_{AB} C \dfrac{(y_{A1} - y_{A2})}{L}$
Step 3: Substitute values
$N_A = (1 \times 10^{-5})(100)\dfrac{(0.8 - 0.2)}{0.02}$
$N_A = (1 \times 10^{-5})(100)\dfrac{0.6}{0.02}$
$N_A = (1 \times 10^{-3}) \times 30$
$N_A = 3 \times 10^{-2} \text{ mol m}^{-2} \text{s}^{-1}$
Step 4: Calculate molar flow rate
$\dot{n}_A = N_A \times A$
$\dot{n}_A = (3 \times 10^{-2}) \times (0.02)$
$\dot{n}_A = 6 \times 10^{-4} \text{ mol s}^{-1}$
Final Answer
$\boxed{6 \times 10^{-4} \text{ mol s}^{-1}}$
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).
- For a binary mixture of components A and B, $N_A$ and $N_B$ denote the total molar fluxes of components A and B, respectively. $J_A$ and $J_B$ are the corresponding molar diffusive fluxes. Which of the following is true for equimolar counter-diffusion in the binary mixture?
- 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
