Diffusion Rate Dependency on Droplet Size
The rate of diffusion ($W$) of a substance into a droplet can depend on various factors, including the surface area, concentration gradient, and diffusion coefficients. For spherical droplets diffusing from surroundings, the rate is often related to the surface area ($A \propto d^2$, where $d$ is diameter) or potentially other geometric factors.
However, based on the provided options and correct answer, we infer that the diffusion rate in this specific scenario is assumed to be directly proportional to the droplet's diameter ($d$).
This assumption leads to the relationship: $W \propto d$.
Calculating Diffusion Rate for the Larger Droplet
Let $d_1$ be the diameter of the smaller droplet and $W_1$ be its diffusion rate.
Let $d_2$ be the diameter of the larger droplet, given as $d_2 = 2d_1$. Let $W_2$ be its diffusion rate.
Using the proportionality $W \propto d$:
- For the smaller droplet: $W_1 = k \cdot d_1$, where $k$ is the constant of proportionality.
- For the larger droplet: $W_2 = k \cdot d_2 = k \cdot (2d_1)$.
To find $W_2$ in terms of $W_1$, we can express $k$ from the first equation as $k = W_1 / d_1$. Substitute this into the second equation:
$ W_2 = \left( \frac{W_1}{d_1} \right) \cdot (2d_1) $
Simplifying the expression:
$ W_2 = 2 W_1 $
Thus, the rate of diffusion of $CO_2$ into the larger droplet is $2W_1$.
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 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).
