Toluene Diffusive Flux in a Spherical Vessel
The question asks to identify the location where the diffusive flux of toluene is maximum within a partially filled spherical storage vessel under steady-state conditions.
Understanding Diffusive Flux
- Diffusive flux is the rate at which a substance moves due to concentration differences. It is driven by the concentration gradient, represented mathematically as $J \propto -\nabla C$, where $J$ is the flux and $C$ is the concentration.
- Maximum flux occurs where the concentration gradient is steepest.
Analyzing Flux Location
- The toluene evaporates from the liquid surface, creating a high concentration of toluene vapor just above the liquid.
- The vessel has a vent at the top, acting as an exit point for the toluene vapor.
- Under steady-state conditions, toluene vapor diffuses from the liquid surface towards the vent and escapes.
- The largest concentration difference driving diffusion exists between the toluene-rich vapor inside the vessel near the vent and the essentially toluene-free environment outside the vent.
- Therefore, the concentration gradient is steepest, and consequently, the diffusive flux is maximum at the opening of the vent.
Evaluating Options
- The liquid surface is the source, but the flux increases as it moves towards the escape route.
- The mid-plane is far from both the source and the primary escape route.
- The vent represents the interface with the largest concentration drop and the pathway for escape, leading to the maximum flux.
- Points far away from the vent will have significantly lower flux due to reduced concentration gradients.
Based on the principles of diffusion and mass transfer in steady state, the maximum diffusive flux of toluene occurs at the vent.
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
