Question

Consider one-dimensional mass transfer in a moving medium that consists of species $A$ and $B$ with $\rho ={\rho }_A+{\rho }_B=$ constant. Mark these statements as being True or False. (a) The rates of mass diffusion of species $A$ and $B$ are equal in magnitude and opposite in direction. (b) $D_{AB}=D_{BA}$. (c) During equimolar counterdiffusion through a tube, equal numbers of moles of $A$ and $B$ move in opposite directions, and thus a velocity measurement device placed in the tube will read zero. (d) The lid of a tank containing propane gas (which is heavier than air) is left open. If the surrounding air and the propane in the tank are at the same temperature and pressure, no propane will escape the tank and no air will enter.

Short answer

Identify whether each statement is True or False and provide an explanation: (a) The rates of mass diffusion of species A and B are equal in magnitude and opposite in direction. Answer: True Explanation: Since the system has a constant density, any mass that diffuses from species A would mean a corresponding mass loss in species B, and vice versa. Hence, their rates of mass diffusion have the same magnitude but opposite directions. (b) $D_{AB}=D_{BA}$. Answer: True Explanation: The diffusivity coefficients ($D_{AB}$ and $D_{BA}$) are a measure of how fast species A diffuses into species B and vice versa. As per Fick's law of diffusion, these coefficients are equal under identical conditions. (c) During equimolar counterdiffusion through a tube, equal numbers of moles of A and B move in opposite directions, and thus a velocity measurement device placed in the tube will read zero. Answer: True Explanation: In the case of equimolar counterdiffusion, the number of moles moving in each direction is equal, leading to zero net mass velocity. Therefore, any velocity measurement device placed in the tube would read zero. (d) The lid of a tank containing propane gas (which is heavier than air) is left open. If the surrounding air and the propane in the tank are at the same temperature and pressure, no propane will escape the tank and no air will enter. Answer: False Explanation: Propane, being heavier than air, will tend to diffuse downwards due to gravity. At the same time, air will be able to enter the tank from the open lid. Thus, there will be a gas exchange between propane and air, regardless of the same temperature and pressure conditions.

Step-by-step solution

Statement (a)

The rates of mass diffusion of species A and B are equal in magnitude and opposite in direction. This statement is True. Since the system has a constant density, any mass that diffuses from species A would mean a corresponding mass loss in species B, and vice versa. Hence, their rates of mass diffusion have the same magnitude but opposite directions.

Statement (b)

The statement $D_{AB}=D_{BA}$ is True. The diffusivity coefficients ($D_{AB}$ and $D_{BA}$) are a measure of how fast species A diffuses into species B and vice versa. As per Fick's law of diffusion, these coefficients are equal under identical conditions.

Statement (c)

During equimolar counterdiffusion through a tube, equal numbers of moles of A and B move in opposite directions, and thus a velocity measurement device placed in the tube will read zero. This statement is True. In the case of equimolar counterdiffusion, the number of moles moving in each direction is equal, leading to zero net mass velocity. Therefore, any velocity measurement device placed in the tube would read zero.

Statement (d)

The lid of a tank containing propane gas (which is heavier than air) is left open. If the surrounding air and the propane in the tank are at the same temperature and pressure, no propane will escape the tank and no air will enter. This statement is False. Propane, being heavier than air, will tend to diffuse downwards due to gravity. At the same time, air will be able to enter the tank from the open lid. Thus, there will be a gas exchange between propane and air, regardless of the same temperature and pressure conditions.

Key concepts

Mass Diffusion

Mass diffusion is a fundamental phenomenon where particles, such as atoms, ions, or molecules, spread from regions of higher concentration to regions of lower concentration. This natural process is driven by the concentration gradient, propelling species towards a uniform distribution. Imagine pouring a drop of food coloring into a glass of water. Initially, the color is most intense where the drop enters the water, but over time, the color spreads throughout the glass until it's even, which is an everyday example of mass diffusion.

Understanding mass diffusion is crucial in various disciplines, including environmental sciences, engineering, and material science, as it helps us predict how pollutants spread in the environment, how different substances interact in industrial processes, and how to design better batteries and fuel cells, where diffusion plays a key role in the effectiveness of these devices.

Fick's Law of Diffusion

where $J$ is the diffusion flux (amount of substance per unit area per unit time), $D$ is the diffusivity or diffusion coefficient, and $\frac{\text{d}\rho }{\text{d}x}$ is the concentration gradient. The direction of mass transfer is from higher to lower concentration, which is why there is a negative sign. This law is crucial for engineers and scientists to calculate the rate of diffusion under various conditions and to design systems where controlled diffusion is necessary, such as in drug delivery systems.

An understanding of Fick's law also implies that factors such as the medium's temperature and porosity can significantly affect diffusion rates. For instance, increasing the temperature typically increases the rate of diffusion, as particles move more quickly and spread out faster.

Equimolar Counterdiffusion

Equimolar counterdiffusion occurs when two species diffuse through a stationary medium at equal rates but in opposite directions, without any overall change in molar concentration. This is a special case of diffusion that often occurs in gas mixtures when no bulk flow is present, for example, in a closed container where two different gases are allowed to mix.

A classic example to visualize equimolar counterdiffusion is a tube with hydrogen on one end and nitrogen on the other. If allowed to diffuse, the lighter hydrogen will move faster; however, due to the stoichiometry of the counterdiffusion, an equal number of nitrogen atoms will diffuse towards the hydrogen. There's no net movement of mass or change in pressure because the molar flow of each gas is balanced. Devices measuring flow or velocity in the tube during this process would indicate no movement, corroborating the 'zero net mass velocity' characteristic of equimolar counterdiffusion.

Diffusivity Coefficients

Diffusivity coefficients, often represented with the symbol $D$, are values that quantify the ease with which different species can diffuse through a particular medium. These coefficients are determined experimentally or estimated using predictive models and are fundamental parameters in Fick's law. Factors such as temperature, pressure, and the nature of the medium affect these coefficients.

For instance, in a binary system consisting of species $A$ and $B$, the diffusivity $D_{AB}$ measures how quickly $A$ diffuses into $B$, and vice versa for $D_{BA}$. Under the same conditions, $D_{AB}$ and $D_{BA}$ are equal due to the reciprocal nature of diffusion forces and collision frequencies. Understanding and calculating the correct diffusivity coefficients is crucial for engineers and scientists when predicting how a substance will behave under specific conditions, such as in air purification processes, industrial mixing, or environmental modeling of contaminant spread.