Question

Consider steady one-dimensional mass diffusion through a wall. Mark these statements as being True or False. (a) Other things being equal, the higher the density of the wall, the higher the rate of mass transfer. (b) Other things being equal, doubling the thickness of the wall will double the rate of mass transfer. (c) Other things being equal, the higher the temperature, the higher the rate of mass transfer. (d) Other things being equal, doubling the mass fraction of the diffusing species at the high concentration side will double the rate of mass transfer.

Short answer

a) Other things being equal, the higher the density of the wall, the higher the rate of mass transfer. b) Other things being equal, doubling the thickness of the wall will double the rate of mass transfer. c) Other things being equal, the higher the temperature, the higher the rate of mass transfer. d) Other things being equal, doubling the mass fraction of the diffusing species at the high concentration side will double the rate of mass transfer. Answer: a) False, b) False, c) True, d) True

Step-by-step solution

(Statement a)

(Statement (a) states that, other things being equal, the higher the density of the wall, the higher the rate of mass transfer. This statement is False. The rate of mass transfer depends on the diffusion coefficient, which is not directly related to the density of the wall. Increasing the density will not directly lead to an increase in the rate of mass transfer.)

(Statement b)

(Statement (b) states that, other things being equal, doubling the thickness of the wall will double the rate of mass transfer. This statement is False. The rate of mass transfer through a wall is inversely proportional to its thickness, according to Fick's first law of diffusion. If the thickness of the wall is doubled, the rate of mass transfer will decrease, not increase.)

(Statement c)

(Statement (c) states that, other things being equal, the higher the temperature, the higher the rate of mass transfer. This statement is True. Temperature affects the diffusion coefficient, which is directly related to the rate of mass transfer. As the temperature increases, the diffusion coefficient usually increases, leading to a higher rate of mass transfer.)

(Statement d)

(Statement (d) states that, other things being equal, doubling the mass fraction of the diffusing species at the high concentration side will double the rate of mass transfer. This statement is True. According to Fick's first law of diffusion, the rate of mass transfer is directly proportional to the concentration difference between the two sides of the wall. If the concentration at the high-concentration side is doubled, the rate of mass transfer will also double.)

Key concepts

Fick's First Law of Diffusion

Understanding Fick's first law of diffusion is essential when studying mass transfer phenomena. This law connects the rate at which particles, such as atoms or molecules, diffuse, and the concentration gradient driving this diffusion process. It states that the rate of transfer of mass of a component across a unit area is proportional to the concentration gradient. Mathematically, the law is expressed as \begin{displaymath} J = -D \frac{dC}{dx}, \begin{displaymath} where \(J\) is the diffusion flux, which measures the amount of substance that will flow through a unit area during a unit time interval. \(D\) is the diffusion coefficient, and \(\frac{dC}{dx}\) is the concentration gradient. The negative sign indicates that diffusion occurs from regions of high concentration to regions of low concentration. This concept was reinforced in the solution to the exercise by clarifying that increasing the wall's thickness would decrease, not increase, the rate of mass transfer.

This law can be applied broadly, from predicting how quickly a sugar cube will dissolve in water to understanding how gases exchange in the lungs. In our exercise scenario, it was vital to identify that Fick's Law implies an inverse relationship between the thickness of a wall and the rate of mass transfer, leading to the conclusion that statement (b) is false.

Diffusion Coefficient

The diffusion coefficient, often symbolized by \(D\), plays a crucial role in the discussion of mass transfer. It quantifies the ease with which particles move through a medium. Factors affecting \(D\) include the nature of the diffusing substance, the medium through which diffusion occurs, and the temperature and pressure conditions. The coefficient pertains to how quickly a diffusing species such as a gas, liquid, or solute can move through a solvent or membrane.

Higher values of the diffusion coefficient mean that the substance will diffuse more rapidly. This property is intrinsically tied to temperature, which was highlighted in the exercise as statement (c) and noted true—increased temperatures result in increased diffusion coefficients and therefore elevate the rate of mass transfer. Consequently, the diffusion coefficient is not merely a fixed property but is also sensitive to environmental conditions.

Concentration Difference

The concentration difference, or gradient, is a driving force for diffusion. It is the variation in concentration of a substance across a distance. Fick's first law emphasizes that diffusion is proportional to this difference; when there's a larger difference in concentration between two areas, there's a greater flux and thus a greater rate of mass transfer. The exercise illustrated this by showing that doubling the concentration difference, according to statement (d), would double the rate of mass transfer, marking it as true.

As an example, if you have a chamber filled with a perfume on one side and pure air on the other, the perfume molecules will rapidly move towards the area with less perfume concentration, diffusing until the concentration is uniform throughout the chamber. Understanding concentration gradients is fundamental in designing processes ranging from pharmaceuticals to environmental engineering.

Temperature Effect on Mass Transfer

Temperature has a significant effect on mass transfer. This is most notably seen in its influence on the diffusion coefficient, as mentioned earlier. As temperature increases, the kinetic energy of particles also increases. This higher energy leads to more rapid movement and a greater ability to overcome resistance within the medium, which enhances the rate of diffusion. In other words, at higher temperatures, molecules are more 'agitated' and can move more freely.

This relationship is reflected in statement (c) of our exercise, which was rightly identified as true. Temperature changes can dramatically influence processes such as the mixing of substances in chemical reactions, the freshness of stored food, or cooling in electronic devices. Therefore, controlling temperature is often critical in managing mass transfer in industrial and environmental systems.