Question

The basic equation describing the diffusion of one medium through another stationary medium is (a) \(j_{A}=-C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) (b) \(j_{A}=-D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) (c) \(j_{A}=-k \frac{d\left(C_{A} / C\right)}{d x}\) (d) \(j_{A}=-k \frac{d T}{d x}\) (e) none of them

Short answer

a) \(j_{A} = -C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) b) \(j_{A} = -D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) c) \(j_{A} = -k \frac{d\left(C_{A} / C\right)}{d x}\) d) \(j_{A} = -k \frac{d T}{d x}\) e) None of them Answer: a) \(j_{A} = -C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\)

Step-by-step solution

Option (a)

\(j_{A} = -C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) In this equation, \(j_{A}\), the flux, is directly proportional to the negative gradient of the molar ratio. It takes into consideration the gradient with respect to the position '\(x\)' and a proportionality constant, \(D_{A B}\). This option seems to represent Fick's first law appropriately.

Option (b)

\(j_{A} = -D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) This expression is similar to option (a) but without the term 'C', which represents a concentration. Fick's law states that the flux depends on the total concentration, so this option is incorrect.

Option (c)

\(j_{A} = -k \frac{d\left(C_{A} / C\right)}{d x}\) This option has the same format as option (b) but uses the proportionality constant 'k' instead of the diffusion coefficient \(D_{A B}\). Since Fick's law specifically mentions the diffusion coefficient, this option is incorrect.

Option (d)

\(j_{A} = -k \frac{d T}{d x}\) In this option, the gradient is of temperature with respect to position '\(x\)', rather than concentration or molar ratio. This does not represent Fick's first law of diffusion, which is based on concentration gradients.

Option (e)

None of them Since we have already analyzed each option, and option (a) seems to represent Fick's first law correctly, there is no need to consider this option. In conclusion, the correct equation describing the diffusion of one medium through another stationary medium is given by:

Answer

Option (a) \(j_{A} = -C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\)

Key concepts

Diffusion Coefficient

The diffusion coefficient, denoted as \(D_{AB}\), plays a crucial role in Fick's First Law of diffusion. It is indicative of how easily particles, like molecules or atoms, can spread through another medium. The higher the diffusion coefficient, the faster the diffusion process occurs. This parameter is specific to the types of substances involved; for instance, gases tend to have higher diffusion coefficients compared to liquids. Understanding this helps predict how substances will behave in various environments.

Several factors influence the diffusion coefficient:

• Temperature: Typically, as temperature increases, so does the diffusion coefficient, because particles move more vigorously at higher temperatures.
• Medium Viscosity: A more viscous medium slows down the diffusion, resulting in a lower diffusion coefficient.
• Particle Size: Smaller particles tend to diffuse more quickly, influencing the diffusion coefficient.

By considering these factors, scientists and engineers can control and optimize the diffusion processes in various applications.

Concentration Gradient

The concentration gradient represents the difference in concentration of a substance between two regions. Fick's First Law of diffusion tells us that diffusion occurs down the concentration gradient—from regions of higher concentration to regions of lower concentration. This movement continues until equilibrium is reached.

Understanding the concept of a concentration gradient allows us to predict and explain how substances move in different contexts. For a solid grasp of its significance:

• Imagine a sugar cube dissolving in water. Initially, the sugar concentration near the cube is higher than the surrounding water, forming a concentration gradient.
• Diffusion will naturally occur as sugar molecules move from the area of higher concentration (around the cube) to the area of lower concentration (the rest of the water).

Concentration gradients are essential in fields ranging from biology, where they influence processes like oxygen diffusion in tissues, to chemical engineering, where they help in designing effective separation processes.

Flux

In the context of Fick's First Law, "flux" refers to the rate at which particles move through a unit area per unit of time. It is denoted by \(j_{A}\) and embodies the concept of movement through a medium due to a concentration gradient. The flux is directly proportional to the concentration gradient, highlighting that a steeper gradient results in a higher rate of diffusion.

The formula for flux, based on Fick's First Law, is given by:\[j_{A} = -C D_{AB} \frac{d\left(C_{A} / C\right)}{d x}\]Here, the negative sign indicates that the movement occurs from higher to lower concentration. Understanding flux and its dependence on factors such as the diffusion coefficient and concentration gradient helps in quantifying how rapidly substances will diffuse.

To visualize this:

• Think of heat transfer as an analogy, where heat flows from a hotter area (higher energy concentration) to a cooler area.
• The flux of particles in diffusion works similarly, moving from areas of abundance to areas of scarcity until equilibrium is reached.

In summary, flux is a critical concept in evaluating and predicting diffusion behavior.