Question

Fick's law of diffusion is expressed on the mass and mole basis as \(\dot{m}_{\mathrm{diff}, A}=-\rho A D_{A B}\left(d w_{A} / d x\right)\) and \(\dot{N}_{\mathrm{diff}, A}=\) \(-C A D_{A B}\left(d y_{A} / d x\right)\), respectively. Are the diffusion coefficients \(D_{A B}\) in the two relations the same or different?

Short answer

Answer: The diffusion coefficients in the mass-based and molar-based expressions of Fick's law of diffusion are the same.

Step-by-step solution

Rewrite the mass-based expression in terms of mole fraction #

To express the mass flux \(\dot{m}_{\mathrm{diff}, A}\) in terms of mole fraction, we can use the relationship between mass concentration and molar concentration: \(\rho = C*M\), where \(\rho\) is the mass concentration, \(C\) is the molar concentration, and \(M\) is the molar mass. Since \(w_A = \frac{m_A}{m_A + m_B}\), we can express the derivative of \(w_A\) as \(d w_A / d x = \frac{1}{(M_A + M_B)} * d C_A / d x\). Substituting this into the mass-based expression, we get: \(\dot{m}_{\mathrm{diff}, A}=-\rho A D_{A B} \left(\frac{1}{M_A + M_B} \frac{d C_A}{d x}\right)\). Now, let us rewrite it in terms of mole fraction: \(y_A = \frac{C_A}{C_A + C_B}\).

Rewrite the molar-based expression in terms of mass fraction #

We can express the derivative of \(y_A\) as \(dy_A /dx =\frac{1}{C_{total}} \frac{d C_A}{d x} - \frac{(C_A + C_B)}{C_{total}^2} \frac{d C_{total}}{d x}\). Substituting this into the molar-based expression, we have: \(\dot{N}_{\mathrm{diff}, A}=-C A D_{A B}\left(\frac{1}{C_{total}} \frac{d C_A}{d x} - \frac{(C_A + C_B)}{C_{total}^2} \frac{d C_{total}}{d x}\right)\). Now we have both expressions in terms of mole fractions.

Compare the diffusion coefficients #

Comparing the modified mass-based expression and the modified molar-based expression, we can see that the term involving the diffusion coefficients (\(D_{AB}\)) is the same in both expressions: \(-\rho A D_{A B} \left(\frac{1}{M_A + M_B} \frac{d C_A}{d x}\right) = - C A D_{A B}\left(\frac{1}{C_{total}} \frac{d C_A}{d x} - \frac{(C_A + C_B)}{C_{total}^2} \frac{d C_{total}}{d x}\right)\). Since we can derive one expression from the other, we can conclude that the diffusion coefficients \(D_{AB}\) are the same in both expressions.

Key concepts

Diffusion Coefficients

Diffusion coefficients are key to understanding how substances spread or move through different media. They are symbolized as \(D_{AB}\) in Fick's law of diffusion. This coefficient essentially measures how easily molecules of a species \(A\) move in a medium comprising species \(B\).

A few factors affect diffusion coefficients:

• Temperature: Higher temperatures increase diffusion rates, as molecules move more rapidly.
• Medium: The nature of the medium, whether it's air, liquid, or solid, also influences how freely molecules can move.
• Size of Molecules: Larger molecules tend to diffuse more slowly than smaller ones.

In the given exercise, we see diffusion coefficients in both mass and mole basis expressions. Importantly, these coefficients remain the same in both formulations. This is because \(D_{AB}\) remains constant when shifting between mass and mole fractions, ensuring the consistency of diffusion measurement across different contexts.

Mass Fraction

Mass fraction plays a crucial role in expressing the concentration of a particular component in a mixture. It is often denoted as \(w_A\), where \(A\) represents the species of interest. The mass fraction is the ratio of the mass of a specific component in relation to the total mass of all components in the mixture.

Here's how to think about mass fraction:

• The mass fraction is unitless, making it easy to apply across different systems and scales.
• It ranges from 0 to 1, where 0 indicates that the component is not present, and 1 means the mixture is purely that particular component.
• Since it involves mass, it links closely with real, tangible quantities in systems like industrial processes and chemical reactions.

In the context of Fick's law, mass fraction helps in representing concentrations from a mass perspective. This can be especially helpful in engineering applications where mass conservation is crucial.

Mole Fraction

The mole fraction, denoted as \(y_A\), represents another way to express the concentration of a species within a mixture. Instead of focusing on mass, the mole fraction considers the number of moles present. It provides a way to express concentration from a chemical standpoint.

Important aspects of mole fraction include:

• Similar to the mass fraction, the mole fraction is also a unitless quantity.
• It offers accurate concentration descriptions, particularly in chemical contexts, as reactions often depend on the number of participating molecules.
• Mole fractions add up to 1 across all components in a mixture, facilitating easy comparison and calculation.

In Fick's law of diffusion, using mole fractions emphasizes the role of chemical stoichiometry and reaction mole balances, offering a versatile way to engage with chemical processes.