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

The diffusion coefficients in both the mass basis and mole basis relations of Fick's law of diffusion are the same.

Step-by-step solution

Analyze mass basis equation

The mass basis equation of Fick's law of diffusion is given as: \(\dot{m}_{\mathrm{diff}, A}=-\rho A D_{A B}\left(d w_{A} / d x\right)\) Here, \(\dot{m}_{\mathrm{diff}, A}\) represents the rate of mass transfer of component A by diffusion (kg/s), \(\rho\) is the density of the mixture (kg/m³), \(A\) is the cross-sectional area (m²), \(D_{AB}\) is the diffusion coefficient on a mass basis (m²/s), \(w_A\) is the mass fraction of component A, and \(x\) denotes the spatial direction (m).

Analyze mole basis equation

The mole basis equation of Fick's law of diffusion is given as: \(\dot{N}_{\mathrm{diff}, A}=\) \(-C A D_{A B}\left(d y_{A} / d x\right)\) Here, \(\dot{N}_{\mathrm{diff}, A}\) represents the rate of molar transfer of component A by diffusion (mol/s), \(C\) is the molar concentration of the mixture (mol/m³), \(A\) is the cross-sectional area (m²), \(D_{AB}\) is the diffusion coefficient on a mole basis (m²/s), \(y_A\) is the mole fraction of component A, and \(x\) denotes the spatial direction (m).

Compare the diffusion coefficients

Let's compare the two diffusion coefficients by analyzing their units: - Diffusion coefficient on a mass basis (\(D_{AB}\) in the mass basis equation): m²/s - Diffusion coefficient on a mole basis (\(D_{AB}\) in the mole basis equation): m²/s Both diffusion coefficients have the same units, which indicates that they represent the same quantity.

Conclusion

The diffusion coefficients \(D_{AB}\) in both the mass basis and mole basis relations of Fick's law of diffusion are the same.