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title: Fick's Law
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# Fick's Law
Unit 5, Lecture 43

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## Fick's Law of Diffusion

- Course: Heat and Mass Transfer (DI05019071)
- Unit 5: Mass Transfer
- Lecture 43: Fick's Law

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Welcome to Lecture 43. Today we will discuss Fick's Law, which is the foundational governing equation for molecular diffusion, much like Fourier's Law is for heat conduction.
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## Lecture Agenda

- 1. Introduction to Fick's First Law
- 2. The Mathematical Equation
- 3. Diffusion Coefficient (Diffusivity)
- 4. Characteristics of Diffusivity in different phases
- 5. Fick's Law: Mass vs. Molar Basis
- 6. General Mass Diffusion Equation

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We will formally state Fick's First Law, examine the diffusion coefficient, look at how the law can be written in mass or molar terms, and then outline the general diffusion equation.
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## Introduction to Fick's First Law

- Proposed by Adolf Fick in 1855.
- It states that the mass diffusion flux of a constituent in a mixture is proportional to the concentration gradient.
- Diffusion always occurs in the direction of decreasing concentration.
- It perfectly mirrors Fourier's Law of heat conduction.

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Adolf Fick recognized that diffusion behavior was identical to heat conduction. Therefore, he formulated his law based entirely on the mathematical framework established by Joseph Fourier for heat transfer.
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## Fick's First Law (Mass Basis)

- The equation for mass diffusion flux (j_A) in the x-direction:
- j_A = - D_AB * (d(ρ_A) / dx)
- Where:
- - j_A = Mass diffusion flux of species A (kg/m^2·s)
- - D_AB = Diffusion coefficient (or mass diffusivity) of species A in species B (m^2/s)
- - d(ρ_A) / dx = Concentration gradient of species A (kg/m^4)

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Notice the negative sign. It indicates that mass diffuses in the direction of decreasing concentration, from high to low. The coefficient D_AB dictates how easily A diffuses into B.
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## Diffusion Coefficient (Diffusivity)

- Denoted as D_AB (diffusion of A into B).
- Unit: m^2/s (Same unit as thermal diffusivity, α, and kinematic viscosity, ν).
- It is a property of the specific binary mixture (A and B), the temperature, and the pressure.
- A higher D_AB means mass diffuses more rapidly.

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The diffusion coefficient is analogous to thermal conductivity in heat transfer. Interestingly, its units are meter squared per second, exactly the same as thermal diffusivity and kinematic viscosity.
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## Characteristics of Diffusivity

- **In Gases**: Diffusion is rapid. D_AB is relatively high (approx. 10^-5 m^2/s). D_AB increases with temperature and decreases with pressure.
- **In Liquids**: Diffusion is much slower due to close molecular packing (approx. 10^-9 m^2/s).
- **In Solids**: Diffusion is extremely slow (approx. 10^-14 m^2/s). Depends heavily on the solid structure (crystalline, amorphous).

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Molecules in gases are far apart and move quickly, leading to high diffusion rates. In liquids, they are tightly packed, reducing diffusion by orders of magnitude. In solids, it's a very slow creep.
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## Fick's Law (Molar Basis)

- Chemical engineers often prefer the molar basis.
- Molar diffusion flux (J_A):
- J_A = - D_AB * (dC_A / dx)
- Where:
- - J_A = Molar diffusion flux (kmol/m^2·s)
- - C_A = Molar concentration (kmol/m^3)
- - dC_A / dx = Molar concentration gradient (kmol/m^4)

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The structure of the equation is exactly the same, but we swap mass flux for molar flux, and mass concentration for molar concentration. The diffusion coefficient D_AB remains exactly the same.
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## Fick's Law using Mass/Mole Fractions

- If total mixture density (ρ) is constant:
- j_A = - ρ * D_AB * (dm_A / dx)
- If total molar concentration (C) is constant (typical for ideal gas mixtures at constant T and P):
- J_A = - C * D_AB * (dx_A / dx)
- This form is highly useful in derivation and problem solving.

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Often, the total density or total molar concentration of the mixture is roughly constant. In these cases, we can express Fick's Law in terms of mass fraction or mole fraction gradients.
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## Total Absolute Flux Equation

- Recall from last lecture: Total Absolute Flux = Diffusion Flux + Bulk Flux
- n_A = j_A + ρ_A * v
- Substituting Fick's Law for j_A:
- n_A = - D_AB * (dρ_A / dx) + ρ_A * v
- Similarly for Molar Flux:
- N_A = - D_AB * (dC_A / dx) + x_A * (N_A + N_B)

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This is a critical concept. Fick's law only gives us the diffusion flux. If the bulk mixture is also moving, we must add the convective transport term to find the total rate of mass transfer.
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## General Mass Diffusion Equation

- Analogous to the general heat conduction equation (Heat Equation).
- Derived from species conservation over a control volume.
- Governing Equation (1D, stationary medium, no bulk flow):
- ∂C_A / ∂t = D_AB * (∂^2 C_A / ∂x^2) + R_A
- (Where R_A is the rate of generation of species A due to chemical reactions).

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By applying the conservation of mass to a control volume and substituting Fick's law, we arrive at the general mass diffusion equation. It mirrors the heat conduction equation, where R_A acts like internal heat generation.
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## Summary

- Fick's First Law relates diffusion flux linearly to the concentration gradient.
- The negative sign indicates mass moves from high to low concentration.
- Diffusivity (D_AB) depends on the mixture, temperature, and pressure, and is highest in gases.
- The total absolute mass flux includes both Fickian diffusion and bulk fluid motion.

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To summarize, Fick's law is our primary tool for quantifying molecular diffusion. We now have the mathematical machinery to solve actual mass transfer problems.
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## Next Lecture Preview

- Topic: Steady state diffusion through a plain membrane
- Key questions to ponder:
- - How do we apply Fick's law to a solid membrane?
- - What happens to the concentration profile across a membrane in steady state?
- - How can we use the concept of 'mass transfer resistance' to solve problems quickly?

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Next time, we will apply Fick's Law to a real-world scenario: steady-state diffusion through a stationary solid membrane, and we'll introduce the concept of mass transfer resistance.
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