Mass Diffusivity: Equation And Applications

What Is Mass Diffusivity

Mass diffusivity, abbreviated as DD, is the rate or measure at which the particles or molecules of a substance spread out in another substance, typically in a fluid system. It is a physical parameter that determines how easily a substance spreads out from a concentrated region to a diluted region. Diffusion is caused by molecular random motion and concentration gradients. Mass diffusivity is particularly relevant in various industries and scientific areas, including chemical engineering, biology, and environmental science.

Diffusivity Equation (Fick's Law)

The most widely used model that accounts for mass diffusion is Fick's Law of Diffusion. Fick's Law connects the diffusion flux (the amount of substance diffusing through a unit area within one unit of time) with the concentration gradient.

Fick's first law of diffusion equation is:

J=−D⋅(dC/dx)

Where:

•J is the diffusion flux (mol/m²·s), or the rate of diffusion.

•D is the mass diffusivity (m²/s), a measure of how readily a substance diffuses.

•dC/dx is the concentration gradient (mol/m³·m), or how the concentration of the diffusing substance varies over distance.

The minus sign shows that the flux is from high to low concentration, in accordance with the natural drift of diffusion to reduce concentration gradients. Fick's Law assumes a steady-state diffusion process, where the concentration gradient is not altering.

For non-steady-state diffusion (where the concentration alters with time), Fick's second law is used:

∂C*∂t=D*(∂²*C/∂x²)

This equation represents time-dependent concentration change and is common in applications such as diffusion in living organisms or in transient heat or mass transfer in engineering.

Factors Affecting Mass Diffusivity

Mass diffusivity (D) characterises the rate at which a substance diffuses through a medium and is dependent on several key factors:

1. Temperature

Diffusivity is greater at higher temperatures due to increased molecular motion. The diffusion coefficient of oxygen in water, for instance, increases from 2.0 × 10⁻⁹ m²/s at 25°C to 3.0 × 10⁻⁹ m²/s at 50°C, demonstrating an approximate 50% increase in the rate of transport of molecules.

2. Viscosity of the Medium

Increased viscosity slows down diffusion. For example, glucose diffuses in water at a rate of 6.7 × 10⁻¹⁰ m²/s, while in glycerol, a more viscous fluid, the diffusivity is 2.2 × 10⁻¹¹ m²/s, indicating how medium resistance impedes molecular flow.

3. Molecular Size and Mass

Large molecules take more time to diffuse. Sodium ions (Na⁺, with a diameter of 0.102 nm) diffuse in water at 1.33 × 10⁻⁹ m²/s, but a protein like bovine serum albumin (~66 kDa) diffuses at only 6 × 10⁻¹¹ m²/s, illustrating how weight and size directly impact mobility.

4. Concentration Gradient

Diffusion follows Fick's first law: more extensive concentration differences result in faster diffusion. In an example application, for oxygen diffusion within a microfluidic channel, the flux can increase from 10⁻⁷ mol/m²·s with a 0.1 mol/m³ gradient to 10⁻⁶ mol/m²·s with a 1 mol/m³ gradient, demonstrating very linear scaling with the gradient.

5. Nature of the Diffusing Substance

Chemical properties such as polarity and solubility affect diffusion. For example, hydrophobic molecules such as benzene diffuse in water at 1.2 × 10⁻⁹ m²/s, while polar molecules such as ethanol diffuse at 1.24 × 10⁻⁹ m²/s depending on the interaction of the molecule with the solvent.

6. Medium Properties

The nature, porosity, density, and phase of the medium determine diffusivity. Gas-phase diffusivity is typically orders of magnitude larger than in liquids; for example, CO₂ diffuses in air at 1.6 × 10⁻⁵ m²/s but in water at only 1.9 × 10⁻⁹ m²/s. Effective diffusivity in porous media is reduced by tortuosity, relevant to applications such as gas separation in membranes.

Applications of Mass Diffusivity

Mass diffusivity is a critical parameter in many scientific and industrial applications:

1.      Chemical Engineering: Diffusion is the driving force for many operations such as mixing, separation, and reaction kinetics. The diffusion rate impacts the efficiency of chemical reactions, especially catalytic reactions, in reactors.

2.      Pharmaceutical Industry: Mass diffusivity is crucial in designing drug delivery systems. Controlled release formulations are based on understanding how drugs diffuse across membranes or other barriers within the body.

3.      Biological Systems: In biology, mass diffusivity has a critical role in explaining processes such as oxygen and nutrient transport within cells and tissues and the diffusion of signalling molecules in organisms.

4.      Environmental Science: Diffusion is of vital importance for pollutant spreading in the air and water. Simulation of how substances diffuse in natural systems enables prediction of environmental impacts and remediation design.

5.      Material Science: Diffusivity plays an important role in processes such as sintering, coating, and material production where materials are diffused into substances for altering their properties.

Values of Diffusion Coefficients

Diffusion coefficients vary significantly depending on the substance as well as the medium. For example:

•Water: Typical substances' diffusivity in water ranges from 10⁻⁹ to 10⁻⁶ m²/s.

•Air: Diffusivity of gases like oxygen or carbon dioxide in air tends to be higher, ranging from 10⁻⁵ to 10⁻⁴ m²/s.

•Solids: Diffusivity in solids is typically much lower, ranging from 10⁻¹⁵ to 10⁻¹⁰ m²/s.

Table 1: Diffusion Coefficients in Water

Table 2: Diffusion Coefficients in Air (at 25°C)

Table 3: Diffusion Coefficients in Solids (at 1000°C)

Table 4: Diffusion Coefficients in Polymers

Table 5: Diffusion Coefficients in Gases (at 1 atm and 25°C)

For more information, please check Stanford Advanced Materials (SAM).

Frequently Asked Questions

1. In what way does mass diffusivity differ from thermal diffusivity?

Mass diffusivity is the diffusion of particles through a medium, whereas thermal diffusivity is the diffusion of heat through a substance. Both are transport phenomena, but one involves mass transfer and the other involves heat transfer.

2. How does molecular weight affect the diffusivity of a substance?

Generally, heavier molecules diffuse at a slower rate than lighter molecules, as their larger size and mass restrict their mobility in a medium.

3. Is mass diffusivity ever constant in a system?

Mass diffusivity is usually considered constant in most cases, especially in steady-state situations. However, in cases involving non-homogeneous systems or temperature gradients in the system, diffusivity may vary.