13.2 Transport properties: diffusion, viscosity, and thermal conductivity

Diffusion, Viscosity, and Thermal Conductivity

Defining and Differentiating Transport Properties

Transport properties describe how particles, momentum, and energy move through a gas. Kinetic theory connects these macroscopic flows to the microscopic behavior of molecules, giving us quantitative tools to predict how gases behave under non-equilibrium conditions.

There are three core transport properties to know:

• Diffusion is the net movement of particles from regions of high concentration to regions of low concentration, driven by a concentration gradient. It transports mass.
• Viscosity measures a fluid's resistance to flow or deformation, caused by internal friction between adjacent fluid layers moving at different velocities. It transports momentum.
• Thermal conductivity quantifies the rate at which heat flows through a material in response to a temperature gradient. It transports energy.

The unifying idea: each property involves molecules carrying something (mass, momentum, or kinetic energy) from one region to another through random molecular motion and collisions.

Applications and Examples of Transport Properties

• Diffusion plays a role in many natural and industrial processes:
  • Oxygen diffusion across alveolar membranes in the lungs drives gas exchange during respiration
  • Nutrient and waste transport in living cells depends on diffusion over short distances
  • Many chemical reactions are diffusion-limited, meaning reactants must diffuse together before they can react
• Viscosity matters throughout fluid dynamics and engineering:
  • Motor oil viscosity determines how effectively it lubricates engine components at different temperatures
  • Blood viscosity affects flow through vessels and can change in conditions like sickle cell anemia, where deformed red blood cells increase resistance
• Thermal conductivity governs heat transfer and insulation design:
  • High-conductivity metals (copper at ~400 W/m·K, aluminum at ~235 W/m·K) are used in heat sinks and exchangers
  • Low-conductivity materials (fiberglass, foam) minimize heat loss in buildings and appliances

Fick's Laws for Diffusion

Fick's First Law and the Diffusion Coefficient

Fick's first law relates the particle flux to the concentration gradient. It says that particles flow from high to low concentration at a rate proportional to how steep the gradient is:

$J = -D \nabla c$

• $J$ is the diffusive flux (particles per unit area per unit time)
• $D$ is the diffusion coefficient
• $\nabla c$ is the concentration gradient
• The negative sign ensures the flux points in the direction of decreasing concentration

The diffusion coefficient $D$ tells you how fast particles spread through a medium. It depends on several factors:

• Temperature: Higher temperatures increase molecular speeds, so $D$ generally increases with temperature.
• Particle size: Larger particles diffuse more slowly, giving a smaller $D$.
• Medium: Diffusion through a dense medium is slower than through a dilute gas.

Fick's Second Law and Diffusion Scenarios

Fick's second law describes how the concentration profile evolves over time. It's derived by combining Fick's first law with the continuity equation:

$\frac{\partial c}{\partial t} = D \nabla^2 c$

This partial differential equation says the local rate of concentration change equals $D$ times the curvature of the concentration profile. Where the concentration profile curves most sharply, the concentration changes fastest.

Two important regimes:

• Steady-state diffusion ($\partial c / \partial t = 0$): The concentration profile doesn't change with time. The flux is constant throughout the system. A classic example is diffusion through a membrane with fixed concentrations on each side.
• Non-steady-state diffusion: The concentration profile evolves over time toward equilibrium. Think of a drop of dye spreading in water or a pollutant dispersing in the atmosphere.

Solving diffusion problems requires specifying:

1. Boundary conditions that fix the concentration or flux at the edges of the system
2. Initial conditions that describe the concentration distribution at $t = 0$

These conditions, combined with Fick's second law, determine the full time-dependent concentration profile.

Factors Influencing Viscosity and Thermal Conductivity

Temperature and Molecular Mass Effects

Temperature has a strong effect on gas transport properties, and the direction of the effect is opposite to what you might expect from liquids. In gases, both viscosity and thermal conductivity increase with temperature. Why? Higher temperature means faster molecules, more frequent collisions, and more efficient transfer of momentum and energy between adjacent layers.

Molecular mass also matters. Lighter molecules (like helium) move faster at a given temperature and collide more frequently, so they transfer momentum and energy more efficiently. Heavier molecules (like xenon) move more slowly, resulting in lower viscosity and thermal conductivity.

Pressure, Intermolecular Forces, and Molecular Structure

Pressure has surprisingly little effect on viscosity and thermal conductivity in the dilute gas regime. This is a key prediction of kinetic theory: increasing pressure raises the number density (more molecules to carry momentum) but decreases the mean free path (each molecule carries momentum a shorter distance). These two effects nearly cancel. At very high pressures, though, intermolecular forces become significant and this cancellation breaks down.

Intermolecular forces enhance transport properties. Attractive interactions (dipole-dipole forces, hydrogen bonding) make collisions more effective at transferring momentum and energy. Ammonia, with its strong hydrogen bonding, has higher viscosity and thermal conductivity than nitrogen at the same conditions.

Molecular shape and size play a role too:

• More spherical molecules (like methane) tend to have lower viscosity than elongated molecules (like ethane), because elongated shapes create more drag between layers
• Larger molecules have bigger collision cross-sections, which affects both the mean free path and the efficiency of momentum transfer

Transport Coefficients from Kinetic Theory

Calculating Viscosity and Thermal Conductivity

Kinetic theory connects transport coefficients to three microscopic quantities: gas density, average molecular speed, and mean free path. The key results are:

Viscosity:

$\eta = \frac{1}{3} \rho \langle v \rangle \lambda$

Thermal conductivity:

$\kappa = \frac{1}{3} \rho \langle v \rangle \lambda c_v$

where:

• $\rho$ is the gas density
• $\langle v \rangle$ is the mean molecular speed, given by $\langle v \rangle = \sqrt{\frac{8k_BT}{\pi m}}$
• $\lambda$ is the mean free path
• $c_v$ is the specific heat capacity at constant volume (per unit mass), which depends on the degrees of freedom of the molecule

Mean Free Path and Limitations of Kinetic Theory

The mean free path $\lambda$ is the average distance a molecule travels between collisions. It's calculated as:

$\lambda = \frac{1}{\sqrt{2} \pi d^2 n}$

• $d$ is the effective molecular diameter (the collision cross-section parameter)
• $n$ is the number density, related to pressure and temperature by the ideal gas law: $n = P / k_B T$

At standard conditions, typical mean free paths for common gases are on the order of tens to hundreds of nanometers.

Limitations to keep in mind:

• These kinetic theory formulas assume an ideal gas with no intermolecular forces and treat molecules as hard spheres. This works well for dilute gases at low to moderate pressures.
• For real gases, especially at high pressures or low temperatures where intermolecular forces matter, more sophisticated approaches like Chapman-Enskog theory are needed. Chapman-Enskog theory systematically accounts for the details of intermolecular potentials and yields transport coefficients that agree much better with experiment.
• The simple kinetic theory results are still valuable because they correctly capture the qualitative trends (temperature dependence, molecular mass dependence, pressure independence) and provide order-of-magnitude estimates.