.Transport Properties
The phrase "transport properties" implies that something is moving or being transported. We will mention three, but discuss only one transport property in detail.
Mass transport is called "diffusion,"
energy transport is called "thermal conductivity,"
momentum transport is called "viscosity."
We will discuss these three properties in one dimension. That is, we will consider that whatever is being transported is being transported in the z-direction of ordinary x, y, z space and not worry about what is happening in the other two directions. Three-dimensional versions of our equations are available, but the fundamental principles are the same as in one dimension and the math is easier so we will stay in one dimension. All of these transport properties are described by the same form of equation. In all cases there is the flux of some quantity, which we will call, Jz, indicating that the flux is in the positive z direction,
(1) .
So the flux is the amount of mass or energy or momentum passing through a unit area (square meter) per second.
Then there is a gradient of some other quantity, which drives the flux. For diffusion there is a gradient in the density (which we write as "number density," that is, the number of molecules per unit volume). If the density varies from place to place (along the z axis) material will diffuse from high density regions to low density regions trying to make the density the same everywhere. We will indicate the gradient by d(quantity)/dz. The bulk transport equation, then, relates the flux to the gradient. It says that the flux is proportional to the gradient. That is, for diffusion,
(2) 
,
where a gradient in the number density drives a mass flux and D is called the diffusion coefficient. This is called "Fick’s law of diffusion" (sometimes Fick’s first law of diffusion).
For heat conductivity a temperature gradient drives a heat (or energy) flux,
(3) 
,
and
k is called the thermal conductivity (or the coefficient of thermal conduction).Viscosity is a little harder to visualize, but let’s try anyway. We have a large (
¥ ) surface perpendicular to the z axis (call it the xy plane) and there is a gas flowing along the surface in the x direction. The molecules right at the surface of the plane are not flowing at all because they are in contact with the stationary surface. But the bulk gas is flowing so as we move away from the surface in the positive z-direction the velocity of the molecules increases because they are being dragged along by the flowing gas. This increase in velocity as we move away from the plane means that the gas is transporting momentum in the z-direction. At any given plane (at some value of z) the plane above it is passing down momentum to try to speed the plane up and the plane below it is passing up momentum in the opposite direction trying to slow it down. The net result is,(4) 
,
and
h is called the coefficient of viscosity.Our task is to see if we can find out how these various coefficients depend on the parameters of our model of the gas, things like m, T, p, and etc.
Let’s try diffusion first:
Consider a plane at z = 0. There are molecules crossing this plane from the
- z side of the plane (moving in the positive z direction) and from the + z side moving in the negative z direction. The net flux of molecules is the flux from the - x side minus the flux from the + z side. Let’s assume that the molecules coming from the - z side originate (on the average) at a plane located at z = - l , and the molecules coming from the + z side originate from a plane located at z = + l . (Recall that l is the mean free path. That is, l is the average distance a molecule moves without making a collision with another molecule.) Then, assuming that l is small,(5a,b,c,d) 
Comparing Equation 5d with the bulk equation, Equation 2 we see that the diffusion coefficient is,
.
The assumption that the molecules crossing the plane at z = 0 originate from planes at z =
± l is a fairly crude approximation. In reality we should integrate the flux from planes ranging from z = 0 to z = ± ¥ . A careful treatment of the hard-sphere gas (which our model is) would give a corrected result of 
,
which is not too far off from our approximate result since,
.
Kinetic molecular theory expressions for the other two transport properties can be derived in a manner similar to the one used here. One considers the net flux of either energy (as heat) or momentum across the plane, z = 0, where the fluxes originate from planes at z = ± l .
Copyright 2000, W. R. Salzman