# Diffusion

### The Equipartition Theorem

A basic axiom for us will be the Equipartition Theorem. It states that the thermal energy of a system of particles is evenly divided over all of its degrees of freedom: there are k T / 2 (where k is Boltzmann's constant, equal to 1.381 * 10-23 J / K) Joules per degree of freedom. This means that temperature is a measure of thermal energy, and that thermal energy is essentially kinetic energy, since every molecule has

• one degree of freedom per translational direction: x, y and z;
• one degree of freedom per axis of rotation: zero for atoms, which we take to be point particles and which therefore have zero moment of inertia; two for diatomic molecules, which we model as two point particles joined by a rod, and three for all molecules with more than two atoms;
• one vibrational degree of freedom for every bond:

• and one angular degree of freedom for every pair of bonds:

Thus an atom has 3 degrees of freedom, a diatomic molecule has 6 degrees of freedom and in general, the number of degrees of freedom grows much more quickly than the number of atoms in the molecule.

We can use the Equipartition Theorem to find the average velocity of, for example, an oxygen molecule. Since it is diatomic, it has not only three translational degrees of freedom, but two rotational and one vibrational as well. Therefore its total kinetic energy is 6 k T / 2. To find its velocity, however, we only consider the translational degrees of freedom. By equating 3 k T / 2 to m v2 / 2, we find that at room temperature (293 K) its average velocity is 478 m / s! Of course, it does not travel for long before colliding with another molecule and changing direction: only about 7 * 10-12 s! What does this imply about the nature of its travels?

### Random Walks

Due to the number of particles and the frequency of collision, we describe the path of any given particle as a random walk:

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As you can see, this motion does not result in efficient transport; instead, it leads us to the concept of diffusion. Diffusion is the gradual movement of molecules along a concentration gradient, from regions of higher concentration into regions of lower concentration. Suppose that we uncover a cup of coffee in a closed room with no air circulation. The aroma of the coffee, carried in part by molecules of 2 - Furylmethanethiol (C5 H6 O S, molecular weight 114), will begin to diffuse throughout the room, albeit slowly. If we imagine a thin spherical shell of area A and thickness Δx surrounding the cup, the number of molecules of 2 - Furylmethanethiol passing through the shell per unit time will be

ΔN / Δt = D A ΔC / Δx,
where D is the diffusion constant and ΔC is the change in concentration across the shell (C is measured in molecules per unit volume). The diffusion constant has units of m2 / s, increases with temperature and decreases with the size of the molecule, its shape factor (which measures asphericity, or "oblongness") and the viscosity of the medium. Proteins typically have a diffusion constant on the order of 10-10 m2 / s at room temperature in water. The left hand side of this equation is called the diffusion current.

By means of unit analysis, we can see that the average distance traveled by an arbitrary molecule in time t is proportional to (D t)1/2 (the proportionality constant being 61/2). Since the diffusion constant of 2 - Furylmethanethiol is around 2.25 * 10-7 m2 / s, we see that in one minute the aroma should diffuse only a few millimeters! Obviously, diffusion is not the reason smells waft across a room; air currents do the job much more quickly.

When trying to deduce a scaling relationship, it is necessary to have a single equation which contains exactly two interrelated variables. In the event that three variables are interrelated through two or more equations, one variable must be eliminated. Any single equation which contains three interrelated variables is insufficient to deduce the relation; in that case a different equation is necessary.

In the following applet, you will examine the scaling behavior of the various aspects of diffusion. Answers must be exact; this means that answers less than one must be entered as fractions (ie., 1/2).

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In the next section, we find how your body loses excess heat energy.