So far, we have been interested in macroscopic objects. We are now going to examine microscopic objects, but taken in huge numbers so that they can be described by macroscopic quantities. To understand their collective behavior, we need to treat them statistically, since there are far too many to treat individually. This means that we will be concerned primarily with average behavior. Luckily, when the number of objects is of the order of Avogadro's Number (NA = 6.022 * 1023, or 1 mol), the deviations from average behavior tend to cancel each other out, making the average a very meaningful quantity.
These average quantities we will be concerned with are called state variables. As their name implies, their values depend only on the current state of the system, and not on the path taken to that state: they have no memory of their past values. Three of the most important state variables are temperature, pressure and volume. Temperature and pressure are intrinsic state variables, since their value does not depend on the size of the system. Volume is an extrinsic state variable, since its value does depend on the size of the system. We will measure all temperatures in Kelvin (K = Celsius + 273.15). Zero K is called absolute zero, since it is the lowest possible temperature. A temperature difference of 1K is equal to a temperature difference of 1C. We will not be concerned with pressure all that much, since most physiological functions assume a constant pressure equivalent to that of the atmosphere. We will typically measure volume in liters (1 l = 1000 cm3 = 10-3 m3).
In order to become used to thinking statistically, let us find the volume of one mol of air using its density and gram molecular weight (the mass of one mol of a substance in grams is numerically equal to its molecular weight). If air is 78.08% N2 (with molecular weight 28), 20.95% O2 (32) and .93% Ar (40), the average molecular weight of air is 28.94 g / mol. Since the density of air at standard temperature and pressure is 1.293 kg / m3, one mol of air then occupies 22.4 liters. This implies that on average, each molecule in the air has a volume in which it can move before hitting another molecule of about 3.72 * 10-26 m3. By taking the cube root, we obtain an order of magnitude estimate of its mean free path l of 3.34 * 10-9 m: the average distance it travels before hitting another molecule.
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
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?
Due to the number of particles and the frequency of collision, we describe the path of any given particle as a random walk:
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
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.
If two quantities are proportional, we say that one scales with the other. If y is proportional to x2, we say that y scales
as the square of x: if x increases by a factor of 2, y must increase by a factor of 4. In an arbitrary equation involving multiple variables,
we can relate any two of them, holding all of the others constant.
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).
In the next section, we find how your body loses excess heat energy.
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The Equipartition Theorem
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.
Random Walks
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.