Statistical Systems

Thermodynamics is the dynamics of heat. It brings together concepts from nearly every subject we have studied so far. In addition, it has the benefits of being fascinating and not a little fun, because it allows us to understand much that we take for granted in our everyday lives.

Statistical systems are systems with large numbers of particles (atoms and / or molecules). By large, we mean on the order of 6.022 x 10 23 ("Avogadro's Number", designated NA ; one mol). Clearly such a system is far too complex to treat the individual motions of every particle in it. Instead, we must treat it "statistically", by considering "mean" (average) values of measurable quantities, and the deviations we might expect from them. Fortunately, when the number of particles is that large, means are very descriptive of the system as a whole.

The measurable quantities 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. In thermodynamics, all temperatures are measured 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.

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% N 2 (with molecular weight 28), 20.95% O 2 (32) and .93% Ar (40), the average molecular weight of air is 28.94 g / mol. Since the density of air is 1.293 kg / m 3, one mol of air then occupies 22.4 liters (.0224 m 3). 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 x 10 - 26 m 3. By taking the cube root, we find its "mean free path" l is 3.34 x 10 - 9 m: the average distance it travels before hitting another molecule.

A basic axiom for us will be the "Equipartition Theorem". It relates the energy of the particles in the system to the macroscopic temperature by stating that the average energy is k T / 2 (where k is Boltzmann's constant) per degree of freedom (ie., translational, rotational, vibrational). We can equate thermal to kinetic energy 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 one rotational and one vibrational as well (see Chapter 6). Therefore its total kinetic energy is 5 k T / 2. To find its velocity, however, we only consider the translational degrees of freedom. By equating 3 k T / 2 to m v 2 / 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 x 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" (try the Java simulation). The percentage of particles which have moved a distance x away from their starting point is proportional to exp(-x 2 / 4Dt), where D is the diffusion constant (with dimensions of area / time; D depends on the particle and the medium through which it diffuses; see Chapter 2). This is a "Gaussian" (or random) distribution: the so-called "bell curve". We define the diffusion "flux" J as the number of particles passing through a unit area per unit time:

J = D DC / Dx,

where C is the concentration (with dimensions of number / volume). Recalling that the diffusion constant is proportional to the inverse of the drag coefficient, and considering the flux as a kind of velocity, the concentration gradient then plays the part of a force. This implies that the concentration is analogous to energy, as we will see below. We can also define the diffusion "current"

DN / Dt = J A,

where A is the area and N is the number of particles passing through A per unit time.

The average distance travelled by any molecule in time t is Sqrt (6 D t), with D = v l / 3. Consider the diffusion of 2 - Furylmethanethiol (C 5 H 6 OS, molecular weight 114), which is one of the ingredients in the aroma of coffee. Using the mean free path in air and the equipartition theorem to find v, we see that in one minute the molecule can only be expected to travel about 9 mm! Conversely, one might expect it to travel 3 m across a room in about two months! Obviously, diffusion is not the reason smells waft across a room (convection is; see Section C below).

Problem

The next section is about chemical potential energy.



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1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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