Figure 2.9

peratures.

Energy distribution of electrons in a metal at three different tem-

2.20 Boltzmann's Law

A very useful result from statistical mechanics is Boltzmann's law, which describes the concentration of particles in a gas as a function of their potential energy and their temperature. This law is used in a number of chapters in this book.

Consider a force, F (derived from a potential), acting on each atom and aligned along the x-direction. nF is, of course, the total force acting on a cubic meter of gas, and, if we restrict ourselves to a rectangular prism of base area, A, and height, dx, the force is nFdxA, and the pressure on the base is nFdx. In equilibrium, this pressure must balance the gas pressure, kTdn.

from which, we obtain Boltzmann's equation n « exp [—W°) ■ (2.102)

Example

Atmospheric pressure

Each molecule in air has a potential energy, mgAh, relative to a plane Ah meters closer to the ground.

According to Boltzmann's law, the concentration of molecules must vary as n = no exp —Tf-j. (2.103)

Note that kT/mg has the dimensions of length. It is called the scale height, H:

and, if T is independent of h,

(Continued)

In an isothermal atmosphere, the air pressure falls exponentially with height.

Taking the mean mass of the molecules in air as 29 daltons, and knowing that to convert daltons to kg it suffices to divide by Avogadro's number, we find that the mass of a representative air molecule is about 48 x 10-27 kg. Consequently, the scale height of Earth's atmosphere (isothermal at 300 K) is

Appendix: Symbology

We will try to adopt the following convention for representing thermodynamic quantities such as

G, free energy,

S, entropy, and U, internal energy.

1. Capital letters indicate the quantity associated with an arbitrary amount of matter or energy.

2. Lowercase letters indicate the quantity per unit. A subscript may be used to indicate the species being considered. For example, the free energy per kilomole of H2 will be represented by Tj^.

g = free energy per kilogram. g = free energy per kilomole.

g = free energy per kilogram, at 1 atmosphere pressure. g* = free energy per kilomole, at 1 atmosphere pressure. gf = free energy of formation per kilomole.

~gj = free energy of formation per kilomole, at 298 K, 1 atmosphere, that is, at RTP (Standard Free Energy of Formation).

For more information on some topics in this chapter, read: Reference

Cengel, Y. A., and M. A. Boles, Thermodynamics, An Engineering Approach, McGraw-Hill, 1994.

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