Modern Thermodynamics

- Chapter 1

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This is a wonderful result because it relates temperature to molecular motion, in agreement with Robert

Boyle's intuition. It shows us that the average kinetic energy of a molecule equals 3kBT/2. It is an

important step in our understand the meaning of temperature at the molecular level.

From (1.6.8) we see that the total kinetic energy of one mole of a gas equals 3RT/2. Thus, for

monatomic gases whose atoms could be thought of as point particles that have neither internal structure

nor potential energy associated with intermolecular forces (He and Ar are examples), the total molar

energy of the gas is entirely kinetic; this implies Um = 3RT/2. The molar energy of a gas of polyatomic

molecules is larger. A polyatomic molecule has additional energy in its rotational and vibrational motion.

In the nineteenth century, as kinetic theory progressed, it was realized that random molecular collisions

result in equal distribution of energy among each of the independent modes of motion. According to this

equipartition theorem, the energy associated with each independent mode of motion equals (kBT/2). For

a point particle, for example, there are three independent modes of motion, corresponding to motion along

each of the three independent spatial direction, x, y and z. According to the equipartition theorem, the

average kinetic energy for motion along the x direction, mvxavg2/2 = kBT/2; similarly for the y and z

directions, making the total kinetic energy 3(kBT/2) in agreement with (1.6.8). For a diatomic molecules,

which we may picture as two spheres connected by a rigid rod, there are two independent modes of

rotational motion in addition to the three modes of kinetic energy of the entire molecule. Hence, for a

diatomic gas the molar energy Um = 5RT/2, as we noted in the context of eqn. (1.4.8). The independent

modes of motion are often called degrees of freedom.

Maxwell-Boltzmann Velocity Distribution

A century after Bernoulli's Hydrodynamica was published, kinetic theory of gases began to make

great inroads into the nature of the randomness of molecular motion. Surely molecules in gas move with

different velocities. According to (1.6.8), the measurement of pressure only tells us the average of the

square of the velocities. It does not tell us what fraction of molecules have velocities with a particular

magnitude and direction. In the later half of the nineteenth century, James Clerk Maxwell (1831-1879)

directed his investigations to the probability distribution of molecular velocity that specifies such

details. We shall denote the probability distribution of the molecular velocity v, by P(v). The meaning of

P(v) is as follows:

P(v)dvx dvy dvz = the fraction of the total number of molecules whose velocity

vectors have their components in the range (vx, vx +d vx ),