Modern Thermodynamics
- Chapter 1
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 ),
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