Modern Engineering Thermodynamics

728 CHAPTER 18: Introduction to Statistical Thermodynamics
HOW DID IT ALL BEGIN?
The development of statistical thermodynamics began in the late 19th century, shortly after William Thomson (1824–1907)
and Rudolf Clausius (1822–1888) unified classical thermodynamics in the 1860s. Starting from basic mechanics principles,
James Clerk Maxwell (1831–1879) developed a simple molecular interpretation of ideal gas behavior called the kinetic theory
of gases, which led many physicists to conclude that all thermodynamic phenomena could be fully explained from mechanics
principles. However, the mechanical approach was never able to predict the classical thermodynamic laws of the conservation
of energy and positive entropy production; consequently, thermodynamics has held its own as an independent science.
In the 1870s, Ludwig Boltzmann (1844–1906) made great progress in the understanding of entropy, when he postulated
that a mathematical relationship existed between entropy and mathematical probability by arguing that equilibrium states
are not simply inevitable but merely highly probable states of molecular order.
Between 1900 and 1930. quantum mechanics blossomed under Max Planck (1858–1947), Albert Einstein (1879–1955),
Peter Debye (1884–1966), Niels Bohr (1885–1962), Enrico Fermi (1901–1954), Erwin Schrodinger (1887–1961), and
many others. It was only natural that their results be extended into the thermodynamic area whenever possible; thus
evolved the new area of “quantum statistical thermodynamics,” which is still an important research area.
equations of state are very useful when dealing with a substance for which empirically derived thermodynamic
tables and charts do not yet exist but the basic molecular structure of the substance is known.
In this chapter, we survey the main engineering results of statistical thermodynamics by treating its two main
components, kinetic theory and quantum statistical thermodynamics, as separate topics. The goal is the development
of thermodynamic property relationships and equations of state of engineering value.
18.2 WHY USE A STATISTICAL APPROACH?
To begin with, we should explain why we resort to a statistical approach rather than simply a molecular
approach. Suppose we have a large number of particles N in a box. To find out what happens inside the box
without using a statistical analysis, we would have to follow the motion of each of the N individual particles.
The motion of each particle must satisfy Newton’s second law, and because of collisions and long-range forces
between particles, each particle could conceivably influence the motion of every other particle in the box. Let F ij
be the force exerted on particle i by particle j. Then, the sum of all the forces on particle i duetoalltheother
j particles must equal the mass m i of particle i times its acceleration a i ,or
N−1
∑F ij = m i a i = m i
j=1
dV i
dt
d
= m 2 x i
i
dt 2
(18.1)
where the terms V i and x i are the time-dependent velocity and position vectors of particle i. Since each of the
N particles must obey Eq. (18.1), and particle-particle interactions couple all N Eqs. (18.1) together, they must
all be solved simultaneously. Also, since Eq. (18.1) is a vector equation, there are really 3N scalar second-order
coupled differential equations to be solved.
For a typical gas at standard temperature and pressure, N ≈ 10 20 molecules/cm 3 . Therefore, if we were to try to
follow the molecules of the gas contained in one cubic centimeter at STP using the methods of classical
mechanics, we would need to solve about 3 × 10 20 scalar second-order coupled differential equations, each containing
10 20 terms. This is impossible today, even with the fastest digital computers. Hence, we must abandon
the approach of applying the equations of classical mechanics to each particle in the system. Instead of formulatingatheorybasedonknowingtheexactpositionof
each particle in time and space, we develop a theory
based on knowing only the average behavior of the particle.
18.3 KINETIC THEORY OF GASES
The elements of kinetic theory were developed by Maxwell, Boltzmann, Clausius, and others between 1860 and
1880. Though kinetic theory results are currently available for solids, liquids, and gases, we are concerned only
with the behavior of gases. The following eight assumptions underlie the kinetic theory of gases:
1. The gas is composed of N identical molecules moving in random directions.
2. There is always a large number of molecules (N ≫ 1) in the system.