620 14 Classical Mechanics and the Old Quantum Theory14.1 Introduction
Part 1 of this textbook presents the study of the macroscopic properties of material
systems of many molecules, based on thermodynamics. This part of the textbook
presents the study of individual atoms and molecules. This study is based on quan-
tum mechanics. Classical mechanics pre-dated quantum mechanics, and this chapter
presents both classical mechanics and the so-called Old Quantum Theory, which
consists of several theories contrived to explain the failure in the late 1800s of classical
mechanics to describe or explain certain molecular phenomena.
Classical mechanics was invented by Sir Isaac Newton to describe and predict the
motions of objects such as the planets as they move about the sun. Although classical
mechanics was a great success when applied to objects much larger than atoms, it
was a complete failure when applied to atoms and molecules. It was superseded by
quantum mechanics, which has enjoyed great success in explaining and predicting
atomic and molecular properties. However, quantum mechanics was built upon classical
mechanics, and someone has said that if classical mechanics had not been discovered
prior to quantum mechanics, it would have had to be invented in order to construct
quantum mechanics.Sir Isaac Newton, 1642–1727, was a
great British mathematician and
physicist who was also one of the
inventors of calculus.
There are some important differences between classical mechanics and quantum
mechanics. Classical mechanics, like thermodynamics, is based on experimentally
grounded laws, while quantum mechanics is based on postulates, which means unproved
assumptions that can be accepted only if their consequences agree with experiments.
However, thermodynamics, classical mechanics, and quantum mechanics are all math-
ematical theories. Galileo once wrote “The book of nature is written in the language of
mathematics.” We will review some of the mathematics that we use as we encounter
it, and there are a few mathematics topics presented in the appendixes. There are also
several books that cover the application of mathematics to physical chemistry.^1
Mathematical functions play an important role in thermodynamics, classical
mechanics, and quantum mechanics. Amathematical functionis a rule that delivers
a value of adependent variablewhen the values of one or moreindependent vari-
ablesare specified. We can choose the values of the independent variables, but once
we have done that, the function delivers the value of the dependent variable. In both
thermodynamics and classical mechanics, mathematical functions are used to represent
measurable properties of a system, providing values of such properties when values
of independent variables are specified. For example, if our system is a macroscopic
sample of a gas at equilibrium, the value ofn, the amount of the gas, the value ofT,
the temperature, and the value ofV, the volume of the gas, can be used to specify
the state of the system. Once values for these variables are specified, the pressure,
P, and other macroscopic variables are dependent variables that are determined by
the state of the system. We say thatPis astate function. The situation is somewhat
similar in classical mechanics. For example, the kinetic energy or the angular momen-
tum of a system is a state function of the coordinates and momentum components
of all particles in the system. We will find in quantum mechanics that the princi-
pal use of mathematical functions is to represent quantitites that are not physically
measurable.(^1) See for example Robert G. Mortimer,Mathematics for Physical Chemistry, 3rd ed., Academic Press,
San Diego, CA, U.S.A., 2005; James R. Barrante,Applied Mathematics for Physical Chemistry, 3rd ed.,
Pearson Prentice-Hall, Upper Saddle River, NJ, 2004; Donald A. McQuarrie,Mathematical Methods for
Scientists and Engineers, University Science Books, Sausalito, CA, 2003.