1.1 Introduction 5
need to be able to apply mathematical methods. There are several books that cover the
application of mathematics to problems in physical chemistry.^1
Arithmeticis the principal branch ofnumerical mathematics. It involves carrying
out operations such as addition, subtraction, multiplication, and division on actual
numbers. Geometry, algebra, and calculus are parts ofsymbolic mathematics, in which
symbols that represent numerical quantities and operations are manipulated without
doing the numerical operations. Both kinds of mathematics are applied in physical
chemistry.Mathematical Functions
A mathematical function involves two kinds of variables: Anindependent variableis
one to which we can assign a value. Amathematical functionis a rule that delivers the
value of adependent variablewhen values are assigned to the independent variable or
variables. A function can be represented by a formula, a graph, a table, a mathematical
series, and so on. Consider theideal gas law:PVnRT (1.1-1)In this equationPrepresents the pressure of the gas,Vrepresents its volume,nrep-
resents the amount of substance in moles,Trepresents the absolute temperature, and
Rstands for theideal gas constant. The ideal gas law does a good but not perfect
job of representing the equilibrium behavior of real gases under ordinary conditions.
It is more nearly obeyed if the pressure of the gas is made smaller. A gas that is at a
sufficiently low pressure that it obeys the ideal gas law to an adequate approximation
is called adilute gas.Anideal gasis defined to obey this equation for all pressures
and temperatures. An ideal gas does not exist in the real world, and we call it amodel
system. A model system is an imaginary system designed to resemble some real system.
A model system is useful only if its behavior mimics that of a real system to a useful
degree and if it can be more easily analyzed than the real system.
We can solve the ideal gas law forVby symbolically dividing byP:V
nRT
P(1.1-2)
The right-hand side of Eq. (1.1-2) is a formula that represents a mathematical function.
The variablesT,P, andnare independent variables, andVis the dependent variable.
If you have the numerical values ofT,P, andn, you can now carry out the indicated
arithmetic operations to find the value ofV. We can also solve Eq. (1.1-1) forPby
symbolically dividing byV:P
nRT
V(1.1-3)
We have now reassignedVto be one of the independent variables andPto be the
dependent variable. This illustrates a general fact:If you have an equation containing(^1) Robert G. Mortimer,Mathematics for Physical Chemistry, 3rd ed., Academic Press, San Diego, CA,
U.S.A., 2005; James R. Barrante,AppliedMathematicsforPhysicalChemistry, 3rd ed., Pearson Prentice Hall,
Upper Saddle River, NJ, 2004; Donald A. McQuarrie,Mathematical Methods for Scientists and Engineers,
University Science Books, 2003.