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
Arithmetic is the principal branch of numerical mathematics. It involves carrying
out operations such as addition, subtraction, multiplication, and division on actual
numbers. Geometry, algebra, and calculus are parts of symbolic 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: An independent variable is
one to which we can assign a value. A mathematical function is a rule that delivers the
value of a dependent variable when 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 the ideal gas law:
PV nRT
(1.1-1)
In this equation P represents the pressure of the gas, V represents its volume, n represents the amount of substance in moles, T represents the absolute temperature, and
R stands for the ideal 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 a dilute gas. An ideal gas is defined to obey this equation for all pressures
and temperatures. An ideal gas does not exist in the real world, and we call it a model
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 for V by symbolically dividing by P:
V
nRT
P
(1.1-2)
The right-hand side of Eq. (1.1-2) is a formula that represents a mathematical function.
The variables T , P, and n are independent variables, and V is the dependent variable.
If you have the numerical values of T , P, and n, you can now carry out the indicated
arithmetic operations to find the value of V . We can also solve Eq. (1.1-1) for P by
symbolically dividing by V :
P
nRT
V
(1.1-3)
We have now reassigned V to be one of the independent variables and P to 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, 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, 2003.
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
Arithmetic is the principal branch of numerical mathematics. It involves carrying
out operations such as addition, subtraction, multiplication, and division on actual
numbers. Geometry, algebra, and calculus are parts of symbolic 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: An independent variable is
one to which we can assign a value. A mathematical function is a rule that delivers the
value of a dependent variable when 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 the ideal gas law:
PV nRT
(1.1-1)
In this equation P represents the pressure of the gas, V represents its volume, n represents the amount of substance in moles, T represents the absolute temperature, and
R stands for the ideal 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 a dilute gas. An ideal gas is defined to obey this equation for all pressures
and temperatures. An ideal gas does not exist in the real world, and we call it a model
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 for V by symbolically dividing by P:
V
nRT
P
(1.1-2)
The right-hand side of Eq. (1.1-2) is a formula that represents a mathematical function.
The variables T , P, and n are independent variables, and V is the dependent variable.
If you have the numerical values of T , P, and n, you can now carry out the indicated
arithmetic operations to find the value of V . We can also solve Eq. (1.1-1) for P by
symbolically dividing by V :
P
nRT
V
(1.1-3)
We have now reassigned V to be one of the independent variables and P to 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, 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, 2003.