CHM 3410 - Physical Chemistry I
Chapter 1 - Supplementary Material
1. Even and odd functions
A function is an even function if f(-x) = f(x) for all x, and an odd function if f(-x) = - f(x) for all x. Examples of even functions include x2, cos(x), and exp(-ax2). Examples of odd functions include x and sin(x). Some functions are neither even nor odd, as, for example, x2 + x and exp(x).
A useful property for integrals of even and odd functions is the following:
ò-aa f(x) dx = 2 ò0a f(x) dx if f(x) is an even function (1.1)
ò-aa f(x) dx = 0 if f(x) is an odd function (1.2)
2. Partial derivatives and partial derivative relationships (see also pp. 42-43, 91-93 of Atkins)
For f(x), a function of one variable, the rate of change of the function can be written as
df = (df/dx) dx (2.1)
where (df/dx) is the derivative of the function, found using standard techniques from calculus.
By analogy with the above the rate of change of a function of two variables, f(x,y), can be written as
df = (¶f/¶x)y dx + (¶f/¶y)x dy (2.2)
where (¶f/¶x)y and (¶f/¶y)x are the partial derivatives of the function f with respect to x and with respect to y. Note that we use ¶f/¶x to indicate a partial derivative, and list the independent variables being held constant outside of the parentheses enclosing the partial derivative. Equation 2.2 can be generalized in a straightforward manner to functions of more than two variables.
The partial derivative of a function is found by treating all variables except the one for which the partial derivative is being taken as if they are constants. Thus, finding the partial derivative of a function of several variables is no more difficult than finding the normal derivative of a function of one variable.
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EXAMPLE: If f(x,y) = 4x3 + 2xy2 + 7y + 1, then: (¶f/¶x)y = 12x2 + 2y2 (¶f/¶y)x = 4xy + 7
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The following four relationships (which can be derived) are often useful in manipulating partial derivatives (note that x, y, z, and w are variables)
#1 (¶y/¶x)z = 1/(¶x/¶y)z (2.3)
#2 (¶x/¶y)z (¶y/¶z)x (¶z/¶x)y = -1 (2.4)
#3 (¶y/¶x)z = - (¶z/¶x)y/(¶z/¶y)x (2.5)
#4 (¶z/¶x)w = (¶z/¶x)y + (¶z/¶y)x (¶y/¶x)w (2.6)
The reason we are discussing partial derivatives is that thermodynamic functions are generally functions of several variables. A complete discussion of thermodynamics requires a working knowledge of the basics of partial derivatives.
3. Taylor series expansion of a function
Subject to the usual mathematical restrictions, a function f(x) can be expressed by expansion about a point a. The technical term for this process is a Taylor series expansion. The expansion is given by the following relationship
f(x) = f(a) + (x – a) (df/dx)a + (x – a)2 (d2f/dx2)a + (x – a)3 (d3f/dx3)a + … (3.1)
2! 3!
= ån=0¥ (x – a)n (dnf/dxn)a (3.2)
n!
A few useful Taylor series expansions are given below
ex = 1 + x + (x2/2!) + (x3/3!) + ... ; all values of x (3.3)
ln(1 + x) = x - (x2/2) + (x3/3) - ... ; -1 < x < 1 (3.4)
sin(x) = x - (x3/3!) + (x5/5!) - ... ; all values of x (3.5)
cos(x) = 1 - (x2/2!) + (x4/4!) - ... ; all values of x (3.6)
1/(1 + x) = 1 - x + x2 - x3 + ... ; -1 < x < 1 (3.7)