CHEM 331 Physical Chemistry Revision 1.1 An Overview of Mathematics for Thermodynamics It is assumed you come to this Physical Chemistry I course with a solid background in Calculus. However, that background most likely has as its emphasis definitions and proofs rather than applications. Here we need to use calculus as a tool. In other words, thermodynamics, the main focus of this course, is an application with which we are chiefly concerned. The mathematics takes second place. You need to be reasonably comfortable with the mathematics itself so that you can focus on this application, that which in this course takes first place in our attention. Below you will find a very brief summary of some of the important results of calculus needed to complete this course. Most of this summary should be in the form of a review. If you find some of it unclear, you should certainly review the associated material from your coursework in calculus. Because, in the end, you want to keep in mind, it is Chemistry we are studying here. Functions Functions of a Single Variable A function f is a rule that assigns to each number in its domain D a unique number in its Range. This gives us an ordered pair of numbers (x,y), where x is an independent variable for which there corresponds a unique value of the dependent variable y. We will denote this as: y = f(x) For example, a function might be: f(x) = x 2 where x is any real number In this case, the domain of f is (-, ) and its range is [0, ). Functions of Several Variables In a similar manner, we can define a function of several variables. If a number is uniquely determined by the set of numbers x 1, x 2,, x n, we say that is a function of x 1, x 2,, x n. For example: P = where R is a constant, is a function of the independent variables n, T and V. We could write this as P(n, T, V).
In this course, we will write generic functions of one variable as f(x) and those of several variables as f(x,y,z). Derivatives We will often be interested in how some physical property of a system changes when some other property changes. Drawing on the above example, how does the pressure P of an Ideal Gas change when the temperature T changes? Mathematically we determine this by differentiating the function P(n, T, V) with respect to T. The derivative of a function f(x) is defined as: This definition yields the following results: 1. If f(x) = x n, then 2. If f(x) = e x, then 3. If f(x) = ln x, then 4. If f(x) = sin x, then 5. If f(x) = cos x, then The following three rules concerning differentiation are worth recalling: 1. Product Rule If the functions f(x) and g(x) are both differentiable at x, then the derivative of
their product f(x)g(x) is: 2. Quotient Rule Similarly, the derivative of f(x)/g(x) is: 3. Chain Rule If y = f(z) and z = g(x), then For example, suppose y = z 3 and z = x 2, then As another example, suppose y = (x 3 + 2x + 1) 2, then In this last example, y = z 2 and z = x 3 + 2x + 1. For each of the following functions, determine df/dx. 1. (3x -1) (2x + 5) 2. x e -x 3. 4. 5.
Inverses Suppose f is a differentiable function that has an inverse f -1. Then f -1 is itself differentiable. Using our standard notation: y = f(x) the inverse can be written as: x = f -1 (y) The derivatives of f and f -1 can then be written as: We can write the function f as: y = f(f -1 (y)) and Differentiation of both sides of this identity with respect to y yields: 1 = f ' (f -1 (y)) (f -1 )'(y) = f '(x) (f -1 )'(y) = This allows us to write the derivative of the inverse of f as: 6. For the function y = determine explicitly and and confirm the validity of the above formula for the derivative of inverses. Higher Derivatives If the function f is differentiable, we can define its second derivative as: Higher derivatives are also possible.
Maxima and Minima Suppose a function f has local maxima or local minima, as pictured below. Calculus: One and Several Variables, Part 1; 3 rd ed. S.L. Salas and Einar Hille These critical points are such that f ' (x) is zero when x = critical point. Formally: If f ' (a) = 0 and f '' (a) > 0, then f has a local minimum at a. If f ' (a) = 0 and f '' (a) < 0, then f has a local maximum at a. If f ' (a) = 0 and f '' (a) = 0, then it is possible f has an inflection point at a. Partial Derviatives Suppose we have a function of two variables; z = f(x, y). Then z can change if either x or y changes and the function may have two first partial derivatives. These derivatives are defined as: = The mechanics of carrying out a partial derivative are the same as those employed when evaluating an ordinary derivative; the variable that is not changing is treated as though it is constat. For example, if z = xy - y, then
= x - 1 For each of the following functions, determine both first partial derivatives. 7. z = x 2 y -xy 2 + y 3 8. z = x e -y 9. z = ln(1 + xy + x 2 ) Higher Derviatives If the function f is differentiable in both x and y, then higher partial derivatives may exist. Second partial derivatives of the following form can be determined: If z and its derivatives are continuous, then For each of the following functions, determine the second mixed partial. 10. z = x 2 y -xy 2 + y 3 11. z = x e -y
12. z = ln(1 + xy + x 2 ) Total Differentials An explicit function of several variables, say z = f(x,y), has a total differential of the form: The Chain Rule Consider a function of several variables z = f(x, y) where x and y themselves depend parametrically on another variable u. Then, 13. For the function z = x 2 + y 2 where x = cos u and y = sin u, determine Exact Differentials Suppose we have a differential of the form: du = M(x,y) dx + N(x,y) dy If we have a function f(x, y) such that df(x, y) = du, then In this case the differential du is said to be Exact. If the differential is exact, then integration of the differential will be independent of the Path taken in going from (x 1, y 1 ) to (x 2, y 2 ) and we will find: In this case the function u depends only on the state of the system and we refer to u as a State Function.
If the differential u is not exact (note the change of notation), then In this case the integration of u depends upon the path y = y(x) taken from point (x 1, y 1 ) to (x 2, y 2 ). The integral is then referred to as a Line or Path Integral: u Since the integral is a path integral, it does not make sense to write the integral as u because u depends on more than just its initial and final states. As an example, we will examine the differential: u = xy dx + xy dy In this case, M(x,y) = xy and N(x,y) = xy. Testing for exactness, we have: and Therefore, u is not exact. Now consider integrating u from (0,0) to (1,1) along two different paths. Path 1 involves going from (0,0) to (0,1) holding x constant at 0. Then going from (0,1) to (1,1) holding y constant at 1. Path 2 will move along the path y = x. For Path 1, we have: u = = = (0) + ½ (1) = (0) + ½ (1 2 + 0 2 ) = ½ Now, for Path 2, we have: u = = = = = 2/3
Note that the path integral along Path 1 is distinctly different than it is along Path 2. Determine if each of the following differentials is Exact or Inexact. 14. du = (x + y) dx + (x + y 2 ) dy 15. du = (2x e y + e x ) dx + (x 2 + 1) e y dy 16. du = (x 2 + x - y) dx + x dy Manipulating Partial Derivatives In many cases we will be tasked with manipulating a partial derivative into another form. In doing so, the following relationships will be useful. or Taylor Series Finally, consider the Taylor's Series expansion of a function f about a point x = a. f(x) = Very often, this Series converges to f(x), especially if x is reasonably close to a. As an example, let's examine the Taylor's Series expansion of f(x) = e x about the point x = 0. f(x) = e x f(0) = 1 f '(x) = e x f '(0) = 1
f ''(x) = e x f ''(0) = 1 etc. So, e x = 1 + x + + + The first four terms in the Series are plotted below, along with the function e x. You should appreciate that including more terms in the Series provides for a better estimate of the function in the vicinity of x = 0. Calculus: One and Several Variables, Part 1; 3 rd ed. S.L. Salas and Einar Hille Expand each of the following functions to the 3 rd term in the Taylor's Series about x = 0. 17. f(x) = sin x 16. f(x) = Additionally, you should plot, using an appropriate graphing software package, the following: f(0) f(0) + f'(0)x f(0) + f'(0)x + ½ f''(0)x 2
about x = 0. References Bromberg, J. Philip (1984) "Physical Chemistry," 2 nd Ed. Allyn and Bacon, Inc., Boston. Lang, Serge (1979) "Calculus of Several Variables," 2 nd Ed. Addison-Wesley Publishing Company, Reading, Massachusetts. Salas, S.L. and Hille, Einar (1978) "Calculus: One and Several Variables, Part 1," 3 rd Ed. John Wiley & Sons, New York.