Mathematical Skills for General Chemistry

Transcription

1 Mathematical Skills for General Chemistry General Chemistry Staff Revision: August 2003

2 'As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. ' Albert Einstein Do not worry about your difficulties in mathematics. I can assure you mine are still greater. ' Albert Einstein '' All is number Pythagoras (attrib.) Mathematical skills are an essential part of the physical sciences and achieving mathematical fluency is vitally important. Here, we present a practical guide to the basic mathematical skills that will allow you to understand General Chemistry without becoming impeded by the mathematics that is required. It is not a rigorous guide; the Mathematics Department will be able to give you far more detail than what we present here. We hope that, by reading and understanding what is presented here and working the problems that are included, you will master the mathematics and will be able to appreciate the chemistry. We begin by reviewing some of the basic operations and concepts with which we should all be familiar and that will be essential for success in general chemistry. Numbers There are several groups or sets of numbers that are important in the physical sciences. Some of these sets of numbers are intuitively obvious and we use them in everyday life; we use the set of positive whole numbers or positive integers to describe the numbers of people attending a football game or the number of mushrooms that we buy at the grocery store. Other sets of numbers include prime, rational, irrational, imaginary and transcendental numbers. All of types of numbers are used in the sciences at some point to describe and quantitate our model of the physical world. We will meet some of these in General Chemistry; if you progress further into the physical sciences, you will eventually become acquainted with all of them. 1. Whole numbers These are the numbers with which we count and are also known as integers, and a number that is in the series 1,2,3,4,5,6... etc. Zero is sometimes included. Other variants

3 include the negative integers, 1, 2, 3, 4, 5, They are sometimes termed the counting numbers or the natural numbers. In this sense, integers are whole numbers that can be used to identify individual or groups of individual objects. 2. Fractions and ratios Fractions are numbers written as a b. For some types of number, only a fractional representation is possible one half, one quarter and so on are only expressible as fractions. We may express them in several different fractional forms. For instance, 1 2 can also be written as or 4 8 ; the representation 1 2 is the lowest form of the fraction. 3. Rational numbers These are numbers that can be written as a fraction composed of two integers. So a number p is real when p= q r so long as q and r are integers. All integers are therefore rational as the can be written as a fraction with r=1 ; in a fraction, the q term is called the numerator and the r term is called the denominator. 4. Irrational numbers This is the first set of numbers that are more difficult to understand. Irrational numbers cannot be expressed in the form of a fraction composed of two integers. Proofs of the irrational nature of certain numbers need not concern us at the moment but they can be found in most texts on pure mathematics or the theory of numbers. Examples of irrational numbers include 2, 3 and so on. 5. Transcendental numbers This set of numbers are even more abstract. Briefly, rational and irrational numbers can always act as roots of an algebraic equation. Transcendental numbers are defined as those which do not have these properties. Examples of transcendental numbers are,e. For more on this topic, see below in the section on the geometrical relationships of numbers and graphical applications. We can think of a number as one of the basic mathematical 'building blocks'from which we can construct more complicated mathematical and scientific structures, as long as we understand the rules by

4 which these structures are assembled from the more basic units. Our first task is to write down some of the basic operations that we can perform with numbers, which constitute arithmetic. The basic operations or manipulations of numbers in arithmetic are addition multiplication subtraction division Addition is simply the grouping together of two sets of numbers to form a second, larger set. Subtraction is the opposite of addition it is the removal of a set of numbers from the larger set to give a smaller set of numbers. We can show addition and subtraction in the form of a diagram: = = 3 Illustration 1 Addition and subtraction Multiplication is the successive addition of the same set of numbers; 3 4 can be thought of as four added to itself three times or equally, three added to itself four times. Whichever way we attempt this, we will achieve the same result 12. Division is partition of a set of numbers into two or more equal and smaller sets of numbers. If we divide 12 by 4, we find that we will have four equal sets that contain 3 members. We then say that 12 divided by four is three. These four actions that we can perform on numbers constitute the basic operations of arithmetic.

5 Algebra One of the problems with working only with numbers is that numbers are not general. To make a cake, it is simple to follow the recipe; what happens if one wishes to make a cake of any given size? Here lies the rub. To do this, or any other trivial operation, we would need to write down all the possible quantities for all possible sizes of cake, which would be a very tedious and lengthy business, given that there are few prescriptions on the maximum size of a cake. You are probably thinking that this is ridiculous if I want to bake a cake twice as big, I just double the quantities of the ingredients. That is obviously true but in making that assumption, you have implicitly resorted to algebra. Algebra is the method we employ to talk about the general relationships between general numbers. It is here that we encounter the first special type of number the prime numbers. We define a prime number as one which is an integer and is the product of only itself and 1. This means that no other whole numbers, except 1 and the prime, when multiplied together will give the prime number. We say that a prime has no factors other than itself and 1. Returning to the number 12, we notice that there are several ways of generating the number 12 viz and so on. We see that 1, 12, 2, 6, 3 and 4 are all factors of 12. Indeed, we can extend the list by noticing that 2 and 2 are factors of 4 and so the simplest factors of 12 are 1, 2, 3 and 12. The conclusion of all of this is that 12 is not prime. A number such as 13 is only divisible by 1 and by itself and we say that 13 is prime. Examples of primes are 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and so on. Incidentally, the number of primes and the rule for the distribution of primes in amongst the real numbers is one of the great mysteries of mathematics. When we write algebraic relationships, we are writing down the relationship between one number and another number or numbers. We write letters to represent these numbers and we term these letters the variables. We also assume that the variables are prime, so that any relationship that we write is completely general. Why should this be so? Why are primes so important for algebra? Remember that a prime has no factors other than 1 and itself. We cannot write a prime as a quotient a fraction or as a product of two

6 other numbers. If we could arbitrarily state that any general variable could be expressed in this manner, we could decide to divide say by 3 in a calculation simply to make it simpler, which would mean that we are arbitrarily changing the relationships between the variables. We assume that all variables are therefore prime if the relationship that we write is true for a prime, then it is also true for the other types of real or integer numbers that we are going to encounter. Exercise 1: (a) Write the following numbers in terms of their smallest factors: (i) 4 (ii) 8 (iii) 2 (iv) 10 (v) 15 (vi) 35 (vii) 72 (viii) 100 (ix) 1000 (x) 54 (b) By writing down the factors for the following numbers, determine which of these numbers are prime (i) 3 (ii) 17 (iii) 23 (iv) 88 (v) 55 (vi) 56 (vii) 57 (c) Numbers can be defined as odd or even. Define in words an odd number and an even number. Now mathematically define odd and even in this context, using an equation. (d) Is the statement All primes can be odd or even true or false? Explain your reasoning, referring to your definition of odd and even. (e) What is the smallest common factor, greater than one, between the following pairs of numbers? In cases that you think posses no common factors, show that this is true by factorizing both numbers completely. (i) 3 and 12 (ii) 27 and 81 (iii) 23 and 8 (iv) 14 and 58 (v) 6 and 88 (vi) 55 and 121 (f) Is 1 a prime number? Justify your answer. We use a specific type of notation for algebra, so that we can express clearly the relationships between variables in a universal way. Addition and subtraction take the usual symbols: and. There are several ways of writing multiplication and division and these are shown below. The result of addition is called the sum, the result of multiplication is termed the product and the result of division is the quotient. For two variables a and b we can write the following arithmetical relationships: Addition or summation: a b Multiplication: ab, a b or a b

7 Division: a b From our definition of multiplication, the product, a/b or a b Subtraction: a b ab has two factors, a and b. As a and b are numbers, then the product is also a number; we can call that number x. We can now write the relationship between a, b and x as an algebraic, mathematical statement: x=ab The symbol = is the equals or equality sign. It is very important as it shows that the quantity on the left hand side (LHS) and the right hand side (RHS) are the same. Let us explore that a little more closely and think about what that means. This relationship which we have written between a, b and x as means that x is ab and because we say this, we equate or say that these two mathematical objects are equal. The relationship that we have written is an equation, meaning a statement that shows the equality between two mathematical objects. The equality is very important for this reason. It is also important logically as if what we have written is true, then whatever operation we perform on one side of the equality sign, we must perform on the other side. If we do not do this, then our original statement is not true. write Let us think about the relationship that we have written for a, b and x in light of this idea. We can and therefore, by dividing both sides by a or x=ab x a =ab a x a =a a b Now we ask, what is the result of dividing a number by itself? Clearly this must be 1. So, and so the final result is that x a =a a b=1 1 b=1 b=b x a =b We can perform a similar operation for b. We have rewritten out original equation, x=ab, in order to find an expression for b in terms of x and a. The essential truth of our original equation is preserved, no matter how we write it, as long as we treat both sides of the equation equally. We call the

8 manipulation of equations in this manner rearrangement and we have rearranged terms of x and a. x=ab to find b in Exercise 2 (a) Write the following fractions in their lowest terms: (i) 4 8 (ii) 4 6 (iii) 54 9 (iv) (v) 7 7 (b) Rearrange the following equations to make the given variable the subject of the equation: (i) y=4 x for x (ii) st=4 x y for t (iii) c=2 r for r (iv) E=8 ħ c t for c (v) d=4 fch mv for v More Algebraical Rules The rules of arithmetic are intuitively obvious though the formal, written form may not be as obvious. The first rule is that of addition. It is clearly obvious that the order of addition is unimportant and so, for two numbers m and n. m n=n m Similarly, the order of multiplication is unimportant and so mn=nm These rules are true for numbers, but are not necessarily true for other more complicated mathematical objects. Examples that you may meet in the future, though not in General Chemistry, are matrices and operators. A common technique to write algebraic relationships in simple and compact form is to use parentheses '(...)'or brackets '[...]'to clarify the mathematical meaning. We can then state two rules that we must follow with relation to brackets or parentheses. For a bracket such as a b c = a b c unimportant. which is simply common sense. The order of addition using bbrackets or parentheses is

9 Another rule is important when we use brackets or parentheses. We write a m n =am an Similarly, a b m n =am an bm bn We introduce some notation at this point which makes writing complex algebraic expressions a great deal easier. When we multiply two numbers, a and b, together we write it as ab. What happens when a and b equal? The answer is aa. This is clumsy, especially when we come to write a multiplied by itself say 15 times. In order to simplify this state of affairs, we use indices or exponents to indicate how many times we multiply the number by itself. So we write aa as a 2, the index or exponent 2 showing that a is multiplied by itself once. For the situation alluded to above, where a is multiplied by itself fourteen times, we would write this as a 15. Another way of talking about the same issue is to say that a number is raised to the power n; in the last example we would say that a is raised to the power 15. Some powers have special names the power 2 is the square of the number, the power 3 is the cube of the number. Example 1: Expand a a b as fully as possible We take each term inside the parenthesis and multiply it in turn by the term outside the parenthesis. The first product is a a, the second is a b. So the total expansion is just the sum of these two terms, i.e. answer is a a b = a a a b We don t write the products with a multiplication sign; instead we use indices and so the final Example 2: Expand a a b = a a a b =a 2 ab t 1 t 2 3 as fully as possible: We can approach this problem in the same way. The expansion can be written is several ways. We will show only one here t 1 t 2 3 =t t t 2 3 =t t 2 t 3 t 2 3 =t 3 3 t t 2 3 Conventionally, we write the result in descending powers and so the final answer is

10 t 1 t 2 3 =t 3 t 2 3t 3 Example 3: Expand the following expression as far as possible and then calculate it s value when g = 3 and f = 5 : g 1 g 2 2 f 1 f 2 f 1 This looks quite complicated when we look at it the first time. However, we can immediately simplify the problem by spotting that the term f 1 occurs in the numerator and the denominator. We can cancel this as any number divided by itself is 1. So the expression becomes g 1 g 2 2 f 2 Now we expand the parenthesis out term by term. Again, there are several different ways of securing the right result. I will show only one here: g 1 g 2 2 f 2 =[g g 2 2 g 2 2 ] f 2 =[g 3 2g g 2 2] f 2 =fg 3 2fg fg 2 2f 2g 3 4g 2g 2 4 =fg 3 2g 3 fg 2 2g 2 2fg 2f 4g 4 I have used extra parenthesis here to keep the order of the terms correct and I have also grouped them in decrease size of index from left to right, which is the standard method of writing expressions like this. Try out the other possibilities for yourself in terms of the order of the multiplication of parenthesis. You should get the same answer. Now we are asked numerically calculate the answer for the situation when g = 3 and f = 5. We can either do this from the long form that we have calculated above or we can use the short, initial version that we were given. I will use the initial version but check that the final, long version is correct too. When g = 3 and f = 5: Then So the final calculation becomes g 1 = 1=4 g 2 2 =3 2 2=9 2=11 f 2 =5 2=7 g 1 g 2 2 f 2 = =28 11 = = =308 By using a parenthesis at the third line, we didn t even need to use a calculator! When we work with parentheses or brackets, we can use three identities, simple memorized

11 general rules, to simplify our work. These are a b 2 =a 2 2ab b 2 a b 2 =a 2 2ab b 2 a 2 b 2 = a b a b When we come to look at the connection between geometry and algebra, we will see that there is a very simple derivation and explanation of these rules. Indices Imagine two numbers p and q such that p=a 3 and q=a 5 Now imagine that p and q are multiplied together. How do we write the answer in terms of the indices that we have used to define p and q? We just add the indices together i.e. pq=p q=a 3 a 5 = a a a a a a a a =a a a a a a a a =a 8 =a 3 5 When the index is negative, we write the answer as the reciprocal of the number: b 1 = 1 b b 4 = 1 b 4 This leads to the definition of the case when the index is 0; for the number m m 2 m 2 =m 2 2 =m 0 m 2 m 2 = m2 1 = m2 m 2 m 2=1 m 0 =1 so we can say that any number raised to the power 0 is defined as 1 Given that we can square a number, i.e. raise it to the power 2, can we find an inverse function? The answer is yes and it is termed the square root. This is written as a fractional index: Square root: 1 a 2 = a Cube root: 1 a 3 = 3 a and in general a n = n a We now have the rules in hand to deal with very large numbers. For the the two numbers 1

12 p=a 3 and q=a 5 if a takes the value 7, then p = 343 and q = and so pq = a 3.a 5 = a 8 = 5,764,801 which is a sizable calculation by hand, though it is relatively tractable if somewhat laborious. Extraction of the 9 th root of the cube of the answer would be an extremely tough task by hand. Help is at hand however. Using the properties of indices, a system of calculation for complex problems with large numbers was developed. This is the system of logarithms. We write an expression in the following form n=a m. In this case, n is the number we are interested in, a is the base and m is the logarithm of n. The common bases for logarithms that we use are 10 and the transcendental number 'e'. Logarithms of base 10 (a = 10) are termed the common logarithms and are written either as m=log n,m=lg n or m=log 10 n. Natural logarithms are logarithms to base e and are written m=ln n,m=log e n. That a number of which e is a type arises in a completely natural manner from considerations of the limit of certain sums and this deeper appreciation is beyond the scope of our discussion. Numerically, e,to 5 figures has the value e= (the value for e may be remembered in the following manner: Andrew Jackson, a native Tennessean, was the 7th president of the United States of America and was elected in 1828.) Graphs of y = e x and y = 10 x and are shown in figure 2. The rules of using logarithms follow those for indices. To multiply two numbers together, we add the logarithms of the numbers. To divide the numbers, we subtract the logarithms. To find the square root of a number, we multiply the logarithm of the number by ½. In general to find the n th root, we multiply the number by 1 / n. Table 1 shows a summary of these rules. These rules are also independent of the base. For the work we will be doing in general chemistry, we need not be able to change base or understand at a fundamental level the nature of logarithms but we will need to be able to manipulate them as a matter of routine.

13 Illustration 2 A graph of y = 10 x, between x = 0 to x = 4 (above) and x = 2 to x = 2 (below) llustration 3 A graph of y = e x, between x = 0 to x = 6 (above) and x = 2 to x = 2 (below) Operatio n Logarithmic representation Logarithmic operation Operation Logarithmic representation Logarithmic operation xy log xy log x log y x log x x y log x y log x log y x log n x 1 2 log x 1 n log x Table 1 Basic logarithmic operations

14 Exercise 3 (a) Without using a calculator, write down the logarithms for the following numbers to the given bases (i) 10 in base 10 (ii) 1000 in base 10 (iii) e in base e (iv) in base 10 (v) y in base 10 (v) x in base 10 (vi) (b) in base (vii) 1 in base 10, base e, base and base 2 6 Calculate the following products logarithmically, giving the answers in index form: (i) (ii) 1,000 1,000,000,000,000,000,000 (iii) e 5.11 e 2 (iv) (v) (vi) (vii) 3 e 4.1 e Logarithms and exponents, when used in equations can be confusing, because exponents and indices do not obey the same rules as ordinary variables. Take a complex equation such as e A kt =k 1 C 1 C 2 where ' A 'is the variable for which we wish to solve, k, k 1, and T are constants and C 1 and C 2 are other variables. How can we simplify this equation to be able to find an expression for A? Note that the subscript ' A 'is simply a label for and is not a variable in itself. Apparently, we cannot do this, as we cannot manipulate exponential indices in the same way that we can manipulate ordinary variables. The RHS is well behaved in the sense that the variables and constants are normal algebraic terms. The exponent does not behave in this manner and so, because we have to treat both sides of the equality identically, we need to manipulate the equation into a form where we can rearrange it. Although exponents are different, we can implicitly change variable to the logarithm and then treat these in the normal way. If we take the logarithm of both sides, we can write e A kt =k 1 C 1 C 2 ln e A kt =ln k 1 C 1 C 2

15 Recall, from Table 1, that from the definition of the logarithm, and that from the rules of indices and logarithms. So, ln e A kt = A kt ln k 1 C 1 C 2 =ln k 1 ln C 1 ln C 2 A kt =ln k 1 ln C 1 ln C 2 which is considerably simpler than the original equation as A is now an ordinary algebraic variable. This type of equation is very common in General Chemistry and logarithmic equations are important in chemical kinetics, thermodynamics and the properties of gases. You will encounter these equations mainly in Chemistry 130, though we touch on them in Chemistry 120. Geometry and Algebra The language that we have used to describe certain operations is clearly related to the description of the shapes and volumes that we are used to in the real world a 2 is the square of a a 3 is the cube of a The vocabulary that we use for some descriptions certainly implies a phenomenological connection on a qualitative level between algebra and geometry. The idea that we can use geometry, not only to illustrate certain numbers but also to construct algebraic relationships is about 3000 years old, and emerged in the explosion of thought that occurred in the West in Ancient Greece. The first comprehensive condensation that has survived is the work of Euclid, though Euclid apparently condensed a great deal of the material in his works from other sources something like writing a General Chemistry text book. In this section, we are going to show how we can use geometry to represent numbers and the relationships between numbers. In the next section, on graphs and functions, we will show how this relationship does not stop at numbers but can be extended even further.

16 We have already used a geometrical representation of a mathematical operation when we discussed the square of a number. Indeed, the language that is used to describe certain mathematical operations is rooted in geometry, alluding to the root of the word geometry, meaning measurement of the earth. Let us first examine the square. A square is a plane figure that has four sides of equal length and angles that total 360 o and are all equal to each other. Self evidently, these angles are 90 o. The area of a square is found, from consideration of the area of the two triangles into which the square may be formed on joining opposite corners, by simply multiplying the lengths of the two sides together. Thus for a square of side 2, the area is 4 and in general, the area of a square of side l is l 2. For a cube of side l the volume is l 3. In this respect, we are defining definite physical attributes volume and area in abstract terms. One of the key concepts in this discussion in the representation of a number as a length of a segment of the number line. The number line is a straight line on which we choose an arbitrary point and define it as the origin, O. We then mark a small segment of the line and choose this as the measure of unit length. The distance between O and the end of this segment of unit length we mark as 1. The line then appears as follows: This length of the line gives us a geometrical definition of the number 1. Any multiple of 1, i.e. any other integer, whether defined explicitly or not, is then simply a multiple of the length of the line 1, in the same direction as the line O1. The variable x is therefore represented by a line of length x beginning at O. The geometrical representation of algebraic relationships can be illuminating; recall the identity for the square of the sum, i.e. x y 2 =x 2 2 xy y 2 This identity is very easily illustrated in the following manner, simply from a knowledge of the area of a rectangle. First, we mark a length on the line and label it x together with another line, which we label y. We then have a figure that looks like

17 with the total line length being (x + y). If we now form a square whose length is the (x + y) we obtain the following figure: where the area of the square in bold is just the length of one side multiplied by itself i.e. A square = x y 2 We now draw perpendicular lines so as to divide the area of the square up into individual rectangles and squares as shown below, together with the area of each one: The area of the initial square is simply the sum of the areas of the internal squares and rectangles i.e. A square =x 2 xy xy y 2 =x 2 2 xy y 2 But A square = x y 2 and so x y 2 =x 2 2 xy y 2