Appendix I: CommonMathematical Operations in Chemistry

A. Scientific NotationA number written in scientific notation consists of a decimal part, a number that is usuallybetween 1 and 10, and an exponential part, 10 raised to an exponent, n.

1.2

Exponential partDecimal part

10�10�

Exponent (n)

Each of the following numbers is written in both scientific and decimal notation.

A positive exponent means 1 multiplied by 10 n times.

A negative exponent means 1 divided by 10 n times.

To convert a number to scientific notation, we move the decimal point to obtain a num-ber between 1 and 10 and then multiply by 10 raised to the appropriate power. For exam-ple, to write 5983 in scientific notation, we move the decimal point to the left three placesto get 5.983 (a number between 1 and 10) and then multiply by 1000 to make up for mov-ing the decimal point.

Since 1000 is we write

5983 = 5.983 * 103

103,

5983 = 5.983 * 1000

10-3=

1

10 * 10 * 10= 0.001

10-2=

1

10 * 10= 0.01

10-1=

1

10= 0.1

(-n)

103= 1 * 10 * 10 * 10 = 1000

102= 1 * 10 * 10 = 100

101= 1 * 10

100= 1

6.7 * 103= 6700 6.7 * 10-3

= 0.0067

1.0 * 105= 100,000 1.0 * 10-5

= 0.000001

A-1

A-2 Appendix I : Common Mathemat ica l Operat ions in Chemist ry

We can do this in one step by counting how many places we move the decimal point toobtain a number between 1 and 10 and then writing the decimal part multiplied by 10raised to the number of places we moved the decimal point.

0.00034 3.4 10�4��

If the decimal point is moved to the left, as in the previous example, the exponent is pos-itive. If the decimal is moved to the right, the exponent is negative.

5983 5.983 103��

To express a number in scientific notation:

1. Move the decimal point to obtain a number between 1 and 10.

2. Write the result from step 1 multiplied by 10 raised to the number of places youmoved the decimal point.

• The exponent is positive if you moved the decimal point to the left.

• The exponent is negative if you moved the decimal point to the right.

Consider the following additional examples:

Multiplication and DivisionTo multiply numbers expressed in scientific notation, multiply the decimal parts and addthe exponents.

To divide numbers expressed in scientific notation, divide the decimal parts and subtractthe exponent in the denominator from the exponent in the numerator.

Consider the following example involving multiplication:

Consider the following example involving division:

Addition and SubtractionTo add or subtract numbers expressed in scientific notation, rewrite all the numbers so thatthey have the same exponent, then add or subtract the decimal parts of the numbers. Theexponents remained unchanged.

1A ; B2 * 10n;B * 10n A * 10n

= 4.0 * 104

15.6 * 107211.4 * 1032 = a 5.6

1.4b * 107 -3

= 6.3 * 1010 13.5 * 104211.8 * 1062 = 13.5 * 1.82 * 104 +6

1A * 10m21B * 10n2 = aA

Bb * 10m-n

1A * 10m21B * 10n2 = 1A * B2 * 10m+n

0.000000000070 m = 7.0 * 10-11 m

290,809,000 = 2.90809 * 108

B. Logarithms A-3

Notice that the numbers must have the same exponent. Consider the following exampleinvolving addition:

First, express both numbers with the same exponent. In this case, we rewrite the lowernumber and perform the addition as follows:

Consider the following example involving subtraction.

First, express both numbers with the same exponent. In this case, we rewrite the lowernumber and perform the subtraction as follows:

Powers and RootsTo raise a number written in scientific notation to a power, raise the decimal part to thepower and multiply the exponent by the power:

To take the nth root of a number written in scientific notation, take the nth root of thedecimal part and divide the exponent by the root:

B. LogarithmsCommon (or Base 10) LogarithmsThe common or base 10 logarithm (abbreviated log) of a number is the exponent to which10 must be raised to obtain that number. For example, the log of 100 is 2 because 10 must beraised to the second power to get 100. Similarly, the log of 1000 is 3 because 10 must beraised to the third power to get 1000. The logs of several multiples of 10 are shown below.

Because by definition, .The log of a number smaller than one is negative because 10 must be raised to a negative

exponent to get a number smaller than one. For example, the log of 0.01 is because 10-2

log 1 = 0.100= 1

log 10,000 = 4

log 1000 = 3

log 100 = 2

log 10 = 1

= 1.6 * 102 14.0 * 10621>3 = 4.01>3

* 106>3

= 16 * 1013 = 16 * 1012

14.0 * 10622 = 4.02* 106 *2

7.14 * 105-0.19 * 105

7.33 * 105

-1.9 * 104

7.33 * 105

5.16 * 107+0.34 * 107

4.82 * 107

+3.4 * 106

4.82 * 107

A-4 Appendix I : Common Mathemat ica l Operat ions in Chemist ry

must be raised to to get 0.01. Similarly, the log of 0.001 is because 10 must be raisedto to get 0.001. The logs of several fractional numbers are shown below.

The logs of numbers that are not multiples of 10 can be computed on your calculator. Seeyour calculator manual for specific instructions.

Inverse LogarithmsThe inverse logarithm or invlog function is exactly the opposite of the log function. For ex-ample, the log of 100 is 2 and the inverse log of 2 is 100. The log function and the invlogfunction undo one another.

The inverse log of a number is simply 10 rasied to that number.

The inverse logs of numbers can be computed on your calculator. See your calculatormanual for specific instructions.

Natural (or Base e) LogarithmsThe natural (or base e) logarithm (abbreviated ln) of a number is the exponent to which e(which has the value of 2.71828…) must be raised to obtain that number. For example, theln of 100 is 4.605 because e must be raised to 4.605 to get 100. Similarly, the ln of 10.0 is2.303 because e must be raised to 2.303 to get 10.0.

The inverse natural logarithm or invln function is exactly the opposite of the ln func-tion. For example, the ln of 100 is 4.605 and the inverse ln of 4.605 is 100. The inverse lnof a number is simply e rasied to that number.

The invln of a number can be computed on your calculator. See your calculator manualfor specific instructions.

Mathematical Operations Using LogarithmsBecause logarithms are exponents, mathematical operations involving logarithms are simi-lar to those involving exponents as follows:

log an= n log a ln an

= n ln a

log a

b= log a - log b ln

a

b= ln a - ln b

log1a * b2 = log a + log b ln1a * b2 = ln a + ln b

invln 3 = e3= 20.1

invln x = ex

invlog 3 = 103= 1000

invlog x = 10x

invlog(log 100) = 100

invlog 2 = 100

log 100 = 2

log 0.0001 = -4

log 0.001 = -3

log 0.01 = -2

log 0.1 = -1

-3-3-2

D. Graphs A-5

C. Quadratic EquationsA quadratic equation contains at least one term in which the variable x is raised to the sec-ond power (and no terms in which x is raised to a higher power). A quadratic equation hasthe following general form:

A quadratic equation can be solved for x using the quadratic formula:

Quadratic equations are often encountered when solving equilibrium problems. Belowwe show how to use the quadratic formula to solve a quadratic equation for x.

As you can see, the solution to a quadratic equation usually has two values. In any realchemical system, one of the values can be eliminated because it has no physical signifi-cance. (For example, it may correspond to a negative concentration, which does not exist.)

D. GraphsGraphs are often used to visually show the relationship between two variables. For example, inChapter 5 we show the following relationship between the volume of a gas and its pressure:

x = 1.43 or x = 0.233

=

5 ; 3.6

6

=

-1-52 ; 21-522 - 41321122132

x =

-b ; 2b2- 4ac

2a

3x2- 5x + 1 = 0 1quadratic equation2

x =

-b ; 2b2- 4ac

2a

ax2+ bx + c = 0

0 160 320 480 640 800 960 1120

Pressure (mmHg)

Vol

um

e (L

)

0

100

200

300

400

500

The horizontal axis is the x-axis and is normally used to show the independent variable.The vertical axis is the y-axis and is normally used to show how the other variable (called thedependent variable) varies with a change in the independent variable. In this case, the graphshows that as the pressure of a gas sample increases, its volume decreases.

Volume versus Pressure A plot of the volume of a gas sample––as measured in a J-tube––versuspressure. The plot shows that volume and pressure are inversely related.

Many relationships in chemistry are linear, which means that if you change one vari-able by a factor of n the other variable will also change by a factor of n. For example, thevolume of a gas is linearly related to the number of moles of gas. When two quantities arelinearly related, a graph of one versus the other produces a straight line. For example, thegraph below shows how the volume of an ideal gas sample depends on the number ofmoles of gas in the sample:

A-6 Appendix I : Common Mathemat ica l Operat ions in Chemist ry

�x

�y

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Number of moles (n)

Vol

um

e (L

)0

5

10

15

20

25

30

35

A linear relationship between any two variables x and y can be expressed by the followingequation:

where m is the slope of the line and b is the y-intercept. The slope is the change in y divid-ed by the change in x.

For the graph above, we can estimate the slope by simply estimating the changes in y andx for a given interval. For example, between and ,and we can estimate that Therefore the slope is

In several places in this book, logarithmic relationships between variables can be plottedin order to obtain a linear relationship. For example, the variables and t in the fol-lowing equation are not linearly related, but the natural logarithm of and t are lin-early related.

A plot of versus t will therefore produce a straight line with and y-intercept = ln[A]0 .

slope = -kln[A]t

y = mx + b

ln[A]t = -kt + ln[A]0

[A]t

[A]t

m =

¢y

¢x=

18 L

0.80 mol= 23 mol>L

¢y = 18 L.¢x = 0.80 mol1.2 molx = 0.4 mol

m =

¢y

¢x

y = mx + b

Volume versus Number of Moles The volume of a gas sample increases linearly with thenumber of moles of gas in the sample.