Math and Significant Figure Review

Math and Significant Figure Review Math Review and Significant Figures For more help see Chapter 1 and Appendix A in your text book. Much of the material covered in Chemistry 1A requires the tools associated with well developed and practiced algebra and problem solving skills. In order to pass this course you will also need to be familiar with the use of a basic scientific calculator. You should remember to bring your calculator to each lecture and laboratory session. You will specifically need to: • Perform algebraic manipulations. • Use conversion factors to change units of a value. • Have the ability to extract information from a word problem and solve it using unit based dimensional analysis methods. • Perform calculations including logarithms and anti‐logarithms. • Perform calculations involving terms with exponents. • Express results with correct units and the correct number of significant figures. Significant Figures: (see Chapter 1 & Appendix 1 in your text) When performing scientific calculations, there are two types of numbers you will encounter: Exact numbers and inexact numbers. Exact numbers generally arise from definitions; inexact numbers arise from measurements. The exactness of any measured number is expressed by the number of significant figures or digits. Examples: Exact numbers: numbers whose values are known exactly and are assumed to contain no uncertainty. 12 eggs = 1 dozen eggs 100 cm = 1 meter 1 gallon = 4 quarts Exact numbers often come from definitions within a system of units like the metric system or British Standard units. Inexact numbers: All measured numbers have an associated uncertainty and are “inexact.” The uncertainties can stem from instrumental/equipment limitations, human error, or in some cases the slight variations in the property being measured. When converting between systems of measure, most equalities are inexact. Measured numbers are usually reported in such a way that only the last digit is uncertain. Consider the mass of an object on a balance capable of measuring to the nearest 0.0001 gram. You could report the mass of the object as 1.3276 ± 0.0001 gram to indicate the uncertainty lies in the digit 6. However, scientists have come to the understanding that allows us to assume that the uncertainty in the number lies only in the last digit. As a result, we can drop the ± notation. In a properly reported measured value, all of the digits are significant, even the uncertain ones. Our previous weight, 1.3276 grams, contains 5 significant digits. The ambiguity in significant figures becomes an issue with regard to the use of zeros. This stems from the fact that zeros can be significant digits (or figures) in the number or decimal place locators. For example, consider the zeros in 0.004560. How many significant figures are contained in the value? The zeros may or may not be significant depending on where they appear in the value. We will use a set of guidelines to assist us in determining if zeros are significant: 1. Zeros occurring between nonzero digits are always significant. Ex. 106 cm contains 3 SIG FIGS. 2. Zeros occurring at the beginning of a number, merely locating the decimal place, are never significant. Ex. 0.0234 grams contains 3 SIG FIGS. 3. Zeros occurring at the end of a value AND after the decimal point are always significant. Ex. 0.0600 meters contains 3 SIG FIGS. 4. Zeros at the end of a value not containing a decimal point represent an ambiguous situation. The zeros may or may not be significant. Ex. 100 kg has 1 SIG FIG. 100. has 3 SIG FIGS. and 100 has 2 SIG FIGS. In item (4), we avoid the ambiguity by using scientific notation to denote the value. 1.00 × 102 kg indicates 3 significant figures 1.0 × 102 kg indicates 2 significant figures Calculations and Significant Figures: When performing scientific calculations, one often uses more than one measured (inexact) value in a formula or series of calculations. To calculate the volume of a cylinder we need measurements of radius and height to utilize the equation V = r2h. Since the number of significant figures may be different in the height when compared to the radius one must evaluate the number of significant figures that can be legitimately reported in the result of the calculation. To avoid an error, one must follow these guidelines: When performing mathematical operations of multiplication and division, the least certain measurement limits the certainty of the calculated value: Example: 3 sig. figs. × 5 sig. figs. ÷ 2 sig. figs. should have only 2 sig. figs. in the result V = (0.53mm)2 × (253mm) = 71 mm3 The answer is limited to two significant figures because the radius value only contained two significant figures. For addition and subtraction the answer is limited by the value containing the least number of digits to the right of the decimal place. Example. 20.6 +1.432+ 65 = 87.032 (calculator given value) The final answer would be given as 87 to reflect the fact that 65 has no digits to the right of the decimal place. 20.6 + + 1.432 65 = 87.032 No sig. figs. beyond decimal. Uncertainty lies in the 10ths place so one must round to the unit’s value. When performing addition/subtraction calculations, the result may have more or less sig. figs. depending on the decimal places. 6 . 4 (2 sf) + 4 . 3 (2 sf) 10 . 7 (3 sf) More Complicated Operations: When calculations involve addition/subtraction and multiplication/division, determining the number of sig. figs. is not so straight forward… Example: (23.51 − 12.006) × (0.0112 × 31.00) ÷ (5.1 + 4.99) First rewrite the calculation as follows: (23.51 − 12.006) × (0.0112 × 31.00) (5.1 + 4.99) Now complete each of the calculations in the parentheses while noting the sig. figs.: 4 sf 3 sf (11.504) × (0.34720) = 0.395856 (3 sf) = 0.396 (10.09) 3 sf Rounding numbers: Many times your calculator will provide you with a number containing more than the correct number of significant figures. We must correctly round off the value when dropping the extra significant figures. Use the following guidelines in rounding numbers: 1. If the leftmost digit to be dropped is less than 5, the number preceding it is left unchanged. Ex. 3.453 rounded to 3 significant figures would be 3.45 2. If the leftmost digit to be dropped is 5 or greater, the number preceding it is increased by 1 value. Ex. 5.635 rounded to 3 significant figures would be 5.64 (Note: Although you may come across or learned other ways of handling rounding when the digit to be dropped is a 5, we will follow the methods used in your text. See page 40.) Logarithms and anti‐logs Logarithms, or "logs", are a way of expressing one number in terms of a "base" number that is raised to some power. Common logs are done with base ten, but some logs ("natural" logs) are done with the constant "e" (2.718 281 828) as their base. The log of any number is the power to which the base must be raised to give that number. We are most familiar with base 10 since any number greater than zero can be expressed as 10x. For example, log(10) = 1 (because 101 =10), log(100) = 2 (because 102 = 100), and log(2) is 0.3 (because 100.3 ≈ 2). Logs can easily be found for either base on your calculator. Usually there are two different buttons, one saying "log", which is base ten, and one saying "ln", which is a natural log, base e. It is always assumed, unless otherwise stated, that "log" means log10. The opposite of a log is the antilog, which means to raise the base to that number. Antilogs "undo" logarithms. Observe the following example: log(100) = 2 antilog(2) = 100 Logs are read aloud as "log", "natural log", "ln", or "log base whatever". Logs are commonly used in chemistry. The most prominent example is the pH scale. The pH of a solution is the −log([H+]), where square brackets mean concentration. There are two major kinds of equations that you will have to solve using logs. In one kind, you will know the log of a number and have to find the number by taking anti‐logs, which means raising the base to a power. The other kind gives you the variable in the exponent, and you have to take logs to isolate it. Solving these kinds of problems depends on knowing another property of logs: if the log of a number with an exponent is taken, then the log of that number is multiplied by whatever was in the exponent. log(am) = m×log(a) or ln(am) = m×ln(a) log(23) = 3×log(2) or ln(23) = 3×ln(2) Before we go any further, let's review some definitions that can show the relationship of exponential notation and logarithms. x0 = 1 x a × xb = x (a+b) xa = x (a‐b) b x 1 1 x −1 = , x −2 = 2 , etc. x x log(b) = c means 10c = b log(100) = 2 means 102 = 100 For the logarithm of a measured number, the number of digits after the decimal point equals the number of sig. figs. in the original number. For antilogs, the number of sig. figs. in the answer equals the number of digits past the decimal point in the original number. e5.4763 = 2.390 ln(25.67) = 3.2453 4 sig. figs. 4 sig. figs. past decimal point in answer 4 sig. figs.past decimal point 4 sig. figs. total in answer. If you do not know how to use the log or exponent functions on your calculator, see your instructor. Scientific Notation: Scientific notation is an alternative to conventional (decimal) notation. Any number can be represented in either system. The decision to use one over the other is mainly a matter of using fewer characters. For numbers that are very large or very small, scientific notation is the appropriate form. For example, the number 3,145,000,000,000,000,000,000,000,000 = 3.145 × 1027 Notice that the zeros beyond the last significant digit have disappeared when the number is written in scientific notation. When a number is written correctly in scientific notation, it always has the correct number of sig. figs. Many calculators and computer programs also use scientific notation since the space available in a display is usually at a premium. However, because exponents are problematic to display, they may symbolize the number slightly differently. In Excel, as in many other programs, the number 3.145 × 1027 appears as: 3.145E+27. In order to score well in this course, you must be proficient in your use of scientific notation, rounding, and significant figures. − + × × × ‐16 2 ‐12 = mole − × ×( + × 16 − −