Significant Figures. Accuracy versus Precision 6/20/2014

Transcription

1 Unit I: Measurements A. Significant figures B. Rounding numbers C. Scientific notation D. Using electronic calculators E. Using sig figs in arithmetic operations F. The metric system G. Problem solving with unit analysis H. Derived units I. Practical conversions J. Density K. Applications of using density Significant Figures The digits in a decimal number that are warranted by the accuracy of the means of measurement. Every measurement device has a usefull range of measurement and some level of accuracy associated with it. 1-A Accuracy versus Precision True Value Measured Values 1-A 1

2 Significant Figures Significant figures in a number are all the digits of which we are absolutely certain, plus one additional digit, which is estimated and regarded as uncertain. -Backus, B, CH150 text 1-A Example of measurements/scales 1-A Examples of measurements 2

3 Example of measurements A Note on Units Every measured quantity has units. If the units are not given then the measurement is not correct. In this course an answer given without the proper units is WRONG. 1-A Rules for determining significant figures 1. All non-zero digits are significant 2. Zeros to the left of a nonzero number are NOT significant 3. Zeros between significant digits are significant 4. Zeros at the end of a number andright of a decimal point are significant 5. Zeros to the right of a number and left of an implied decimal are NOT significant 6. Counted numbers, and exact conversions are all considered significant but are exempt from the rules 7. When zeros follow a number with a terminal decimal point, they are significant 1-A 3

4 Rounding Numbers 1. If the digit dropped is >5, round up the final digit 2. If the digit dropped is <5, leave the final digit unchanged 3. If the digit to be dropped =5 (with no following digits) round up if the preceding digit is odd leave as is if the preceding digit is even 1-B Rounding Numbers Lecture problem I-2 (page 26): a. Round L to two significant figs. Step 1: How may digits are significant now? Step 2: Which is the first digit to be dropped? Step 3: Apply the appropriate rounding rule. 1-B Rounding Numbers Lecture problem I-2 (page 26): b. Round 446,500 m to three significant figs. Step 1: How may digits are significant? Step 2: Which is the digit to be dropped? Step 3: Apply the appropriate rounding rule. 1-B 4

5 Scientific Notation Provides a convenient way to express very large or very small numbers using powers of ten 10 = =1 1/10 = ,000,000,000,000 can be written as 1.06 x using scientific notation Likewise can be can be expressed as x C Example 1: Scientific Notation Write 417,500,000 in scientific notation Step 1: Place a decimal to the right of the first non-zero digit, and write the significant figures after the decimal Step 2: Determine what power of 10 needs to be multiplied by to obtain 417,500,000 Step 3: Make sure that the number expressed in scientific notation has the same number of significant digits as in decimal form. 1-C Scientific Notation Lecture Problems I-3 (page 28): a. Express 52,080,000 in scientific notation b. Express in scientific notation 1-C 5

6 Using Electronic Calculators EEor EXPmeans 10 x on most calculators so entering 15, EE, 4 into your calculator will give you: 15 x 10 4 or 150,000. Most calculators have a X 2 button which automatically squares a number The y x or ^button raises a number to an exponent 1-D Significant Figures in Calculations Rule 1: When multiplying or dividing measured numbers, the product or quotient cannot have more significant figures than the value in the operation having the least number of significant figures 1-E Significant Figures in Calculations Rule 2: For adding or subtracting measured numbers, the sum or difference can only be as accurate as the least accurate value in the arithmetic operation In other words the number of decimal placesin the answer must be equal to the least number of places in any of the numbers being added or subtracted (See lecture problem #5 on page 31) 1-E 6

7 Significant Figures in Calculations Rule 3: For combined operations, find the significant figures for each part of the operation, and determine the significant figures of the result based on the least accurate number. Do not round numbers between operations. (See lecture problem #6 on page 32) 1-E The Metric System The modern version of the metric system is known as the SI system. The base units represent independent physical properties or dimensions which can be measured with an appropriate gauge 1-F Prefixes in the SI system Prefix Abbreviation Exponent Number Meaning Giga G 109 1,000,000,000 One billion Mega M 106 1,000,000 One million kilo k 103 1,000 One thousand hecto h One hundred deka da ten Basic Unit (meter, gram, liter, etc.) deci d One tenth centi c One hundredth milli m One thousandth micro µ One millionth nano n One billionth 7

8 Shamelessly borrowed from: ength_units_graph.png The Metric System English to SI conversions you should know 1 inch (in.) 2.54 cm (exact) 1 pound (lb.) 454 g (3 sig figs) 1 quart (qt.) L (3 sig figs) 1 mile (mi.) 1.61 km (3 sig figs) Your text has several other useful conversions on page F Problem Solving with Unit Analysis Problem solving using unit analysis is one of the most important skill you will learn in this class. You will use this technique all throughout future chemistry classes and in all branches of science, engineering, or medicine 1-G 8

9 Unit Factors A unit factor, or conversion factor, shows the relationship between units in a numerator and denominator cm = 1 in. Can be re-written as 100 legs 1 centipede or 1 centipede 100 legs 2.54 cm 1 in or 1 in 2.54 cm 2 sides coin or coin 2 sides 1-G Unit Factors Unit factors come from: Conversion factors known by definition Conversion factors that accurately describe relationships Conversion factors defined by this manual. Consider the molecule diphosphorus pentaoxide P 2 O 5 write as many unit factors as possible relating the molecule to the atom which it is comprised of 1-G Unit Factors Problem solving with unit factors 1. Write down the units of the answer on the right side 2. Write down the given including units on the left 3. Write down the unit factors that apply to the problem 4. Insert the appropriate unit factors between the given and the answer so that all units will cancel except the unit of the answer 5. Calculate the answer paying attention to significant figures where necessary 1-G 9

10 Unit Factors Lecture problems I-7 (pg. 37) a. How many cm are in 7.98 in? b. How many mm are in 9.37 yds? 1-G Prefixes in the SI system Prefix Abbreviation Exponent Number Meaning Giga G 109 1,000,000,000 One billion Mega M 106 1,000,000 One million kilo k 103 1,000 One thousand hecto h One hundred deka da ten Basic Unit (meter, gram, liter, etc.) deci d One tenth centi c One hundredth milli m One thousandth micro µ One millionth nano n One billionth Group Practice Problems How many significant figures in each of the following numbers? m L 1,900 in s Round the following to 3 significant figures mi kg yd. Find the answer and round the correct number of sig figs cm cm cm = How many inches are in m? 10

11 Derived Units Derived units are created by combining base units or other derived units. Common examples include: Area Volume Velocity Pressure Density Energy Power (length x length) (length x length x length) (length / time) (force / area) (mass / volume) (force x length) (energy / time) 1-H Derived Units Frequently derived units deal with area or volume. For Example 1 liter is defined as 1 cubic decimeter, so how many cubic centimeters would be in one liter? 1-H Derived Units Lecture problem I-8 (pg. 41): a. How many cubic millimeters are in cubic miles? b. How many cubic meters are in 18.4 ft 2 c. How many ml are in 1.75 x 10-3 km 3 1-H 11

12 Practical Conversions We sometimes encounter the need to convert units in day to day life: distances, currency exchange rates, etc. Lecture problem I-9 (pg. 43): You fill up your gas tank in a rented car somewhere in Europe and pay Euros for 50.0 L. What were you paying in dollars per gallon? The exchange rate at the time was Euros to the dollar. 1-I Density Density is a physical property of a material and is defined as the ratio of its mass to volume. d = m / v The units for density are g/ml or g/cm 3 (cc) for liquids and solids. For gasses density is usually expressed in g/l 1-J Density The density of a material is not a constant! Pressure and temperature effect the density of solid and liquids only slightly Pressure and temperature have a very large effect on the density of gasses 1-J 12

13 Density Lecture problem I-10 (pg.45): What is the density, in g/cm 3 of a block of metal with a mass of kg and measuring 2.25 cm x 4.3 cm x 12.0 cm? 1-J Density Lecture problem I-11: (pg. 46) What is the mass, in grams, of 75.0 cm 3 of iron? 1-J Density Lecture problem I-12: a. Calculate the volume in quarts of 439 mg of ethyl alcohol b. What is the mass, in lbs., of 2.50 x 10 4 mm 3 of aluminum? 1-J 13

14 Applications of Density Density is used frequently in many of the calculations commonly done in chemistry. The density of a substance can sometimes be used to identify it, or to determine if a substance is pure or not. Because we can relate the mass and volume of a substance through density we can often avoid making difficult measurements. 1-K Group practice If molasses has a density of 1.35 g/ml, what is the mass of 1.20 cups of molasses? (Remember 1 cup is 8 fluid ounces, and 1 fluid ounce = ml) If you have a block of copper that has a mass of grams, what would be its mass if it was gold instead of copper? (see density chart on pg 44) 1-K 14