Chapter 2 Measurements & Calculations Quantity: A thing that can be measured. ex. Length (6.3 ft), mass (35 kg), and time (7.2 s) Measurements can be expressed in a variety of units: Example: length(cm, in, ft, yd, miles) Everyone in the world does not use the same units. For speed: USA miles per hour (mph) Canada & Mexico kilometers per hour (kmph) English Unit Name Symbol Length foot, inch, mile ft., in., mi Weight ounces, pounds oz., lbs. Time minutes, hours min., h. Temperature Celsius, Fahrenheit o C, o F
By international agreement in the 960 s, a set of units called "The International System of Units" was defined for scientific work. AKA: SI Units (Metric Units) SI Units Name Symbol Length meter m Mass kilogram kg Time second s Temperature kelvin K SI Prefixes Used with base SI units to form new SI units that are larger or smaller than the base units by some multiple of 0. 2
Stairway to Metric Conversions Using SI Prefixes 0 9,000,000,000 = billion (Giga) G 0 8 00,000,000 = hundred million 0 7 0,000,000 = ten million 0 6,000,000 = million (Mega) M 0 5 00,000 = hundred thousand 0 4 0,000 = ten thousand 0 3,000 = thousand (kilo) k 0 2 00 = hundreds (hecto) h 0 0 = tens (deka) da d c m µ n (deci) (centi) hundredth = (milli) tenth = thousandth = (micro) 0 ten thousandth = =. = 0-00,000 hundred thousandth = millionth = (nano) =.0 = 0-2 =.00 = 0-3 0,000 00,000 ten millionth = 0 0 = BASE UNIT = ones =.000 = 0-4,000,000 hundred millionth = 0,000,000 billionth = =.0000 = 0-5 =.00000 = 0-6 00,000,000 =.000000 = 0-7,000,000,000 =.0000000 = 0-8 =.00000000 = 0-9 3
Scientific Notation Sometimes it is either too difficult or too ridiculous (or both) to write a number. For example, The speed of light: 300,000,000 m/s The mass of one gold atom: 0.000000000000000000000327 g The solution is to write the number in scientific notation. (not exponential notation) Scientific Notation Expresses a number as a product of numbers between and 0 and an appropriate power of ten. They are written in the following format: #.## x 0? 4
The power of ten depends on the number of places the decimal point needs to be moved. If 0? has a negative exponent this means the number is smaller than. Ex. 0 - = 0. 0-2 = 0.0, etc. If 0? has a positive exponent this means the number is larger than. Ex. 0 = 0 0 2 = 00, etc. Let s try writing numbers in scientific notation: 360000 = 3.6 x 0 5 2000000000 = 2. x 0 0 0.00025 =.25 x 0-4 0.000007 =.7 x 0-6 5
Let s try writing numbers from scientific notation: 5.8 x 0-3 = 0.0058 3.87 x 0 5 = 387000 2. x 0-8 = 0.00000002.5 x 0 9 = 50000000 Uncertainty in Measurement Regardless of the instrument used to obtain a measurement (a simple ruler or a very precise balance), it is important for us to realize that the measurement is never exact. These exact measurements are really estimates. If two people take the same measurement, chances are good that the 2 measurements will not be equal. 6
Why the difference? Both individuals ultimately make an estimate. This estimate seldom is the same from one person to another. Therefore, it is very important to remember that a measurement always carries with it some amount of uncertainty. Let s try reading the graduated cylinder below. 7
What is the true volume of the liquid in the graduated cylinder? We can say with a high level of confidence that the one s and the tenth s place values are and, respectively!!! What about the hundredth s place? Should it be a 4 or 5 or 6??? Since we can t say for sure, we can only come up with a best guess (estimate) of the last integer. The level of uncertainty is dependant on the instrument used. Significant Figures Suppose that I give you a string that measures.0 meters. 8
How accurately could you measure your desk using that string? Which of the following measurements seem reasonable to you based on your string?.6 m?.66 m?.666 m?.6666 m?.66666 m? A reasonable answer can be determined using Significant Figures. The number of significant figures (sig figs) for a measurement is determined by the uncertainty of the measuring instrument. The number recorded in the measurement would include all certain numbers PLUS the first uncertain number. 9
How do we count the number of sig figs for a specific number? Here are the rules: ) Non-zero integers are always significant. Ex. 26, 3, 22.4 2) The 3 zero rules: a) Captive Zeros = zeros in between non-zeros are always significant. Ex. 307 b) Leading Zeros = Zeros preceding non-zero integers are never significant. Ex. 0.00 c) Trailing Zeros = Zeros after a nonzero are only significant if written with a decimal point. Ex. 300 Zeros are not significant 300. Zeros are significant 300.0 - Zeros are significant 0
3) Exact numbers* = They have an unlimited number of sig. figs Do not use these numbers to determine the sig. figs. in a calculation (more on this later). *What are exact numbers? a) A number determined by counting. Ex. I own 2 cars. b) A number coming from a definition. Ex. dozen eggs = 2 eggs
Counting Significant Figures How many sig. figs. are there in each of the following numbers: a) 6000 = b) 0.000092 = 2 c) 0.056 = 3 d) 4.0 x 0-6 = 4 e) 9.0 x 0 3 = 2 f) 7005 = 4 g) 30. = 3 Rounding Numbers Let s review together using sig. figs. Round these number to the indicated number of sig. figs.: ) 0.583 to 2 sig figs = 0.58 2) 354 to 3 sig figs = 3200 3) 3.930 x 0-3 to 3 sig figs = 3.9 x 0-3 4) 0.000337 to 2 sig figs = 0.00033 2
Determining Sig. Figs in Calculations Addition & Subtraction ) Perform the operation (+ or ) 2) Round you answer to the place of the limiting term (the least accurate place). Example: 25.62 957.0 + 5.87 988.49 988.5 Rounded to the place of the limiting term (the least accurate place) Multiplication & Division ) Perform the operation (x or ) 2) Round you answer to the least number of sig. figs. determined by the numbers used in the operation. 3
Example: 25.73 x 5.87 = 5.035 Since 5.87 contains 3 sig figs, the answer should be rounded to 3 sig figs 5 Dimensional Analysis In short, Dimensional Analysis is a technique that allows one to convert one unit to another (for example feet to centimeters). A conversion factor allows us the ability make this transformation. Conversion Factor = Ratio of 2 units. foot For example, 2 inches 2 inches (An equivalent ratio could be foot ) Although these two ratios tell us the same information, they are NOT numerically equivalent! 4
Let s try some calculations: ) Convert 47.9 mg??? g What is the relationship between mg and g? 000 mg g and g 000 mg Which of these two forms will leave us with the units of g and cancel out the units of mg??? g 47.9 mg = 0. 0479 000 mg How many sig. figs. should be used? 3 sig. figs. g Try 0.07 L ml??? 5
Multi-step Dimensional Analysis 2) How many seconds are there in.5 days? (There is more than one way to solve this problem.) What do we need to know to answer this question? day = 24 hr hr = 60 min min = 60 s 24 hr 60 min 60 s.5 days = 29600 day hr min s Since 2 sig. figs. are needed, 29600 s 30000 s or.3 x 0 5 s 6
3) What is the speed of light (3.0 x 0 8 m/s) in miles/hr? (There is more than one way to solve this problem.) 3.0 x 0 s 8 m 60 s 60 min.094 yd 3 ft mile min hr m yd 5280ft 67388.8 miles/hr (Calc. Read-out) We need only 2 sig. figs. 670,000,000 miles/hr or 6.7 x 0 8 miles/hr 7
Density Definition: the amount of matter present in a given volume of that substance. Density = mass volume With simple algebraic re-arranging, we can solve for mass or volume. 8