MEASUREMENT AND CALCULATIONS. Chapter 2

Transcription

1 MEASUREMENT AND CALCULATIONS Chapter 2

2 HOW DO WE ANSWER QUESTIONS IN SCIENCE? Scientific Method A logical approach to solving problems by: Observing and collecting data Formulating hypotheses Testing hypotheses Formulating theories that are supported by data.

3 OBSERVING/COLLECTING DATA Observing using the senses May give us qualitative (descriptive) or quantitative (numerical) information Experimenting carrying out a procedure to make observations and collect data System A specific portion of matter in a given region of space that has been selected for study during an experiment or observation

4 FORMULATING HYPOTHESES Hypothesis A testable statement Serves as a basis for experimentation Often drafted as if-then statements

5 TESTING HYPOTHESES Controls experimental conditions that remain constant Variable experimental conditions that change Results here will either prove your hypothesis or cause it to be discarded or modified

6 THEORIZING Model A physical representation of a system An explanation of how phenomena occur and how data/events are related Theory A broad generalization that explains a body of facts or phenomena

7 UNITS OF MEASUREMENT As part of our data collection, many times we will take measurements Measurements always contain two parts Number Unit A number without a unit does not give you any useful information! Example: Joe is 7. Seven what?

8 INTERNATIONAL SYSTEM OF UNITS (SI UNITS) Uses seven base units: Meters (m) for length Kilograms (kg) for mass** Kelvin (K) for temperature Moles (mol) for amount of substance Ampere (A) for electric current Candela (cd) for luminous intensity Mass: the measure of the quantity of matter Weight: the measure of the gravitational pull on matter Notice that mass is the only base unit that has a prefix!

9 INTERNATIONAL SYSTEM OF UNITS (SI UNITS) Derived Units Units made from a combination of two or more SI base units Examples: Volume Length x width x height (m 3 ) Liquid volume in liters Handy equivalent: 1 ml = 1 cm 3 Density D = MM VV Ratio of mass to volume (g/ml) As the value rises, the objects are heavier

10 DENSITY CALCULATIONS A) If mass is 6.2g and volume is 4.7 ml, what is the density? DD = MM VV =6.2g/4.7mL = 1.3 g/ml B) If the density of a substance is 6.72 g/ml and you have 55.1mL of it, what should the mass be? M = D V = (6.72 g/ml) (55.1 ml) = 370. g

11 DENSITY CALCULATIONS C) If the density of a substance is g/ml and you have 0.451g of it, V =? VV = MM DD = (0.451 g) / (0.824 g/ml) = ml

12 INTERNATIONAL SYSTEM OF UNITS (SI UNITS) Instead of using many units for the same measurement, the SI unit system uses prefixes to change the size of units:

13 PROBLEM SOLVING/DIMENSIONAL ANALYSIS Many of the problems you will be asked to solve during this course involve converting units. The method used for this is called the factor-label method, or dimensional analysis. In order to convert between units, you must first know how they are related to each other. This relationship is referred to as an equivalency statement. For example: 1 meter = 100 centimeters 60 minutes = 1 hour 3 feet = 1 yard

14 DIMENSIONAL ANALYSIS There is an algebraic rule that says a fraction with equivalent values in the numerator and the denominator is equal to one: 10 = 1 10 Since the quantities in an equivalency statement are equal, we can make a fraction from them that is equal to one: 60 minutes = 1 1 hour This is a conversion factor a ratio derived from the equality between two different units

15 DIMENSIONAL ANALYSIS We can use these fractions to set up a problem in which we cancel units to convert values from one unit to another. 3.5 hours =? minutes 1 hour = 60 minutes 60 minutes 3.5 hours 1 hour = 210 minutes NOTICE since we are technically multiplying by a value of one, we haven t changed the value of the number, just the unit it is expressed in!

16 DIMENSIONAL ANALYSIS What if you do not know a direct relationship between what you want to convert? You can use multiple conversion factors: 380 cm =? km 380 cm = km

17 PRACTICE An island has no currency; instead it has the following exchange rate: 50 bananas = 20 coconuts 12 fish = 30 coconuts 1 hammock = 100 fish How many bananas would you need to purchase a hammock? 1 h 100 ff 1 h 30 cc 12 ff 50 bb 20 cc = 625 bb

18 PRACTICE It takes a seamstress 4.56 hours to complete a skirt. A boutique in town would like to buy 25 of these skirts. If she worked without stopping, how long would it take her (in minutes) to complete the skirts for the boutique? 4.56 h = 1 s 60 min = 1 h 25 ss 4.56 h 1 ss 60 mmmmmm 1 h = 6,840 mmmmmm

19 PRACTICE A store sells its own blend of perfume by volume. A girl wants to refill her 0.015L bottle with her favorite fragrance, which costs $1.75 for each milliliter. How much will it cost to fill the bottle? 1 ml = 1.75 d 1 L = 1000 ml LL 1000 mmmm 1 LL 1.75 dd 1 mmmm = d

20 PRACTICE How long (in minutes) will it take to run 1.45 km if the average speed of a runner is 1.85 m/s? 1.85 m = 1 s 1000 m = 1 km 60 s = 1 min 1.45 kkkk 1000 mm 1 kkkk 1 ss 1.85 mm 1 mmmmmm 60 ss = 13.1 mmmmmm

21 PRACTICE Maggie has a job at the local convenience store. She gets paid $7.50/hour for her work. If she wants to purchase a new ipod that costs $199.00, how many SECONDS of work will she have to do in order to purchase the ipod? (Neglect the fact that taxes will be taken out of her pay, and that tax will be charged on her purchase) 95,520 seconds

22 PRACTICE Convert 85.6 deciliters into microliters. 1 L = 10 dl 1 L = 1,000,000 µl 85.6 dddd 1 LL 10 dddd 1,000,000 μμμμ 1 LL = 8,560,000 μμμμ

23 RELIABILITY IN MEASUREMENTS When we make measurements, we must know that they are reliable. There are two benchmarks for reliability: Precision How close multiple measurements are to one another. Accuracy How close a single measurement is to the accepted value.

24 ERASER/DARTBOARD ANALOGY: If I aim at something and I hit my intended target once, that is accuracy. If I hit the same thing each time, regardless of where I have aimed, that is precision. To have both precision and accuracy, you must hit your intended target each time you throw.

25 PERCENT ERROR In chemistry, we often want to make a comparison of how close we have come to an accepted value, also known as percent error. PPPPPPPPPPPPPP eeeeeeeeee = VVVVVVVVVV eeeeeeeeeeeeeeeeeeeeeeee VVVVVVVVVV aaaaaaaaaaaaaaaa VVVVVVVVVV aaaaaaaaaaaaaaaa xx 100 If percent error is negative, the experimental value is LESS THAN the accepted value. If percent error is positive, the experimental value is MORE THAN the accepted value.

26 PERCENT ERROR A g mixture is 85.00g sulfur and 15.00g iron. Using a magnet, you recover 13.75g of iron. What is your percent error? % error = (13.75 g g) x 100% g = % error (rounded to sig figs)

27 TAKING MEASUREMENTS Any time we take a measurement, we must use some type of instrument Temperature thermometers Mass balance (scale) Length ruler Volume graduated cylinder When taking a measurement with a digital instrument, the instrument will estimate your final digit for you. When taking a measurement with a manual instrument, we must include all of the digits we know for certain, plus one estimated digit. We call this rounding to significant figures.

28 TAKING MEASUREMENTS How long would the block below be? 2.15 cm

29 TAKING MEASUREMENTS How many milliliters in this graduated cylinder? We measure to the BOTTOM of the meniscus! 43.0 ml

30 TAKING MEASUREMENTS What is the length of this screw to sig figs? 5.10 cm

31 TAKING MEASUREMENTS What about now? 5.1 cm

32 TAKING MEASUREMENTS And now? 5 cm

33 TAKING MEASUREMENTS Measure the Celsius temperature from this thermometer: 8.0 C

34 TAKING MEASUREMENTS How about this one? -7 C

35 PRACTICE Measure each to the correct number of significant figures: 1.92 cm 52.8 ml 1.0 C

36 SIGNIFICANT FIGURES IN MEASUREMENTS Whenever we take a measurement, we must always include the correct number of significant figures, based upon our instrument. Remember, include all digits you know FOR SURE plus one estimated digit! How do you know which digits are significant if you did not take the measurement yourself?

37 IDENTIFICATION OF SIGNIFICANT FIGURES

38 IDENTIFICATION OF SIGNIFICANT FIGURES Identify the number of significant figures in each of the following measurements: 465 m L s 1000 g 6802 ml

39 IDENTIFICATION OF SIGNIFICANT FIGURES Identify the number of significant figures in: g ml 1900 cm L g

40 IDENTIFICATION OF SIGNIFICANT FIGURES How about these ones? m s g 7200 cm ml L

41 SIGNIFICANT FIGURES IN CALCULATIONS If we multiply 7.7m 5.4m in our calculator, we get m 2. Can a calculated value be more precise than our measured values? No, so we must round our answer manually to make it consistent with our measurements. The way that we round depends upon the operation we have performed.

42 ADDITION/SUBTRACTION Round to the same number of decimal places as the measurement with the least number of decimal places. An easy way to do this is to set up your problem in a column and draw a vertical line next to the last column that has a digit for each entry: m m m m 1. Line up the decimal points 2. Cut off at least precise measurement with a line 3. Look to the right of the line to round up or down In the correct number of sig figs, the answer is m.

43 ADDITION/SUBTRACTION Let s try a subtraction: m m m Your rounded answer is m

44 ADDITION/SUBTRACTION Practice: 61.2 m m m m= 79.2 m 9.44 L 2.11 L 7.33 L 1.36 g g g m 17.3 m m= 17.3 m

45 MULTIPLICATION/DIVISION Round off answer to the least number of sig figs used in any measurement in the calculation. **The position of the decimal point has NOTHING to do with the rounding process!! 7.55m 0.34m (3 sf) (2 sf) = m 2 = 2.6 m g 8.4 = g (5 sf) (2 sf) = 0.29 g

46 MULTIPLICATION/DIVISION Practice: 8.3 m 2.22 m m 2 18 m g g 675 g 5.4 cm 3.21cm cm cm 3 32 cm L L 0.21 L

47 SIG FIGS AND CONVERSION FACTORS It is important to note that conversion factors are generally considered to be exact, so they will NOT affect the certainty of your answer. This means that when you multiply by a conversion factor, the number of digits in the factor will NOT change or limit the number of significant figures that your answer should be given in! Example: mm 100 cccc 1 mm = cccc

48 MIXED OPERATIONS PRACTICE 8.7 g g + 19 g g 43 g 4.32 cm 1.7 cm cm cm L L L L g / ,025g 11,000 g

49 WARM UP Perform each operation and round to the appropriate number of significant figures: 32.5m * 0.021m m 2 =0.68 m cl cl cl =172.0 cl kg kg =92.0 kg s s s = s

50 MIXED OPERATIONS PRACTICE A x x x 10 6 B x 10 7 * 6.82 x x C x x x 10 4 D x 10 6 / 4.92 x x 10 9

51 SCIENTIFIC NOTATION In chemistry, we often use very small or very large numbers: 1 atom = g In order to avoid writing out all these numbers, we use scientific notation. Scientific Notation a number written as the product of a coefficient and a power of ten. Example: 540,000 is written as 5.4 x is the coefficient, 10 5 is the power of ten, or an exponent

52 WRITING NUMBERS IN SCIENTIFIC NOTATION The coefficient is always a number greater than or equal to one AND less than ten: (1 coefficient <10) To get the power of ten, count how many spaces you must move the decimal point to get the coefficient between one and ten: Numbers > 10 have positive exponents (move decimal point to left) Numbers < 1 have negative exponents (move decimal point to right) Regular numbers can be written in scientific notation as well: 5 = 5 x 10 0

53 WRITING NUMBERS IN SCIENTIFIC NOTATION Place the following into scientific notation: 245, x x ,000, x x 10-9

54 PUTTING NUMBERS INTO SCIENTIFIC NOTATION Try some: A. 84,300 B C. 68,200,000,000 D E. 849,000,000,000,000 F A x 10 4 B. 6.2 x 10-6 C x D. 5.2 x E x F x 10-3

55 WRITING NUMBERS IN SCIENTIFIC NOTATION Take the following out of scientific notation: 6.25 x x x x , ,700,

56 TAKING NUMBERS OUT OF SCIENTIFIC NOTATION Try some: A x 10 5 B. 9.3 x 10-4 C. 3.8 x 10 8 D. 7.1 x 10-7 E x 10 3 F. 8.3 x 10-9 A. 548,000 B C. 380,000,000 D E. 2,620 F

57 OPERATIONS USING SCIENTIFIC NOTATION You MUST be able to perform these operations manually (without your calculator!) You will NOT need to worry about keeping track of Sig Figs here; just round the final coefficient to three (3) sig figs If your answer does not follow our standard form for standard scientific notation, you can adjust it: If you make the value of the coefficient bigger, the exponent must get smaller. If you make the value of the coefficient smaller, the exponent must get bigger.

58 MULTIPLICATION To multiply: Multiply the coefficients and add the exponents (3.00 x 10 4 ) (5.00 x 10 2 ) =( ) x 10 (4+2) =15.0 x 10 6 = 1.50 x 10 7

59 DIVISION To divide: Divide the coefficients and subtract the exponents (bottom from top) (6.00 x 10 4 ) (2.00 x 10 2 ) = ( ) x 10 (4-2) = 3.00 x 10 2

60 MULTIPLICATION/DIVISION PRACTICE Practice: 6.8 x x x 10 9 = 3.09 x x x x x x x 10 4 = 1.70 x x x x 10 1 = x 10 14

61 ADDITION/SUBTRACTION To add or subtract, you must first adjust the decimal point so that the exponents in each number are the same. Then, just add/subtract the coefficients.

62 ADDITION/SUBTRACTION (3.00 x 10 4 ) + (2.00 x 10 2 ) = (3.00 x 10 4 ) + (0.02 x 10 4 ) = 3.02 x 10 4 = (300. x 10 2 ) + (2.00 x 10 2 ) = 302 x 10 2 = 3.02 x 10 4 (3.00 x 10 4 ) - (2.00 x 10 2 ) = (3.00 x 10 4 ) - (0.02 x 10 4 ) = 2.98 x 10 4 = (300. x 10 2 ) - (2.00 x 10 2 ) = 298 x 10 2 = 2.98 x 10 4

63 ADDITION/SUBTRACTION PRACTICE 7.2 x x x x x x x x x x x x 10 11

64 PROBLEM SOLVING There are four steps generally followed when one wants to solve a problem: Analyze Read the problem, identify known and unknown quantities Plan Identify equations that might be helpful, find a way to solve Compute Follow your plan to solve the problem Evaluate Make sure your answer makes sense, check your work

65 PROPORTIONS Direct Proportions Dividing one quantity by the other gives a constant value. As y increases/decreases, so must x so that k stays constant yy xx = kk Inverse Proportions Product of two quantities are constant values. As y increases/decreases, x must do the oppositeso that k stays constant xxxx = kk In both of these situations, k is a proportionality constant!

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