Chapter 2. Measurements and Calculations
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1 Chapter 2 Measurements and Calculations
2 Section 2.1 Scientific Notation Measurement Quantitative observation. Has 2 parts number and unit. Number tells comparison. Unit tells scale. If something HAS a unit, and you don t put it in, it is wrong! Put another way, I will deduct 5.4
3 Section 2.1 Scientific Notation Technique used to express very large or very small numbers. Expresses a number as a product of a number between 1 and 10 and the appropriate power of 10. Don t understand scientific notation? Come by my office, TODAY. Or go to the math help room. Don t wait! And Make SURE that you can do this on your calculator!
4 Section 2.1 Scientific Notation Using Scientific Notation Any number can be represented as a product of a number between 1 and 10 and the appropriate power of 10 (either positive or negative). The power of 10 depends on the number of places the decimal point is moved and in which direction.
5 Section 2.1 Scientific Notation Using Scientific Notation The number of places the decimal point is moved determines the power of 10. The direction of the move determines whether the power of 10 is positive or negative.
6 Section 2.1 Scientific Notation Using Scientific Notation If the decimal point is moved to the left, the power of 10 is positive. 345 = If the decimal point is moved to the right, the power of 10 is negative =
7 Section 2.1 Scientific Notation Concept Check Which of the following correctly expresses 7,882 in scientific notation? a) b) c) d)
8 Section 2.1 Scientific Notation Concept Check Which of the following correctly expresses in scientific notation? a) b) c) d)
9 Section 2.2 Units Nature of Measurement Measurement Quantitative observation consisting of two parts. Number Unit Examples 20 grams joule seconds
10 Section 2.2 Units The Fundamental SI Units Physical Quantity Name of Unit Abbreviation Mass kilogram kg Length meter m Time second s Temperature kelvin K Electric current ampere A Amount of substance mole mol
11 Section 2.2 Units Prefixes are used to change the size of the unit.
12 Section 2.3 Measurements of Length, Volume, and Mass Fundamental SI unit of length is the meter.
13 Section 2.3 Measurements of Length, Volume, and Mass Volume Measure of the amount of 3-D space occupied by a substance. SI unit = cubic meter (m 3 ) Commonly measure solid volume in cm 3. 1 ml = 1 cm 3 1 L = 1 dm 3
14 Section 2.3 Measurements of Length, Volume, and Mass Measure of the amount of matter present in an object. SI unit = kilogram (kg) 1 kg = lbs 1 lb = g
15 Section 2.3 Measurements of Length, Volume, and Mass Concept Check Choose the statement(s) that contain improper use(s) of commonly used units (doesn t make sense)? A gallon of milk is equal to about 4 L of milk. A 200-lb man has a mass of about 90 kg. A basketball player has a height of 7 m tall. A nickel is 6.5 cm thick.
16 Section 2.4 Uncertainty in Measurement A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty. Record the certain digits and the first uncertain digit (the estimated number).
17 Section 2.4 Uncertainty in Measurement Measurement of Length Using a Ruler The length of the pin occurs at about 2.85 cm. Certain digits: 2.85 Uncertain digit: 2.85
18 Section 2.5 Significant Figures Rules for Counting Significant Figures 1. Nonzero integers always count as significant figures has 4 sig figs (significant figures).
19 Section 2.5 Significant Figures Rules for Counting Significant Figures 2. Zeros There are three classes of zeros. a. Leading zeros are zeros that precede all of the nonzero digits. These never count as significant figures has 2 sig figs.
20 Section 2.5 Significant Figures Rules for Counting Significant Figures b. Captive zeros are zeros that fall between nonzero digits. These always count as significant figures has 4 sig figs.
21 Section 2.5 Significant Figures Rules for Counting Significant Figures c. Trailing zeros are zeros at the right end of the number. They are significant only if the number contains a decimal point has 4 sig figs. 150 has 2 sig figs.
22 Section 2.5 Significant Figures Rules for Counting Significant Figures 3. Exact numbers have an unlimited number of significant figures. 1 inch = 2.54 cm, exactly. 9 pencils (obtained by counting).
23 Section 2.5 Significant Figures Exponential Notation Example 300. written as Contains three significant figures. Two Advantages Number of significant figures can be easily indicated. Fewer zeros are needed to write a very large or very small number.
24 Section 2.5 Significant Figures Rules for Rounding Off 1. If the digit to be removed is less than 5, the preceding digit stays the same rounds to 5.6 (if final result to 2 sig figs)
25 Section 2.5 Significant Figures Rules for Rounding Off 1. If the digit to be removed is equal to or greater than 5, the preceding digit is increased by rounds to 5.7 (if final result to 2 sig figs) rounds to 3.9 (if final result to 2 sig figs)
26 Section 2.5 Significant Figures Rules for Rounding Off 2. In a series of calculations, carry the extra digits through to the final result and then round off. This means that you should carry all of the digits that show on your calculator until you arrive at the final number (the answer) and then round off, using the procedures in Rule 1.
27 Section 2.5 Significant Figures Significant Figures in Mathematical Operations 1. For multiplication or division, the number of significant figures in the result is the same as that in the measurement with the smallest number of significant figures =
28 Section 2.5 Significant Figures Significant Figures in Mathematical Operations 2. For addition or subtraction, the limiting term is the one with the smallest number of decimal places Corrected ¾¾¾¾ 31.28
29 Section 2.5 Significant Figures Concept Check You have water in each graduated cylinder shown. You then add both samples to a beaker (assume that all of the liquid is transferred). How would you write the number describing the total volume? 3.1 ml What limits the precision of the total volume? 1 st graduated cylinder
30 Section 2.6 Problem Solving and Dimensional Analysis Note: THIS NEXT PART IS A MOST IMPORTANT THING TO LEARN! YOU WILL USE IT ALL SEMESTER 30
31 Section 2.6 Problem Solving and Dimensional Analysis Use when converting a given result from one system of units to another. 1) To convert from one unit to another, use the equivalence statement that relates the two units. 2) Choose the appropriate conversion factor by looking at the direction of the required change (make sure the unwanted units cancel). 3) Multiply the quantity to be converted by the conversion factor to give the quantity with the desired units. 4) Check that you have the correct number of sig figs. 5) Does my answer make sense?
32 Section 2.6 Problem Solving and Dimensional Analysis Example #1 A golfer putted a golf ball 6.8 ft across a green. How many inches does this represent? To convert from one unit to another, use the equivalence statement that relates the two units. 1 ft = 12 in The two unit factors are: 1 ft 12 in and 12 in 1 ft
33 Section 2.6 Problem Solving and Dimensional Analysis Example #1 A golfer putted a golf ball 6.8 ft across a green. How many inches does this represent? Choose the appropriate conversion factor by looking at the direction of the required change (make sure the unwanted units cancel). 6.8 ft 12 in = in 1 ft
34 Section 2.6 Problem Solving and Dimensional Analysis Example #1 A golfer putted a golf ball 6.8 ft across a green. How many inches does this represent? Multiply the quantity to be converted by the conversion factor to give the quantity with the desired units. 6.8 ft 12 in = 82 in 1 ft Correct sig figs? Does my answer make sense?
35 Section 2.6 Problem Solving and Dimensional Analysis Example #2 An iron sample has a mass of 4.50 lb. What is the mass of this sample in grams? (1 kg = lbs; 1 kg = 1000 g) 4.50 lbs 1 kg lbs 1000 g 1 kg 3 = g
36 Section 2.6 Problem Solving and Dimensional Analysis Concept Check What data would you need to estimate the money you would spend on gasoline to drive your car from New York to Los Angeles? Provide estimates of values and a sample calculation. Sample Answer: Distance between New York and Los Angeles: 2500 miles Average gas mileage: 25 miles per gallon Average cost of gasoline: $3.25 per gallon 1 gal $ mi = $ mi 1 gal
37 Section 2.7 Temperature Conversions: An Approach to Problem Solving Three Systems for Measuring Temperature Fahrenheit Celsius Kelvin
38 Section 2.7 Temperature Conversions: An Approach to Problem Solving The Three Major Temperature Scales
39 Section 2.7 Temperature Conversions: An Approach to Problem Solving Converting Between Scales T = T T = T K C C K ( T - 32) ( ) F C = F = C T T 1.80 T
40 Section 2.7 Temperature Conversions: An Approach to Problem Solving Exercise The normal body temperature for a dog is approximately 102 o F. What is this equivalent to on the Kelvin temperature scale? a) 373 K b) 312 K c) 289 K d) 202 K
41 Section 2.7 Temperature Conversions: An Approach to Problem Solving Exercise At what temperature does C = F?
42 Section 2.7 Temperature Conversions: An Approach to Problem Solving Solution Since C equals F, they both should be the same value (designated as variable x). Use one of the conversion equations such as: T C = ( T - 32) F 1.80 Substitute in the value of x for both T C and T F. Solve for x.
43 Section 2.7 Temperature Conversions: An Approach to Problem Solving Solution T C x = = ( T - 32) F 1.80 ( x - 32) 1.80 x = - 40 So 40 C = 40 F
44 Section 2.8 Density Mass of substance per unit volume of the substance. Common units are g/cm 3 or g/ml. mass Density = volume
45 Section 2.8 Density Measuring the Volume of a Solid Object by Water Displacement
46 Section 2.8 Density Example #1 A certain mineral has a mass of 17.8 g and a volume of 2.35 cm3. What is the density of this mineral? mass Density = volume 17.8 g Density = 2.35 cm 3 Density = g/cm
47 Section 2.8 Density Example #2 What is the mass of a 49.6 ml sample of a liquid, which has a density of 0.85 g/ml? mass Density = volume x 0.85 g/ml = 49.6 ml mass = x = 42 g
48 Section 2.8 Density Exercise If an object has a mass of g and occupies a volume of L, what is the density of this object in g/cm 3? a) b) 1.95 c) 30.5 d) 1950
49 Section 2.8 Density Concept Check Copper has a density of 8.96 g/cm 3. If 75.0 g of copper is added to 50.0 ml of water in a graduated cylinder, to what volume reading will the water level in the cylinder rise? a) 8.4 ml b) 41.6 ml c) 58.4 ml d) 83.7 ml
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