Scientific Scientific Measurement Measurement Chapter 3 Chapter 3 Lesson 2 Lesson 2
Dec 31, 2015
Significant Numbers in CalculationsSignificant Numbers in Calculations
A calculated answer cannot be more precise than A calculated answer cannot be more precise than the measuring tool. the measuring tool.
A calculated answer must match the least precise A calculated answer must match the least precise measurement.measurement.
Significant figures are needed for final answers Significant figures are needed for final answers fromfrom
1) adding or subtracting1) adding or subtracting
2) multiplying or dividing2) multiplying or dividing
Adding and SubtractingAdding and Subtracting
The answer has the same number of decimal The answer has the same number of decimal places as the measurement with the fewest places as the measurement with the fewest decimal places.decimal places.
25.25.22 one decimal placeone decimal place
+ 1.+ 1.3434 two decimal placestwo decimal places
26.5426.54
answer 26.5answer 26.5 one decimal placeone decimal place
Learning CheckLearning Check
In each calculation, round the answer to the In each calculation, round the answer to the correct number of significant figures.correct number of significant figures.
A. 235.05 + 19.6 + 2.1 = A. 235.05 + 19.6 + 2.1 =
1) 256.751) 256.75 2) 256.8 2) 256.8 3) 2573) 257
B. 58.925 - 18.2B. 58.925 - 18.2 ==
1) 40.7251) 40.725 2) 40.73 2) 40.73 3) 40.73) 40.7
Multiplying and Dividing
Round (or add zeros) to the calculated Round (or add zeros) to the calculated answer until you have the same number answer until you have the same number of significant figures as the measurement of significant figures as the measurement with the fewest significant figures.with the fewest significant figures.
Learning CheckLearning Check
A. 2.19 X 4.2 =A. 2.19 X 4.2 = 1) 91) 9 2) 9.2 2) 9.2 3) 9.1983) 9.198
B. 4.311 ÷ 0.07 =B. 4.311 ÷ 0.07 = 1)1) 61.5861.58 2) 62 2) 62 3) 603) 60
C. C. 2.54 X 0.00282.54 X 0.0028 = =
0.0105 X 0.060 0.0105 X 0.060
1) 11.31) 11.3 2) 112) 11 3) 0.041 3) 0.041
Conversion FactorsConversion Factors
Fractions in which the numerator and Fractions in which the numerator and denominator are EQUAL quantities expressed denominator are EQUAL quantities expressed in different unitsin different units
Example: 1 in. = 2.54 cm
Factors: 1 in. and 2.54 cm
2.54 cm 1 in.
Learning Check
Write conversion factors that relate each of Write conversion factors that relate each of the following pairs of units:the following pairs of units:
1. Liters and mL1. Liters and mL
2. Hours and minutes2. Hours and minutes
3. Meters and kilometers3. Meters and kilometers
How many minutes are in 2.5 hours?
Conversion factor
2.5 hr x 2.5 hr x 60 min 60 min = 150 min = 150 min
1 hr1 hr
cancel
By using dimensional analysis / factor-label method, By using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the side up, and the UNITS are calculated as well as the
numbers!numbers!
Steps to Problem SolvingSteps to Problem Solving1. Write down the given amount. Don’t forget the units!2. Multiply by a fraction.3. Use the fraction as a conversion factor. Determine if
the top or the bottom should be the same unit as the given so that it will cancel.
4. Put a unit on the opposite side that will be the new unit. If you don’t know a conversion between those units directly, use one that you do know that is a step toward the one you want at the end.
5. Insert the numbers on the conversion so that the top and the bottom amounts are EQUAL, but in different units.
6. Multiply and divide the units (Cancel).7. If the units are not the ones you want for your
answer, make more conversions until you reach that point.
8. Multiply and divide the numbers. Don’t forget “Please Excuse My Dear Aunt Sally”! (order of operations)
Sample Problem
• You have $7.25 in your pocket in You have $7.25 in your pocket in quarters. How many quarters do you quarters. How many quarters do you have?have?
7.25 dollars 4 quarters7.25 dollars 4 quarters
1 dollar1 dollar X = 29 quarters= 29 quarters
You Try This One!You Try This One!
If Jacob stands on If Jacob stands on Spencer’s shoulders, Spencer’s shoulders, they are two and a half they are two and a half yards high. How many yards high. How many feet is that?feet is that?
Learning Check
A rattlesnake is 2.44 m long. How A rattlesnake is 2.44 m long. How long is the snake in cm?long is the snake in cm?
a) a) 2440 cm2440 cm
b)b) 244 cm244 cm
c)c) 24.4 cm24.4 cm
Solution
A rattlesnake is 2.44 m long. How A rattlesnake is 2.44 m long. How long is the snake in cm?long is the snake in cm?
b)b) 244 cm244 cm
2.44 m x 2.44 m x 100 cm 100 cm = 244 cm= 244 cm
1 m
Learning Check
How many seconds are in 1.4 days?
Unit plan: days hr min seconds
1.4 days x 24 hr x ?? 1 day
Wait a minute!
What is What is wrongwrong with the following setup? with the following setup?
1.4 day x 1.4 day x 1 day 1 day x x 60 min 60 min x x 60 sec 60 sec
24 hr 1 hr 1 min24 hr 1 hr 1 min
English and Metric English and Metric ConversionsConversions
• If you know ONE conversion for If you know ONE conversion for each type of measurement, you each type of measurement, you can convert anything!can convert anything!
• You must You must memorizememorize and use these and use these conversions:conversions:
–Mass: 454 grams = 1 poundMass: 454 grams = 1 pound
–Length: 2.54 cm = 1 inchLength: 2.54 cm = 1 inch
–Volume: 0.946 L = 1 quartVolume: 0.946 L = 1 quart
Learning CheckLearning Check
An adult human has 4.65 L of blood. How An adult human has 4.65 L of blood. How many gallons of blood is that?many gallons of blood is that?
Unit plan: L qt gallon
Equalities: 1 quart = 0.946 L
1 gallon = 4 quarts
Your Setup:
Equalities
State the same measurement in two different State the same measurement in two different unitsunits
lengthlength
10.0 in.10.0 in.
25.4 cm25.4 cm
Steps to Problem SolvingSteps to Problem Solving
Read problemRead problem Identify data Identify data Make a unit plan from the initial unit to the Make a unit plan from the initial unit to the
desired unitdesired unit Select conversion factorsSelect conversion factors Change initial unit to desired unitChange initial unit to desired unit Cancel units and checkCancel units and check Do math on calculator Do math on calculator Give an answer using significant figuresGive an answer using significant figures
Dealing with Two UnitsDealing with Two Units
If your pace on a treadmill is 65 meters If your pace on a treadmill is 65 meters per minute, how many seconds will it per minute, how many seconds will it take for you to walk a distance of 8450 take for you to walk a distance of 8450 feet?feet?
What about Square and Cubic What about Square and Cubic units? units?
• Use the conversion factors you already Use the conversion factors you already know, but when you square or cube the know, but when you square or cube the unit, don’t forget to cube the number unit, don’t forget to cube the number also!also!
• Best way: Square or cube the ENITRE Best way: Square or cube the ENITRE conversion factorconversion factor
• Example: Convert 4.3 cmExample: Convert 4.3 cm33 to mm to mm33
4.3 cm4.3 cm33 10 mm 10 mm 33
1 cm 1 cm ( ) =
4.3 cm4.3 cm33 10 1033 mm mm33
1133 cm cm33
= 4300 mm3
Learning CheckLearning Check
• A Nalgene water A Nalgene water bottle holds 1000 bottle holds 1000 cmcm33 of dihydrogen of dihydrogen monoxide monoxide (DHMO). How (DHMO). How many cubic many cubic decimeters is decimeters is that?that?
SolutionSolution
1000 cm1000 cm33 1 dm 1 dm 33
10 cm10 cm( ) = 1 dm= 1 dm33
So, a dmSo, a dm33 is the same as a Liter ! is the same as a Liter !
A cmA cm33 is the same as a milliliter. is the same as a milliliter.
Reading a MeterstickReading a Meterstick
. l. l22. . . . I . . . . I. . . . I . . . . I33 . . . .I . . . . I . . . .I . . . . I44. . cm. . cm
First digit (known)First digit (known) = 2 = 2 2.?? cm2.?? cm
Second digit (known)Second digit (known) = 0.7 = 0.7 2.7? cm2.7? cm
Third digit (estimated) between 0.05- 0.07Third digit (estimated) between 0.05- 0.07
Length reportedLength reported == 2.75 cm 2.75 cm
oror 2.74 cm 2.74 cm
oror 2.76 cm2.76 cm
Known + Estimated DigitsKnown + Estimated Digits
In 2.76 cm…In 2.76 cm…
• Known digitsKnown digits 2 andand 7 are 100% certainare 100% certain
• The third digit 6 is estimated (uncertain)The third digit 6 is estimated (uncertain)
• In the reported length, all three digits In the reported length, all three digits (2.76 cm) are significant including the (2.76 cm) are significant including the estimated oneestimated one
Learning CheckLearning Check
. l8. . . . I . . . . I9. . . .I . . . . I10. . cm
What is the length of the line?What is the length of the line?
1) 9.6 cm 1) 9.6 cm
2) 9.62 cm 2) 9.62 cm
3) 9.63 cm3) 9.63 cm
How does your answer compare with your How does your answer compare with your
neighbor’s answer? Why or why not?neighbor’s answer? Why or why not?
Zero as a Measured NumberZero as a Measured Number
. l3. . . . I . . . . I4 . . . . I . . . . I5. . cm
What is the length of the line?What is the length of the line?
First digitFirst digit 5.?? cm5.?? cm
Second digitSecond digit 55.0? cm.0? cm
Last (estimated) digit isLast (estimated) digit is 5.05.00 cm0 cm
DENSITYExample:
Calculate the density of a metal if 356 grams occupies a volume of 43.1 mL.
mLg26.8
milliliter1.43
grams356
Volume
Massd
DENSITYDENSITY - an important - an important and useful physical propertyand useful physical property
Density mass (g)volume (cm3)
Density mass (g)volume (cm3)
Mercury
13.6 g/cm13.6 g/cm33 21.5 g/cm21.5 g/cm33
Aluminum
2.7 g/cm2.7 g/cm33
Platinum
ProblemProblem A piece of copper has a A piece of copper has a mass of 57.54 g. It is 9.36 cm long, mass of 57.54 g. It is 9.36 cm long, 7.23 cm wide, and 0.95 mm thick. 7.23 cm wide, and 0.95 mm thick. Calculate density (g/cmCalculate density (g/cm33).).
Density mass (g)volume (cm3)
Density mass (g)volume (cm3)
StrategyStrategy1. Get dimensions in common units.1. Get dimensions in common units.
2.2. Calculate volume in cubic centimeters. Calculate volume in cubic centimeters.
3. Calculate the density.3. Calculate the density.
SOLUTIONSOLUTION
1. Get dimensions in common units.1. Get dimensions in common units.
2.2. Calculate volume in cubic centimeters. Calculate volume in cubic centimeters.
3. Calculate the density.3. Calculate the density.
0.95 mm • 1cm
10 mm = 0.095 cm
57.54 g
6.4 cm3 = 9.0 g / cm3
(9.36 cm)(7.23 cm)(0.095 cm) = 6.4 cm(9.36 cm)(7.23 cm)(0.095 cm) = 6.4 cm33
Note only 2 significant figures in the answer!Note only 2 significant figures in the answer!
PROBLEM: Mercury (Hg) has a density PROBLEM: Mercury (Hg) has a density of 13.6 g/cmof 13.6 g/cm33. What is the mass of 95 mL . What is the mass of 95 mL of Hg in grams? In pounds?of Hg in grams? In pounds?
PROBLEM: Mercury (Hg) has a density PROBLEM: Mercury (Hg) has a density of 13.6 g/cmof 13.6 g/cm33. What is the mass of 95 mL . What is the mass of 95 mL of Hg in grams? In pounds?of Hg in grams? In pounds?
StrategyStrategy
1.1. Use density to calc. mass (g) from Use density to calc. mass (g) from volume.volume.
2.2. Convert mass (g) to mass (lb)Convert mass (g) to mass (lb)
Need to know conversion factorNeed to know conversion factor
= 454 g / 1 lb= 454 g / 1 lb
PROBLEM: Mercury (Hg) has a density of PROBLEM: Mercury (Hg) has a density of 13.6 g/cm13.6 g/cm33. What is the mass of 95 mL of Hg?. What is the mass of 95 mL of Hg?PROBLEM: Mercury (Hg) has a density of PROBLEM: Mercury (Hg) has a density of 13.6 g/cm13.6 g/cm33. What is the mass of 95 mL of Hg?. What is the mass of 95 mL of Hg?
First, note thatFirst, note that 1 cm1 cm33 = 1 mL = 1 mL
1.1. Convert volume to massConvert volume to mass
PROBLEM: Mercury (Hg) has a density of 13.6 PROBLEM: Mercury (Hg) has a density of 13.6 g/cmg/cm33. What is the mass of 95 mL of Hg?. What is the mass of 95 mL of Hg?PROBLEM: Mercury (Hg) has a density of 13.6 PROBLEM: Mercury (Hg) has a density of 13.6 g/cmg/cm33. What is the mass of 95 mL of Hg?. What is the mass of 95 mL of Hg?
95 cm3 • 13.6 g
cm3 = 1.3 x 103 g
1.3 x 103 g • 1 lb
454 g = 2.8 lb
2.2. Convert mass (g) to mass (lb)Convert mass (g) to mass (lb)
Learning CheckLearning Check
Osmium is a very dense metal. What is its Osmium is a very dense metal. What is its
density in g/cmdensity in g/cm3 3 if 50.00 g of the metal occupiesif 50.00 g of the metal occupies
a volume of 2.22cma volume of 2.22cm33??
1) 2.25 g/cm1) 2.25 g/cm33
2)2) 22.5 g/cm22.5 g/cm33
3)3) 111 g/cm111 g/cm33
Solution
2) Placing the mass and volume of the osmium 2) Placing the mass and volume of the osmium metal into the density setup, we obtainmetal into the density setup, we obtain
D = D = massmass = = 50.00 g 50.00 g = = volumevolume2.22 cm2.22 cm33
= 22.522522 g/cm= 22.522522 g/cm3 3 == 22.5 g/cm22.5 g/cm33
Volume DisplacementVolume Displacement
A solid displaces a matching volume of A solid displaces a matching volume of water when the solid is placed in water.water when the solid is placed in water.
33 mL33 mL25 mL 25 mL
Learning CheckLearning Check
What is the density (g/cmWhat is the density (g/cm33) of 48 g of a metal if ) of 48 g of a metal if the metal raises the level of water in a graduated the metal raises the level of water in a graduated cylinder from 25 mL to 33 mL? cylinder from 25 mL to 33 mL?
1) 0.2 g/ cm1) 0.2 g/ cm33 2) 6 g/m 2) 6 g/m33 3) 252 g/cm3) 252 g/cm33
33 mL33 mL
25 mL25 mL
Learning CheckLearning Check
Which diagram represents the liquid layers in the Which diagram represents the liquid layers in the cylinder?cylinder?
(K) Karo syrup (1.4 g/mL), (V) vegetable oil (0.91 (K) Karo syrup (1.4 g/mL), (V) vegetable oil (0.91 g/mL,) (W) water (1.0 g/mL)g/mL,) (W) water (1.0 g/mL)
1)1) 2) 2) 3) 3)
K
K
W
W
W
V
V
V
K
Learning CheckLearning Check
The density of octane, a component of The density of octane, a component of gasoline, is 0.702 g/mL. What is the gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane?mass, in kg, of 875 mL of octane?
1) 0.614 kg1) 0.614 kg
2) 614 kg2) 614 kg
3) 1.25 kg3) 1.25 kg
Learning CheckLearning Check
If blood has a density of 1.05 g/mL, how If blood has a density of 1.05 g/mL, how many liters of blood are donated if 575 g many liters of blood are donated if 575 g of blood are given?of blood are given?
1) 1) 0.548 L0.548 L
2) 2) 1.25 L1.25 L
3) 3) 1.83 L1.83 L
Learning CheckLearning Check
A group of students collected 125 empty A group of students collected 125 empty aluminum cans to take to the recycling center. aluminum cans to take to the recycling center. If 21 cans make 1.0 pound of aluminum, how If 21 cans make 1.0 pound of aluminum, how many liters of aluminum (D=2.70 g/cmmany liters of aluminum (D=2.70 g/cm33) are ) are obtained from the cans?obtained from the cans?
1) 1.0 L1) 1.0 L 2) 2.0 L2) 2.0 L 3) 4.0 L3) 4.0 L
SPECIFIC GRAVITY
mlg
000.1
d
d
d.gr.sp cetansubs
water
cetansubs
Note: Specific Gravity has NO units.
SPECIFIC GRAVITY
Example:
The density of gold is 19.3 g/mL. What is its specific gravity?
ml
g3.19dAu
3.19
mLg
000.1
mLg
3.19
mlg
000.1
d.gr.sp Au
Temperature ScalesTemperature Scales• FahrenheitFahrenheit
• CelsiusCelsius
• KelvinKelvin
Anders Celsius1701-1744
Lord Kelvin(William Thomson)1824-1907
Temperature ScalesTemperature Scales
Notice that 1 kelvin = 1 degree Celsius1 kelvin = 1 degree Celsius
Boiling point Boiling point of waterof water
Freezing point Freezing point of waterof water
CelsiusCelsius
100 ˚C100 ˚C
0 ˚C0 ˚C
100˚C100˚C
KelvinKelvin
373 K373 K
273 K273 K
100 K100 K
FahrenheitFahrenheit
32 ˚F32 ˚F
212 ˚F212 ˚F
180˚F180˚F
Calculations Calculations Using Using TemperatureTemperature
• Generally require temp’s in Generally require temp’s in kelvinskelvins
• T (K) = t (˚C) + 273.15T (K) = t (˚C) + 273.15
• Body temp = 37 ˚C + 273 = 310 KBody temp = 37 ˚C + 273 = 310 K
• Liquid nitrogen = -196 ˚C + 273 = 77 KLiquid nitrogen = -196 ˚C + 273 = 77 K
• Generally require temp’s in Generally require temp’s in kelvinskelvins
• T (K) = t (˚C) + 273.15T (K) = t (˚C) + 273.15
• Body temp = 37 ˚C + 273 = 310 KBody temp = 37 ˚C + 273 = 310 K
• Liquid nitrogen = -196 ˚C + 273 = 77 KLiquid nitrogen = -196 ˚C + 273 = 77 K
Fahrenheit FormulaFahrenheit Formula
180°F180°F = = 9°F 9°F == 1.8°F 1.8°F 100°C 100°C 5°C 5°C 1°C1°C
Zero point: 0°C = 32°FZero point: 0°C = 32°F
°F = 9/5 °C + 32°F = 9/5 °C + 32
Celsius FormulaCelsius Formula
Rearrange to find T°CRearrange to find T°C
°F °F = = 9/5 °C + 329/5 °C + 32
°F - 32 = °F - 32 = 9/5 °C ( +32 - 32)9/5 °C ( +32 - 32)
°F - 32°F - 32 = = 9/5 °C9/5 °C
9/5 9/5 9/5 9/5
(°F - 32) * 5/9 = (°F - 32) * 5/9 = °C°C
Temperature ConversionsTemperature Conversions
A person with hypothermia has a body A person with hypothermia has a body temperature of 29.1°C. What is the body temperature of 29.1°C. What is the body temperature in °F? temperature in °F?
°F °F = = 9/5 (29.1°C) + 329/5 (29.1°C) + 32
= = 52.4 + 3252.4 + 32
= = 84.4°F84.4°F
Learning CheckLearning Check
The normal temperature of a chickadee is The normal temperature of a chickadee is 105.8°F. What is that temperature in °C?105.8°F. What is that temperature in °C?
1) 73.8 °C 1) 73.8 °C
2) 58.8 °C2) 58.8 °C
3) 41.0 °C3) 41.0 °C
Learning Check
Pizza is baked at 455°F. What is that in °C?
1) 437 °C1) 437 °C
2) 235°C2) 235°C
3) 221°C3) 221°C
Scientific MethodScientific Method
1.1. State the problem clearly.State the problem clearly.2.2. Gather information.Gather information.3.3. Form a _______________.Form a _______________.4.4. Test the hypothesis.Test the hypothesis.5.5. Evaluate the data to form a conclusion.Evaluate the data to form a conclusion.
If the conclusion is valid, then it becomes a If the conclusion is valid, then it becomes a theorytheory. If the theory is found to be true over . If the theory is found to be true over along period of time (usually 20+ years) with along period of time (usually 20+ years) with no counter examples, it may be considered a no counter examples, it may be considered a lawlaw..
6. Share the results.6. Share the results.