Post on 26-Dec-2015
Chapter 3Chapter 3Observation,Observation,MeasurementMeasurement
and Calculationsand Calculations
MeasurementMeasurement
MeasurementMeasurement – a quantity that has both a number and a unit.
• Measurements are fundamental to the experimental sciences
• Units typically used in the sciences are the International System of Measurements (SI)
In science, we deal with some In science, we deal with some very very LARGELARGE numbers: numbers:
1 mole = 6020000000000000000000001 mole = 602000000000000000000000
In science, we deal with some In science, we deal with some very very SMALLSMALL numbers: numbers:
Mass of an electron = Mass of an electron = 0.000000000000000000000000000000091 kg0.000000000000000000000000000000091 kg
Scientific NotationScientific Notation
Imagine the difficulty of Imagine the difficulty of calculating the mass of 1 mole calculating the mass of 1 mole of electrons!of electrons!
0.00000000000000000000000000000000.000000000000000000000000000000091 kg 91 kg x 602000000000000000000000x 602000000000000000000000
???????????????????????????????????
Scientific NotationScientific Notation
Scientific Scientific Notation:Notation:
A method of representing very large A method of representing very large or very small numbers in the or very small numbers in the form: form:
M x 10M x 10n n
MM is a number between is a number between 11 and and 10 10 nn is an integer is an integer
2 500 000 000
Step #1: Insert an understood decimal pointStep #1: Insert an understood decimal point
.
Step #2: Decide where the decimal Step #2: Decide where the decimal must end must end up so that one number is to its up so that one number is to its leftleftStep #3: Count how many places you Step #3: Count how many places you bounce bounce the decimal pointthe decimal point
123456789
Step #4: Re-write in the form M x 10Step #4: Re-write in the form M x 10nn
2.5 x 102.5 x 1099
The exponent is the number of places we moved the decimal.
0.00005790.0000579
Step #2: Decide where the decimal Step #2: Decide where the decimal must end must end up so that one number is to its up so that one number is to its leftleftStep #3: Count how many places you Step #3: Count how many places you bounce bounce the decimal pointthe decimal pointStep #4: Re-write in the form M x 10Step #4: Re-write in the form M x 10nn
1 2 3 4 5
5.79 x 105.79 x 10-5-5
The exponent is negative because the number we started with was less than 1.
PERFORMING PERFORMING CALCULATIONCALCULATION
S IN S IN SCIENTIFIC SCIENTIFIC NOTATIONNOTATION
ADDITION AND ADDITION AND SUBTRACTIONSUBTRACTION
ReviewReview::Scientific notation Scientific notation expresses a number in the expresses a number in the form:form: M x 10M x 10nn
1 1 M M 1010
n is an n is an integerinteger
4 x 104 x 1066
+ 3 x 10+ 3 x 1066
IFIF the exponents the exponents are the same, we are the same, we simply add or simply add or subtract the subtract the numbers in front numbers in front and bring the and bring the exponent down exponent down unchanged.unchanged.
77 x 10x 1066
4 x 104 x 1066
- 3 x 10- 3 x 1066
The same holds The same holds true for true for subtraction in subtraction in scientific scientific notation.notation.
11 x 10x 1066
4 x 104 x 1066
+ 3 x 10+ 3 x 1055
If the exponents If the exponents are NOT the are NOT the same, we must same, we must move a decimal to move a decimal to makemake them the them the same.same.
4.00 x 104.00 x 1066
+ + 3.00 x 103.00 x 1055 + + .30 x 10.30 x 1066
4.304.30 x 10x 1066
Move the Move the decimal decimal on the on the smallersmaller number!number!
4.00 x 104.00 x 1066
A Problem for A Problem for you…you…
2.37 x 102.37 x 10-6-6
+ 3.48 x 10+ 3.48 x 10-4-4
2.37 x 102.37 x 10-6-6
+ 3.48 x 10+ 3.48 x 10-4-4
Solution…Solution…002.37 x 10002.37 x 10--
66
+ 3.48 x 10+ 3.48 x 10-4-4
Solution…Solution…0.0237 x 100.0237 x 10-4-4
3.5037 x 103.5037 x 10-4-4
Scientific Notation Calculation Summary
Adding and SubtractingYou must express the numbers as the same power
of 10. This will often involve changing the decimal place of the coefficient.
(2.0 x 106) + ( 3.0 x 107)
(0.20 x 107) + (3.0 x 107) = 3.20 x 107
(4.8 x 105) - ( 9.7 x 104)
(4.8 x 105) - ( 0.97 x 105) = 3.83 x 105
Scientific NotationMultiplying
Multiply the coefficients and add the exponents
(xa) (xb) = x a + b
(2.0 x 106) ( 3.0 x 107) = 6.0 x 1013
Dividing(xa) / (xb) = x a - b
(2.0 x 106) / ( 3.0 x 107) = 0.67 x 10-1
Divide the coefficients and subtract the exponents
Nature of MeasurementNature of Measurement
•
Part 1 - Part 1 - number number Part 2 - Part 2 - scale (unit) scale (unit)
Examples: Examples: 2020 grams grams
6.63 x 106.63 x 10-34-34 Joule secondsJoule seconds
Measurement - quantitative Measurement - quantitative observation observation consisting of 2 partsconsisting of 2 parts
Uncertainty in Uncertainty in MeasurementMeasurement
A digit that must be A digit that must be estimatedestimated is called is called uncertainuncertain. A . A measurementmeasurement always has always has some degree of some degree of uncertainty.uncertainty.
Precision and AccuracyPrecision and AccuracyAccuracyAccuracy – measure of how close a – measure of how close a measurement comes to the actual or measurement comes to the actual or truetrue value of whatever is being measured.value of whatever is being measured.
PrecisionPrecision – measure of how close a series of – measure of how close a series of measurements are to one another.measurements are to one another.
Neither accurate nor
precise
Precise but not accurate
Precise AND accurate
Why Is there Uncertainty?Why Is there Uncertainty? Measurements are performed with instruments No instrument can read to an infinite number of decimal places
Which of these balances has the greatest uncertainty in measurement?
Determining ErrorDetermining Error
Accepted ValueAccepted Value – – the correct value based on reliable references
Error(can be +or-)=experimental value – accepted value
Percent error = absolute value of error x 100% accepted value
Experimental ValueExperimental Value – – the value measured in the lab
Significant Figures in Significant Figures in MeasurementMeasurement
In a supermarket, you can use the scales to measure the weight of produce.
If you use a scale that is calibrated in 0.1 lb intervals, you can easily read the scale to the nearest tenth of a pound.
You can also estimate the weight to the nearest hundredth of a pound by noting the position of the pointer between calibration marks.
Significant Figures in Significant Figures in MeasurementMeasurement
If you estimate a weight that lies between 2.4 lbs and 2.5 lbs to be 2.462.46 lbs, the number in this estimated measurement has three digits.
The first two digits (2 and 4 ) are known with certainty.
The rightmost digit (6) has been estimated and involves some uncertainty.
Significant Figures in Significant Figures in MeasurementMeasurement
Significant figures in a measurement include all of the digits that are know, plus a last digit that is estimated.
Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significant figures in the values used in the calculation.
Rules for Counting Rules for Counting SignificantSignificant
FiguresFigures
Nonzero integersNonzero integers always count always count as significant figures. as significant figures.
34563456 hashas
44 sig figs.sig figs.
Rules for Counting Rules for Counting Significant FiguresSignificant Figures
Leading zerosLeading zeros do not count as do not count as significant figuressignificant figures..
0.04860.0486 has has
33 sig figs. sig figs.
Rules for Counting Rules for Counting Significant FiguresSignificant Figures
Zeros at the end of a number Zeros at the end of a number and to the right of a decimal and to the right of a decimal point point are always significant.are always significant.
9.0009.000 has has
44 sig figs sig figs
1.010 1.010 hashas
4 4 sig figssig figs
Rules for Counting Rules for Counting Significant FiguresSignificant Figures
Captive zerosCaptive zeros always count always count as as
significant figures.significant figures.
16.0716.07 has has
44 sig figs. sig figs.
Rules for Counting Rules for Counting Significant FiguresSignificant Figures
Zeros at the rightmost end Zeros at the rightmost end that lie at the left of an that lie at the left of an understood decimal pointunderstood decimal point are are not significant. not significant.
7000 7000 hashas
11 sig fig sig fig
2721027210 has has
44 sig figs sig figs
Rules for Counting Rules for Counting Significant FiguresSignificant Figures
Exact numbersExact numbers have an infinite have an infinite number of significant figures. number of significant figures.
11 inch = inch = 2.542.54 cm, exactlycm, exactly
Rules for Significant Figures in Rules for Significant Figures in Mathematical OperationsMathematical Operations
Multiplication and DivisionMultiplication and Division:: # sig # sig figs in the result equals the number figs in the result equals the number in the in the least precise measurementleast precise measurement used in the calculation. used in the calculation.
6.38 x 2.0 = 6.38 x 2.0 =
12.76 12.76 13 (2 sig figs)13 (2 sig figs)
Rules for Significant Figures Rules for Significant Figures in Mathematical Operationsin Mathematical Operations
Addition and SubtractionAddition and Subtraction: The : The number of decimal places in the number of decimal places in the result equals the number of decimal result equals the number of decimal places in the least precise places in the least precise measurement. measurement.
6.8 + 11.934 = 6.8 + 11.934 =
18.734 18.734 18.7 ( 18.7 (3 sig figs3 sig figs))
Sig Fig Practice #1Sig Fig Practice #1How many significant figures in each of the following?
1.0070 m
5 sig figs
17.10 kg 4 sig figs
100,890 L 5 sig figs
3.29 x 103 s 3 sig figs
0.0054 cm 2 sig figs
3,200,000 2 sig figs
Sig Fig Practice #2Sig Fig Practice #2
3.24 m x 7.0 m
Calculation Calculator says: Answer
22.68 m2 23 m2
100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 4.22 g/cm3
0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2
710 m ÷ 3.0 s 236.6666667 m/s 240 m/s
1818.2 lb x 3.23 ft 5872.786 lb·ft 5870 lb·ft
1.030 g ÷ 2.87 mL 2.9561 g/mL 2.96 g/mL
Sig Fig Practice #3Sig Fig Practice #3
3.24 m + 7.0 m
Calculation Calculator says: Answer
10.24 m 10.2 m
100.0 g - 23.73 g 76.27 g 76.3 g
0.02 cm + 2.371 cm 2.391 cm 2.39 cm
713.1 L - 3.872 L 709.228 L 709.2 L
1818.2 lb + 3.37 lb 1821.57 lb 1821.6 lb
2.030 mL - 1.870 mL 0.16 mL 0.160 mL
Questions
1) 78ºC, 76ºC, 75ºC 2) 77ºC, 78ºC, 78ºC3) 80ºC, 81ºC, 82ºC
The sets of measurements were made of the boiling point of a liquid under similar conditions. Which set is the most precise?
Set 2 – the three measurements are closest together.
What would have to be known to determine which set is the most accurate?
The accepted value of the liquid’s boiling point
QuestionsHow do measurements relate to experimental
science?
Making correct measurements is fundamental to the experimental sciences.
How are accuracy and precision evaluated?
Accuracy is the measured value compared to the correct values. Precision is comparing more than one measurement.
QuestionsWhy must a given measurement always be reported to the
correct number of significant figures?
The significant figures in a calculated answer often depend on the number of significant figures of the measurements used in the calculation.
How does the precision of a calculated answer compare to the precision of the measurements used to obtain it?
A calculated answer cannot be more precise than the least precise measurement used in the calculation.
Question
A technician experimentally determined the boiling point of octane to be 124.1ºC. The actual boiling point of octane is 125.7ºC. Calculate the error and the percent error.
Error = experimental value – accepted value
error = 124.1ºC – 125.7ºC = -1.6ºC
Absolute error / accepted value x 100%
% error = -1.6ºC / 125.7ºC x 100% = 1.3%
Question
Determine the number of significant figures in each of the following
11 soccer playerunlimited
0.070 020 meter5
10,800 meters3
5.00 cubic meters3
QuestionSolve the following and express each answer in scientific
notation and to the correct number of significant figures.
(5.3 x 104) + (1.3 x 104)6.6 x 104
(7.2 x 10-4) / (1.8 x 103)4.0 x 10-7
(104)(10-3) (106)107
(9.12 x 10-1) - (4.7 x 10-2)8.65 x 10-1
(5.4 x 104) (3.5 x 109)18.9 x 1013 or 1.9 x 10 14
End of section 3.1
International Systems of Units• The standards of measurement used in science are
those of the metric system
• All metric units are based on multiples of 10
• Metric system was originally establish in France in 1795
• The International System of Units (SI) is a revised version of the metric system.
• The SI comes from the French name, le Systeme International d’Unites.
• The SI was adopted by international agreement in 1960.
International Systems of Units
There are seven SI base units
SI Base Units
Quantity SI base unit SymbolLength Meter m
Mass kilogram kg
Temperature kelvin K
Time second s
Amount mole mol
Luminous intensity candela cd
Electric current ampere A
Metric Prefixes
Meter (m)
Deka(da) 101
Hecto (hm) 102
Kilo (k) 103
Deci (d) 10-1
Centi (c) 10-2
Milli (m) 10-3
Micro (µ) 10-6
Nano (nm) 10-9
Pico (pm) 10-12
Mega (M)
left
right
Metric Conversions
1.0 decimeter (dm) = ? hectometers
0.001 hectometer (hm)
2.5 hectometer (hm) = ? millimeters
250,000 millimeters (mm)
9.7 centimeters (cm) = ? kiometers
0.000097 kilometers (km)
7.4 grams (g) = ? Milligrams (mg)
7400 milligrams (mg)
Other Common Conversions
1 cm3 = 1ml
1dm3 = 1L
1 inch = 2.54 cm
1kg = 2.21 lb
454 g = 1 lb
4.18 J = 1 cal
1 mol = 6.02 x 1023 pieces
1 GA = 3.79 L
Units of Length
metermeter – the basic SI unit of length or linear measure
Common metric units of length include the centimeter (cm), meter (m), and kilometer (km)
Units of Volume
VolumeVolume -the space occupied by any sample of matter
Volume (cube or rectangle) = length x width x height
The SI unit of volume is the amount of space occupied by a cube that is 1m along each edge. (mm33)
Liter Liter (L) – non SI unit – the volume of a cube that is 10cm along each edge (1000cm1000cm33)
The units milliliter and cubic centimeter are used interchangeably.
1 cm3 = 1ml
1dm3 = 1L
Units of MassCommon metric units of mass include the kilogram,
gram, milligram and microgram.
Weight Weight – is a force that measures the pull on a given mass by gravity.
Weight is a measure of force and is different than mass.
MassMass – measure of the quantity of matter.
Although, the weight of an object can change with its location, its mass remains constant regardless of its location.
Objects can become weightless, but not massless
Units of Temperature
TemperatureTemperature – measure of how hot or cold an object is.
The objects temperature determines the direction of heat transfer.
When two objects at different temperatures are in contact, heat moves from the object at the higher temperature to the object at the lower temperature.
Scientist use two equivalent units of temperature, the degree Celsius and the Kelvin.
Units of Temperature
The Celsius scale of the metric system is named after Swedish astronomer Anders Celsius.
The Celsius scale sets the freezing point of water at 0ºC and the boiling point of water at 100ºC
The Kelvin scale is named for Lord Kelvin, a Scottish physicist and mathematician.
On the Kelvin scale, the freezing point of water is 273.15 kelvins (K), & the boiling point is 373.15 K.
With the Kelvin scale the degree (º) sign is not used
Units of Temperature
A change of 1 º on the Celsius scale is equivalent to one kelvin on the Kelvin scale.
The zero point on the Kelvin scale, 0K, or absolute zero, is equal to -273.15º C.
K = ºC + 273
ºC = K - 273
.
Units of Energy
EnergyEnergy – the capacity to do work or to produce heat.
The joule and the calorie are common units of energy.
The jouleThe joule (J) is the SI unit of energy named after the English physicist James Prescott Joule.
1 calorie1 calorie (cal) - is the quanity of heat that raises the temperature of 1 g of pure water by 1ºC.
1 J = 0.2390 cal
1 cal = 4.184 J
End of Section 3.2
Conversion Factors1 dollar = 4 quarters = 10 dimes = 20 nickels = 100 pennies
Different ways to express the same amount of money
1 meter =10 decimeters =100 centimeters =1000 millimeters
Different ways to express length
Whenever two measurements are equivalent, a ratio of the two measurements will equal 1.
1 m = 100 cm = 1 1m
Conversion factor
Conversion Factors
Conversion factor – a ratio of equivalent measurements
100 cm / 1 m
1000 mm / 1 m
The measurement on the top is equivalent to the measurement on the bottom
Read “one hundred centimeters per meter” and “1000 millimeters per meter”
Smaller number 1 m larger unitLarger number 100 cm smaller unit
Conversion FactorsWhen a measurement is multiplied by a conversion factor,
the numerical value is generally changed, but the actual size of the quantity measured remains the same.
Conversion factors within a system of measurements are defined quantities or exact quantities.
Therefore, they have an unlimited number of significant figures and do not affect the rounding of a calculated answer.
How many significant figures does a conversion factor within a system of measurements have?
Dimensional AnalysisDimensional analysis – a way to analyze and solve
problems using the units, or dimensions, of the measurements.
How many minutes are there in exactly one week?
60 minutes = 1 hour 24 hours = 1 day7 days = 1 week
1 week 7 days 24 hours 60 minutes = 10,080 min 1 week 1 day 1 hour
1.0080 x 104 min
Dimensional AnalysisHow many seconds are in exactly a 40-hr work
week?
60 minutes = 1 hour 24 hours = 1 day7 days = 1 week 60 seconds = 1
minute
40 hr 60 min 60 sec = 144,000 s 1 hr 1 min
1.44000 x 105 s
Dimensional AnalysisAn experiment requires that each student use an
8.5 cm length of Mg ribbon. How many students can do the experiment if there is a 570 cm length of Mg ribbon available?
570 cm ribbon 1 student = 67 students 8.5 cm ribbon
2 sig figs
Dimensional AnalysisA 1.00º increase on the Celsius scale is equivalent to a 1.80º increase on the Fahrenheit scale. If a temperature increases by 48.0ºC, what is the corresponding temperature increase on the Fahrenheit scale?
48.0ºC 1.80ºF = 86.4ºF 1.00ºC
A chicken needs to be cooked 20 minutes for each pound it weights. How long should the chicken be cooked if it weighs 4.5 pounds?
4.5 lb 20 min = 90 min lb
Dimensional AnalysisGold has a density of 19.3 g/cm3. What is the density in kg/m3
19.3 g 1 kg 1 x 106 cm3 = 1.93 x 104 kg / m3 cm3 1000 g m3
There are 7.0 x 106 red blood cell (RBC) in 1.0 mm3 of blood. How many red blood cells are in 1.0 L of blood?
7.0 x 106 RBC 1 x 106 mm3 1 dm3 = 7.0 x 1012
1.0 mm3 dm3 1 L
Dimensional Analysis1.00 L of neon gas contains 2.69 x 1022 neon atoms. How many neon atoms are in 1.00mm3 of neon gas under the same conditions?
2.69 x 1022 atoms 1 L dm3
1.00 L 1 dm3 1 x 106 mm3
2.69 x 1016 atoms in 1.00mm3 of gas
QuestionsWhat conversion factor would you use to convert between these pairs of units?
Minutes to hours
1 hour / 60 minutes
grams to milligrams
1000 mg / 1 g
Cubic decimeters to milliliters
1000 ml / 1 dm3
QuestionsAn atom of gold has a mass of 3.271 x 10-22g. How many atoms of gold are in 5.00 g of gold?
1.53 x 1022 atoms of gold
Light travels at a speed of 3.00 x 1010 cm/sec. What is the speed of light in km/hour?
1.08 x 109 km/hr
QuestionsConvert the following. Express your answers in scientific notation.
7.5 x 104 J to kJ
7.5 x 101 kJ
3.9 x 105 mg to dg
3.9 x 103dg
2.21 x 10-4 dL to µL
2.21 x 101µL
QuestionsMake the following conversions. Express your answers in standard exponential form.
14.8 g to µg
1.48 x 107 µg
3.75 x 10-3 kg to g
3.72 g
66.3 L to cm3
6.63 x 104 cm3
End of Section 3.3
DensityIf a piece of led and a feather of the same volume are weighted, the lead would have a greater mass than the feather.
It would take a much larger volume of feather to equal the mass of a given volume of lead.
Density = mass / volumeD = m / v
Mass is a extensive property (a property that depends on the size of the sample)
Density is an intensive property (depends on the composition of a substance, not on the size of the sample)
Density
A helium filled balloon rapidly rises to the ceiling when released.
Whether a gas-filled balloon will sink or rise when released depends on how the density of the gas compares with the density of air.
Helium is less dense than air, so a helium filled balloon rises.
Density and TemperatureThe volume of most substances increase as the temperature increases.
The mass remains the same despite the temperature and volume changes.
So if the volume changes with temperature while the mass remains constant, then the density must also change with temperature.
The density of a substance generally decreases as its temperature increases. (water is the exception: ice floats because it is less dense than liquid water)
Questions
A student finds a shiny piece of metal that she thinks is aluminum. In the lab, she determines that the metal has a volume of 245cm3 and a mass of 612g. Was is the density? Is it aluminum?
D = 612g / 245cm3 = 2.50g/cm3
D of aluminum is 2.70 g/cm3; no it is not aluminum
A bar of silver has a mass of 68.0 g and a volume of 6.48 cm3. What is the density?
D = 68.0g / 6.48 cm3 = 10.5 g/cm3
Questions
The density of boron is 2.34 g/cm3. Change 14.8 g of boron to cm3 of boron.
D = m / v or v = m / D
V = 14.8 g cm3 = 6.32 cm 3
2.34 g
Convert 4.62 g of mercury to cm3 by using the density of mercury -13.5 g/cm3.
V = 46.2 g cm3 = 0.342 cm 3
13.5 g
Density
D = m / v
v = m / D
m = D · v
End of Chapter 3