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Chapter 2: Data AnalysisSection 1: Units of measurement
Intro problems: D = m V
• Calculate the density of a piece of bone with a mass of 3.8 g and a volume of 2.0 cm3
• A spoonful of sugar with a mass of 8.8 grams is poured into a 10 mL graduated cylinder. The volume reading is 5.5 mL. What is the density of the sugar?
Not so long ago…….
People used all kinds of units to describe measurements:
Their feet
Sundials
The length of their arm
Needless to say, this led to much confusion!
• Scientist needed a way to report their findings in a way that everyone else understood.
• So, in 1795, the French developed a system of standard units, which was updated in 1960
• The revised system is called the Système Internationale d’Unités, which is abbreviated SI
SI Units
A system of standard measures that every scientist uses
It consists of 7 base units which have real measures in the real world
SI Base
Units
Quantity Base unit
Time second (s)
Length meter (m)
Mass kilogram (kg)
Temperature kelvin (K)Amount of substance mole (mol)
Electric current ampere (A)
Luminous intensity candela (cd)
Time
Base unit for time is the secondIt is based on the frequency of microwave
radiation given off by a cesium-133 atom
Length
The SI unit for length is the meter (m).
The distance that light travel through a vacuum
Equals 1/300,000,000 of a second
About 39 inches
Mass
Base unit for mass is the kilogram (kg)
You may see grams (g) or milligrams (mg)
Defined by a platinum-iridium cylinder stored in a bell jar in France
About 2.2 pounds
Temperature You classify an object as
hot or cold by whether heat flows from you to the object or from the object to you.
Heat flows from hot to cold.
Thermometers are used to measure temp.
SI unit of temp is kelvin (K)
In science, the celsius and kelvin scales are most often used.
To convert from celsius to kelvin: add 273
ex: -39º C + 273 = 234 KTo convert from kelvin to celsius: subtract 273
ex: 332 K – 273 = 59ºC
Temperature
Derived Units Not all quantities can be
measured with base units
Volume—the space occupied by an object
-measured in cubic meters (cm3)
-or liters (L) or milliliters (ml)
Density—a ratio that compares the mass of an object to its volume
--units are grams per cubic centimeter (g/cm3)
D = m V
Derived Units
Density equals mass divided by volume.
Example: If a sample of aluminum has a mass of 13.5g and a volume of 5.0 cm3,
what is its density?
Density = mass
volume D = 13.5 g
5.0 cm3
D = 2.7 g/cm3
Suppose a sample of aluminum is placed in a 25 ml graduated cylinder containing 10.5 ml of water. A piece of aluminum is placed in the cylinder and the level of the water rises to 13.5 ml. The density of aluminum is 2.7 g/cm3. What is the mass of the
aluminum sample?
Practice Problems—pg. 29 # 1, 2, 3
Other Derived Quantities
• Velocity or speed- distance an obj travels over a period of time– V = ∆d/ t – Units: m/s
• Force – push or a pull exerted on an object– F = m*a m= mass a= acceleration– Units: Kg * m/s2 = Newton (N)
Metric Prefixes• To better describe the range of possible
measurements, scientists add prefixes to the base units.
• For example: 3,000 m = 3 km (easier to manage)
• Most common prefixes:– King Henry Died by Drinking Chocolate Milk
• Metric prefixes are based on the decimal system
Converting Between Units• To convert b/w units simply move the decimal
place to the right or left depending on the number of units jumped.
• Ex: K he da base d c m
– 24.56 m = 245.6 dm = 2,4560 mm
• May use power of 10 to multiply or divide– Big units to small units Multiply– Small units to big units divide
Section 2.2
Scientific Notation and Dimensional Analysis
Scientific Notation• A way to handle very large or very small
numbers
• Expresses numbers as a multiple of 10 factors
• Structure: a number between 1 and 10; and ten raised to a power, or exponent– Positive exponents, number is > 1– Negative exponents, number is <1
Ex: 300,000,000,000 written in scientific notation is 3.0 x 10 11
Change the following data into scientific notation.a. The diameter of the sun is 1 392 000 km.b. The density of the sun’s lower atmosphere is 0.000 000 028 g/cm3.
Practice probs. Pg. 32 #12, 13
To add or subtract in scientific notation:+ The exponents must be the same before doing the arithmetic+ Add/Subtract numbers, keep the power of 10.
Ex: To add the numbers
2.70 x 107
15.5 x 106
0.165 x 108
Move the decimal to right (make # bigger): subtract from exponent (exp smaller)
Move the decimal to left (smaller #): add to exponent (bigger exp)
Practice probs. Pg. 32 #14
To multiply or divide numbers in scientific notation:
To multiply: multiply the numbers and ADD the exponents
ex: (2 x 103) x (3 x 102)
2 x 3 = 6
3 + 2 = 5
Answer = 6 x 105
To divide: divide the numbers and SUBTRACT the exponents
ex: (9 x 108) (3 x 10-4)
To multiply or divide numbers in scientific notation:
9 3 = 3 8 – (-4) = 12
Answer = 3 x 1012
Practice probs. Pg. 33 #15, 16
Dimensional Dimensional analysisanalysis
• A method of problem-solving that focuses on the units used to describe matter
• Converts one unit to another using conversion factors in a fraction formatfraction format– 1teaspoon = 5 mL 1 tsp or 5 ml
5 ml 1 tsp– 1 km = 1000 m 1 km or 1000 m
1000 m 1 km
• To use conversion factors simply write:1. The number given with the unit 2. Write times and a line (x ______).
3. Place the unit you want to cancel on the bottom.4. Use a conversion factor that contains that unit
5. Use as many conversion factors until you reach your answer
– ex: Convert 48 km to meters:
Dimensional Dimensional analysis analysis cont….cont….
48 km x 1km
= 48,000 m1000m
Conversion factor 1km = 1000 m
Practice: Convert 360 L to ml and to teaspoons:
1. How many seconds are there in 24 hours?
2. How many seconds are there in 2 years?
Practice probs. Pg. 34 #17, 18
You can convert more than one unit at a time:
What is a speed of 550 meters per second in kilometers per minute?
HINTs:Convert one unit at a time!Units MUST be ACROSS from each
other to cancel out!
Section 2.3
How reliable are measurements:
Sometimes an estimate is acceptable and sometimes it is not.
When you are driving to the beach
Miles per gallon your car gets
Your final grade in Chemistry
Okay?
X
When scientists make measurements, they evaluate
the accuracy and precision of the measurements.Accuracy—how close a measured value
is to an accepted value.
Not accurate
Accurate
Precision—how close a series of measurements are to each other
Not preciseNot precise PrecisePrecise
Density Data collected by 3 different students
Accepted density of Sucrose =1.59 g/cm3
Student A
Student B
Student C
Trial 11.54
g/cm3
1.40 g/cm3
1.70 g/cm3
Trial 21.60
g/cm3
1.68 g/cm3
1.69 g/cm3
Trial 31.57
g/cm3
1.45 g/cm3
1.71 g/cm3
Average1.57
g/cm3
1.51 g/cm3
1.70 g/cm3Which student is the most accurate? Which is most Which student is the most accurate? Which is most
precise? What could cause the differences in data?precise? What could cause the differences in data?
It is important to calculate the difference between an accepted value and an
experimental value.
To do this, you calculate the ERROR in data. (experimental – accepted)
Percent error is the ratio of an error to an accepted valuePercent error = error
accepted value x 100
Calculate the percent error for Student A
TrialDensity
(g/cm3)Accepted
valueError
(g/cm3)
1 1.54 1.59
2 1.60 1.59
3 1.57 1.59
Percent error = error x 100
accepted value
First, you must calculate the error!!
Error = (experimental – accepted)
Practice probs. Pg. 38 Practice probs. Pg. 38 #29#29
Significant Figures
Scientists indicate the precision of measurements by the number of digits they report (digits that are DEPENDABLE)
Include all known digits and one estimated digit.
A value of 3.52 g is more precise than a value of 3.5 g
A reported chemistry test score of 93 is more precise than a score of 90
There are 2 different types of numberso Exacto Measured
Exact numbers are infinitely importanto Counting numbers : 2 soccer balls or 4 pizzaso Exact relationships, predefined values 1 foot = 12 inches , 1 m = 100 cm
Measured number = they are measured with a measuring device (name all 4) so these numbers have ERROR.
When you use your calculator your answer can only be as accurate as your worst measurement
Significant Figures
45
Learning Check
Classify each of the following as an exact or a
measured number.
1 yard = 3 feet
The diameter of a red blood cell is 6 x 10-4 cm.
There are 6 hats on the shelf.
Gold melts at 1064°C.
46
Classify each of the following as an exact (1) or a
measured(2) number.
This is a defined relationship.
A measuring tool is used to determine length.
The number of hats is obtained by counting.
A measuring tool is required.
Solution
Measurement and Significant Figures
• Every experimental measurement has a degree of uncertainty.
• The volume, V, at right is certain in the 10’s place, 10mL<V<20mL
• The 1’s digit is also certain, 17mL<V<18mL
• A best guess is needed for the
tenths place.• This guess gives error in data.
Chapter Two47
48
What is the Length?
• We can see the markings between 1.6-1.7cm
• We can’t see the markings between the .6-.7
• We must guess between .6 & .7
• We record 1.67 cm as our measurement
• The last digit an 7 was our guess...stop there
Learning Check
What is the length of the wooden stick?1) 4.5 cm 2) 4.54 cm 3) 4.547 cm
Measured Numbers
• Do you see why Measured Numbers have error…you have to make that Guess!
• All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate.
• To indicate the precision of a measurement, the value recorded should use all the digits known with certainty.
50
Rules for significant figures
1. Non-zero numbers are always significant 72.3 g has__ 2. Zeros between non-zero numbers are 60.5 g
has__ significant
3. Leading zeros are NOT significant 0.0253 g
has __
4. Trailing zeros are significant after a 6.20 g has__ number with a decimal point
Trailing zeros
Leading zeros
100 g has__
Determine the number of significant figures in the following masses:a. 0.000 402 30 gb. 405 000 kg
a. 0.000 402 30 g
b. 405 000 kg
5 sig figs
3 sig figs
To check, write the number in scientific notation
Ex: 0.000 402 30 becomes
4.0230 x 10-
4
and has 5 significant figures
Practice probs. Pg. 39 # 31, 32
Rounding to a specific # of sig figs
When rounding to a specific place using sig figs, use the rounding rules you already know.
ex: Round to 4 sig figs: 32.54321. Count to four from left to right:
1 2 3 4
2. Look at the number to the right of the 4th digit and apply rounding rules
32.54
Practice probs. Pg. 41 #34
Calculations and Sig Figs• Adding/ Subtracting:
– Keep the least amount of sig fig in the decimal portion only.
– Ex:a. 0.011 + 2.0 =
b. 0.020 + 3 + 5.1 =
• Multiplying/ Dividing:– Keep the least amount of sig figs total
– Ex:a. 270/3.33 =
b. 2.3 x 100 =
Calculations and Sig Figs
• Follow your sig figs through the problem, but round at the end– Ex: (3.94 x 2.1) + 2.3418/ .004
Practice probs. Pg. 41 # 35, 36pg. 42 #37, 38