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Measurement
And
Chemical Calculations
Measurement in Every Day Life
Measurement:
• How tall are you?
• What is the temperature outside?
• How many inches of rain did Lubbock get?
Exact number: No uncertainty
• 7 days in a week
• 24 hours in a day
• 100 cents in a dollar
Depending on who is doing the measurement there
could be a difference in the reported value.
Types of Observations and
• We make QUALITATIVE
observations of reactions — changes
in color and physical state.
• We also make QUANTITATIVE
MEASUREMENTS, which involve
numbers.
• Use SI units — based on the metric
system
UNITS OF MEASUREMENT
Use SI units — based on the
metric system
Length
Mass
Time
Meter, m
kg
Kilogram,
Seconds, s
Celsius degrees, ˚C
kelvins, K
Exponential Notation
• Use exponents to represent very large
or very small numbers.
Table 3-1, p. 52
Table 3-2, p. 62
Table 3-3, p. 75
Fig. 3-3c, p. 63
Fig. 3-2, p. 63
Fig. 3-5, p. 63
Fig. 3-6, p. 64
Fig. 3-1, p. 61
Basic Units of Measure
English System:
Units for measurement used in the United States
Metric System:
Units for measurement used in the rest of the world
SI Units:
International system of units based on the metric system
• Mass
• Length
• Temperature
• Time
• Amount
Connecting everything together are numbers and units!
Numbers and Units
Number:
How much of something ie one dollar
Unit:
How much of WHAT ie one dozen eggs, 100 bucks, etc
2 grams
number
•
•
•
•
•
Mass (kilograms)
Length (meters)
Temperature (Kelvin)
Time (seconds)
Amount (moles)
unit
Scientific Notation
Used to express both small and large numbers.
Example:
The mass of a single He atom is
0.000000000000000000000000664 grams
24 leading zeros make this number very small and
very difficult to enter into a calculator!
6.64 x
decimal part:
value between 1 and 10
10-25
exponent:
a whole number
(positive or negative)
exponential
Scientific Notation
Single digit between 1 and 10 before the decimal point
Multiply or divide the number by 10
Exponent equals number of “moved” decimal places
Positive exponent = move left to put into scientific notation
Negative exponent = move right to put into scientific notation
Example:
Express the following numbers in scientific notation.
Using Scientific Notation
Write each of these numbers in scientific notation.
0.000873
To get a number between 1-10, move the decimal how many
places, in what direction?
0.000873 = 8.73 x 10-4
4310000
To get a number between 1-10, move the decimal how many
places, in what direction?
4310000 = 4.31 x 106
Changing Back to Decimals
The size of the exponent shows how to move the decimal
Positive exponent = large number
Negative exponent = small number
When switching back to decimal form,
move in the opposite direction!
3.49 x 10-11 = 0.0000000000349
5.28 x 103 = 5280
6.72 x 10-1 = 0.672
1.29 x 108 = 129000000
Exponents
When two numbers with exponents are multiplied, the product is
the multiple of the base raised to a power equal to the sum of
the exponents.
Example:
10a x 10b = 10a+b
102 x 103 = 105
A Harder Example:
(3.54 x 107)(1.43 x 102)
First, multiply 3.54 x 1.43 to get the value of the base
3.54 x 1.43 = 5.06
Second, add the exponents together
7+2=9
Finally, combine all the numbers together correctly
5.06 x 109 = 5060000000
Exponents
When two numbers with exponents are divided, the product is the
dividend of the base raised to a power equal to the difference of
the exponents.
Example:
10c ÷ 10d = 10c-d
106 ÷ 103 = 10(6-3) = 103
A Harder Example:
(7.35 x 106) ÷ (3.43 x 104)
First, divide 7.35 ÷ 3.43 to get the value of the base
7.35 ÷ 3.43 = 2.14
Second, subtract the exponents
6-4=2
Finally, combine all the numbers together correctly
2.14 x 102 = 214
Combine the Ideas
Very Tricky:
(9.41 x 103)(1.21 x 10-5)
(342)(2.66 x 10-7)
First, multiply 9.41 x 1.21 to get the value of the base on top
9.41 x 1.21 = 11.4 = 1.14 x 101
Second, add the exponents together
3 + -5 + 1 = -1
Third, convert 342 to scientific notation
342 = 3.42 x 102
Fourth, multiply 3.42 x 2.66 to get the value of the base below
3.42 x 2.66 = 9.10
Fifth, add the exponents together
2 + -7 = -5
Sixth, rewrite the problem
1.14 x 10-1
9.10 x 10-5
Seventh, divide 1.14 ÷ 9.10 to get the final base
1.14 ÷ 9.10 = 0.125
Eighth, subtract the exponents
(-1) – (-5) = 4
Finally, combine all the numbers together correctly
0.125 x 104 = 1.25 x 103 = 1250
Adding Numbers with Exponents
If adding/subtracting numbers without a calculator, align the digits
vertically. Adjust the coefficients and exponents so that all
the numbers are raised to the same power.
3.971 x 107 + 1.98 x 104 = ?
39710000
+
19800
39729800
3.971 x 107 + 1.98 x 104 = 3.973 x 107
1.05 x 10-4 – 9.7 x 10-5 = ?
0.000105
- 0.000097
0.000008
1.05 x 10-4 – 9.7 x 10-5 = 8 x 10-6
Significant Figures
In science, it is important to make accurate measurements
and record them correctly.
Every measurement has some degree of uncertainty (or error).
However, depending on the measuring device,
the error can be reduced.
Every digit in a number is known accurately except the last digit,
which is estimated (or uncertain).
Example:
A nut weighs 1.8 grams
We record the weight as 1.8 ± 0.1 g
A nut weighs 1.81 grams
We record weight as 1.81 ± 0.01 g
Showing Uncertainty
Board is ~2/3 the length
of the meter stick, so
length is 0.6-0.7 m
0.6 ± 0.1 m
Lines are added to the meter
stick every tenth of a
meter; the board is
between 60-70 cm
Estimate closest tenth
0.64 ± 0.01 m
Centimeter lines are added to
the meter stick; the board
is between 64-65 cm
Estimate closest tenth
0.642 ± 0.001 m
Millimeter lines are added;
the board is between
64.3-64.4 cm
Best Estimate
0.643 ± 0.001 m
Significant Figures
The location of the decimal point has nothing to do with significant
figures.
0.643 m and 64.3 cm both have 3 sigfigs
Begin counting sigfigs at the first nonzero digit.
All nonzero digits are significant!
345 has 3 sigfigs
All zeros between nonzero digits ARE significant.
305 and 3.05 both have 3 sigfigs
Zeros at the beginning of a decimal number are NOT significant.
0.000643 km has 3 sigfigs
leading zeros are NOT significant
Significant Figures: Decimal Points
Zeros at the end of a large number are NOT significant, though
often ambiguous.
643,000,000 nm has 3 sigfigs
trailing zeros are NOT significant—Use exponential notation to remove
ambiguity
Zeros written at the end of a number AFTER the decimal point
ARE significant.
0.67 has 2 sigfigs while 0.670 has 3 sigfigs
If the final zero were not significant, is should not be recorded.
If you are not sure about a zero, write the number in scientific
notation. All non-significant zeros will be eliminated.
546,000 = 5.46 x 105 = 3 sigfigs
Significant Figures: Practice Time!
How many significant figures are in the following quantities?
1.002 L
36.4 cm
4 sigfigs
3 sigfigs
6.022 x 1023 atoms
4 sigfigs
2.88790 x 108 m/sec
6 sigfigs
0.003440 cm
4 sigfigs
How do we use sigfigs in calculations?
Exact Numbers
Significant figures do NOT apply to exact numbers.
Exact numbers have no uncertainty,
they were not obtained by measurement.
Exact numbers have an infinite number of sigfigs.
Example:
1 foot = 12 inches
1 dozen = 12 objects
1 hour = 60 minutes
Any property based on a measurement is not exact!
Exact Numbers
Which of the following quantities represent exact numbers?
The density of water at 70 oC is 0.97778 g/
mL.
Not exact
14 people are going to the men’s BB game.
Exact
The distance from Lubbock to Amarillo is 124 miles.
Not exact
The width of a human hair is 150 µm wide.
Not exact
There are 60 seconds in a minute.
Exact
Sigfigs: Addition
The number of sigfigs is based on the position of the digits.
15.9994
+ 1.00797
17.00737
Final answer: 17.0074
The numbers being added only have 4 decimal places in common.
5281
Align the decimal places
+ 18.05
+7699
+ 42.9
13040.95
Final answer: 13041
The sigfig stops at the decimal so only 5 numbers are significant.
Rounding Numbers
Should the last significant digit remain the same, or be rounded to
the next highest number?
If the digit AFTER the last sigfig is less than 5,
keep the last sigfig the way it is.
Four sigfigs: 13,672 becomes 13,670
If the digit AFTER the last sigfig is greater than 5,
round the last sigfig to the next highest number.
Four sigfigs: 1.0058 becomes 1.006
Three sigfigs: 3.799 becomes 3.80
If the digit AFTER the last sigfig is 5,
round the last sigfig to the next highest number.
Four sigfigs: 6.7455 becomes 6.746
Sigfigs: Subtraction
The number of sigfigs is based on the position of the digits.
Same rules apply for addition and subtraction!
319.542
- 20.460
0.0639
- 45.6
253.4181
Final answer: 253.4
If there are multiple steps in the calculation,
only round the final answer.
Determine the proper number of sigfigs
at the end of the calculation.
Sigfigs: Multiplication
The number of sigfigs is based on the number of sigfigs of the
quantities being multiplied.
The answer should be limited to the lowest number of
significant digits of the values used.
Exact numbers have an infinite number of sigfigs, so they do not
affect the number of sigfigs in the final answer.
(38.6)(0.009037)(2.00) = 0.6979564
Final answer: 0.698 (3 sigfigs)
0.04201 x 68700 = 2886.087
Final answer: 2890 (3 sigfigs) or 2.89 X 103
Sigfigs: Division
The number of sigfigs is based on the number of sigfigs of the
quantities being divided. Same rules as multiplication!
223.0 = 3.14084507042
71.0
Final answer: 3.14 (3 sigfigs)
-(8.314)(298.15) = -0.02568724456
96,500
Final answer: -0.0257 or -2.57 x 10-2 (3 sigfigs)
What if we combine addition, subtraction,
multiplication or division?
Sigfigs: Everything Together
Evaluate the following expression:
4.32 – 56.92 x (22.87 – 22.73)
Solve the problem in parentheses first.
(22.87 – 22.73) = 0.14
only 2 sigfigs
Next, perform the multiplication.
56.92 x 0.14 = 7.9688
Finally, perform the subtraction.
4.32 – 7.9688 = -3.6488 = -3.65
Final answer
limited to the
hundredth place.
You can limit sigfigs at each step of a calculation, but that may
lead you to a different answer at the end!
Sigfigs: Practice!
Questions:
4.35 + 2.297
5.1 – 1.66
1.97 x 3.904
(8.42 + 11.2) x 1.6
5.11 / 3.0
Answers:
6.647 = 6.65 (decimal places)
3.44 = 3.4 (decimal places)
7.691 = 7.69 (sigfigs)
a.
19.62 (can keep for now)
b. 31.392 = 31 (sigfigs)
1.70 = 1.7 (sigfigs)
Dimensional Analysis
In a problem, identify the given and wanted quantities that are
related by a PER expression.
Example:
How many days are in 28 weeks?
28 weeks x 7 days = 196 days
1 week
conversion factor: written as a fraction; used to change a
quantity of one unit to an equivalent amount of the other unit.
7 days
1 week
1 week
7 days
Dimensional Analysis: Check Units!
Always include units in your calculation setup.
If the units don’t make sense, the answer is wrong!
Example:
How many days are in 14 weeks?
28 weeks x 1 week = 4 weeks2
7 days
days
nonsense units!
The number of days must be larger than the number of weeks!
When setting up a dimensional analysis problem,
make sure the units cancel correctly!
Setting up Calculations
Determine what information is given and what
information is wanted.
If a car travels at an average speed of 74 miles per
hour, how far will it go in 8 hours?
Given: 8 hours
Wanted: miles driven
8 hours x 74 miles = 592 miles
1 hour
How many dollars are in 1,624 quarters?
Given: 1,624 quarters
Wanted: number of dollars
1,624 quarters x 1 dollar = 406 dollars
4 quarters
Proportional Reasoning
Proportional: Any change in either X or Y will result in a
corresponding change in the other.
Y α X
proportionality
constant
Y=mxX
If X increases, Y increases.
We can rearrange the equation to solve for the constant m.
m=Y
X
Proportional Reasoning
Inversely proportional: As one variable is increased,
the other is decreased.
Example:
Explore the relationship between the time (t) it takes to drive a
given distance (d) at a certain speed (s).
Driving at a higher speed means it takes less time to get
somewhere.
s α 1
t
s=dx1
t
180 milesx 1 hour 40= 4.5 hours
miles
180 milesx 1 hour 60= 3 hours
miles
d=sxt
Proportional Reasoning
If the pressure of a sample of gas is held constant, its volume (V)
is directly proportional to the Kelvin temperature (T).
Write an equation for the proportionality between V & T,
where a is the proportionality constant.
V = aT
For 28.6 g of CH4 gas at a pressure of 0.171 atm, V is
observed to be 248 L when T is 290 K. What is the value
of a, and what are its units?
a = V = 248 L = 0.855 L/K
T 290 K
What volume will this gas sample occupy at a
temperature of 392 K?
V = aT = 0.855 L/K x 392 K = 335 L
Metric Units: Mass
The SI unit of mass is the kilogram, kg.
A kilogram is defined as the mass of a platinumiridium cylinder stored in France.
1 kg = 2.2 pounds
1 kg = 1000 grams
1 g = 0.001 kg
Metric Units: Length
The SI unit of length is the meter, m.
A meter is defined as the distance light travels in a
vacuum in 1/299,792,468 second.
1 m = 39.37 inches
2.54 cm = 1 inch
1 km = 1000 meters
1 km = 0.621 miles
1 centimeter (cm) is the width of a fingernail
1 millimeter (mm) is the thickness of a dime
Metric Units: Volume
The SI unit of volume is the cubic meter, m3.
A m3 is too large a volume in the laboratory, so
chemists use the cubic centimeter, cm3.
Liquids and gases are not easy to weigh, so we
measure the volume of space they occupy.
A teaspoon holds approximately 5 cm3.
1 L = 1000 cm3
1 L = 1000 mL
1 L = 1.057 quarts
Volumetric glassware
Conversions with the Metric System
Learn to convert quickly between metric units.
Use dimensional analysis!
How many mm are in 51.5 cm?
Given: 51.5 cm
Wanted: mm
51.5 cm x 1 m x 1000 mm = 515 mm
100 cm
1m
OR combine conversion factors:
51.5 cm x 1000 mm = 515 mm
100 cm
Let’s try another example!
Conversions with the Metric System
Soda is sold in bottles that contain 2.00 L of fluid.
Express the volume in cubic centimeters and in quarts.
Given: 2 L
Wanted: cm3
2 L x 1000 cm3 = 2000 cm3
1L
Given: 2 L
Wanted: quarts
2 L x 1.057 quarts = 2.11 quarts
1L
Check: Liters are larger than cm3 therefore there
should be less liters.
Conversions Between Systems
Learn to convert between the United States Customary System
& the Metric System.
How many inches are in 23.65 cm?
Given: 23.65 cm
Wanted: inches
23.65 cm x 1 inch = 9.311 inches
2.54 cm
How many ounces are in 124.3 grams?
Given: 124.3 g
124.3 g x
Wanted: ounces (oz)
1 lb
x 16 oz = 4.385 oz
453.59 g
1 lb
Temperature
Fahrenheit (°F): Water freezes at 32 °F and boils at 212 °F.
Celsius (°C): Water freezes at 0 °C and boils at 100 °C.
Kelvin (K): Water freezes at 273 K and boils at 373 K.
Temperature Conversions
What are the relationships between temperature scales?
Converting between °F and °C:
Fahrenheit temp
T°F -32 = 1.8 T°C
Celsius temp
What is the temperature in Celsius when the thermometer at a
picnic reads 65 °F?
Given: 65 °F
Wanted: °C
T°C = T°F - 32
1.8
T°C = 65 °F - 32 = 18.3 °C = 18 °C
1.8
Temperature Conversions
What are the relationships between temperature scales?
Converting between °C and K:
Kelvin temp
TK = T°C + 273
Celsius temp
What is the temperature in Celsius when the Kelvin
temperature is 234 K?
Given: 234 K
Wanted: °C
T°C = TK - 273
T°C = 234 K - 273 = -39 °C
The Kelvin scale is also called the absolute temperature scale
because it is based on zero as the lowest possible temperature.
On the Kelvin scale, do not use a degree (°) symbol.
Temperature Conversions
Convert -118 °F to K:
T°C = -118 °F - 32 = -83.3 °C = -83 °C
1.8
TK = -83 °C + 273 = 190 K
Convert 32 K to °F:
T°C = 32 K - 273 = -241 °C
T°F = 1.8 T°C + 32
T°F = 1.8 (-241 °C) + 32 = -401.8 °F = -402 °F
Density
The ratio of mass to volume.
Density =
mass
volume
Density can be thought of as the relative
“heaviness” of a substance.
A block of iron is heavier than a block of
aluminum of the same size, due to
the densities of the two substances.
If you weigh out 6.12 grams of cooking oil
and it takes up a volume of 8.14 mL in the
measuring cup, the density of the oil is:
Density =
6.12 g
8.14 mL
= 0.752 g/mL
Density Problems
In chemistry lab you are asked to identify a piece of
metal. You decide to calculate the density of the metal
to determine its identity.
The piece of metal weighs 198.4 grams. When you drop it
in a cup of water, the metal displaces 18.7 mL of water.
What is the density of the metal? What metal is it?
Density =
198.4 g
18.7 mL
= 10.6 g/mL
Substance
Density (g/mL)
Substance
Density (g/mL)
Water
1.00
Lead
11.34
Aluminum
2.72
Mercury
13.60
Chromium
7.25
Gold
19.28
Nickel
8.91
Tungsten
19.38
Copper
8.94
Platinum
21.46
Silver
10.50
Density Problems
The gasoline in an automobile gas tank has a mass of 80.0 kg
and a density of 0.752 g/cm3. What is the volume in L?
Given: 80.0 kg
80.0 kg x 1000 g
1 kg
Wanted: volume (L)
x
1 cm3
0.752 g
x 1 L = 106 L
1000 cm3
What is the mass of a ball of mercury that has a volume of
1.32 mL and a density of 13.6 g/mL?
Given: 1.32 mL
13.6 g/mL
Wanted: mass
Mass = Volume x Density = 1.32 mL x 13.6 g = 17.9 g Hg
1 mL
Water: A Special Case
Frozen water floats on
liquid water.
Frozen ethanol sinks in
liquid ethanol.
Ice is LESS DENSE than water!
A given volume of ice must have less mass than an equal
volume of liquid water.
Therefore, the molecules in water pack together tighter than
the molecule in ice!
Typically the solid phase is MORE dense than the liquid phase!
Practice At Home!
1. Write each of the following numbers in scientific notation.
a. 56897
b. 123
c. 0.000678
d. 789540
e. 560000000
2. Perform the following calculations using proper sigfigs.
a. 3.65 + 4.2 =
b. 8.6 – 2.34 =
c. 15.6 x 22.34 =
d. (9.7 - 3.48) x 2.3 =
e. (6.0 x103) + (3.2 x104) =
3. Use dimensional analysis to convert between units.
a. Convert 46.2 cm to inches (2.54 cm = 1 in)
b. Convert 27 inches to feet (1 ft = 12 in)
Practice At Home!
1. Rice Krispies comes in a travel
boxes containing 0.88 ounces.
How many grams of cereal is this?
2. An address label has the length of
2.12 inches. What is the length of
the label in cm?
3. Mount McKinley in Alaska is
20,320 ft above sea level. Express
this height in kilometers.
Practice At Home!
1. What is the temperature in Kelvin
when it is 431 oC?
2. What is the temperature in oF
when it is 39 oC outside?
3. What is the temperature in Kelvin
when it is 92 oF outside?
Practice At Home!
1. Calculate the density of air if the mass
of 15.7 L is 18.6 grams.
2. A rectangular block of iron 3.20 cm x
9.87 cm x 11.6 cm has a mass of 2.88
kg. Find its density in g/cm3.
3. Calculate the volume occupied by 32.4
grams of copper, which has a density of
8.94 g/mL.
Practice At Home!
1. A titanium bike has a mass of 3245 g
and a density of 4.50 g/cm3. What is its
volume?
2. An ice cube has a volume of 75.9 cm3
and a density of 0.92 g/cm3. What is its
mass?
3. A glass ball has a mass of 4.5 g and
volume of 1.73 cm3. What is its density
in g/cm3?