CHAPTER 2
QUANTITATIVE MEASUREMENT and CALCULATIONS in CHEMISTRY
Scientific Notation, Accuracy and Precision
Unit Conversions, Derived Units, Density
Heat, Temperature, and Specific Heat
SCIENTIFIC NOTATION
 Necessary in chemistry since many of the numbers we will deal with are extremely large or
extremely small
For example:
 Avogadro’s number = 602 213 674 000 000 000 000 000 atoms in a mole of atoms
 Mass of an electron = 0.000 000 000 000 000 000 000 000 000 000 910 9 kg
Changing a number to scientific notation
1. Move the decimal so that there is one number to the left of the decimal point.
2. Determine the power of 10 by counting the number of places you moved the decimal.
Moving the decimal to the left, the exponent increases. (large #, + exponent)
Moving the decimal to the right, the exponent decreases. (small #, - exponent)
Practice
 Convert the following numbers to scientific notation or rewrite in decimal
form.
1.
530000 L
=
2.
0.00053 L
=
3.
0.00000092 g =
4.
12600000000 mg =
5.
1.87 x 10-5 kg =
6.
5.99 x 103 m =
Scientific notation on a calculator
 Sometimes the (x 10) part of the scientific notation will not
appear on your calculator. The screen will only display the first
factor and the exponent.
Ex. 3.45 x 10-4 would display as 3.45-04 or 3.45 EE -04 or 3.45 E -4
 If your calculator will not leave the number in scientific notation,
you may have to change mode to scientific.
 Use the EE , 2nd EE or EXP key to enter [10 x] followed by the
exponent that you want the 10 raised to. Do not enter (x 10) or
(x 10x) and use the EE key!!!
 Ex. Enter 3.04 x 108 on your calculator and read the display.
ACCURACY and PRECISION
Accuracy and Precision
 All measurements made in lab are subject
to errors (human error and instrumental
error).
 Measurements must be repeated to
validate results.
 Scientists want to be both accurate and
precise. What is the difference?
Accuracy
 A description of how close the experimental
measurement is to the theoretical value.
 Expressed using a % error calculation.
 We would like the % error to be under 10%.
Otherwise, the data is inaccurate.
 % error = [theoretical – experimental] x 100
theoretical
Example:
A student calculated the density of silver to be 10.40
g/cm3 in the lab. The actual density of silver is 10.50
g/cm3.
This result is fairly accurate. However, we must quantify
this claim through a % error calculation.
Calculation of % error =
What if the student calculated 8.40 g/ cm3?
Precision
 the exactness of a measurement
or
the spread of a data set
 describes how close several measurements of
the same quantity are to one another
 expressed using an average deviation
calculation (how much does each of the
measurements deviate from the average?)
 report average measurement  the average
deviation. Ex. 21.05  0.01 grams
Average Deviation
 average deviation =
trial 1- average + trial 2- average +… trial n- average
n
Given an average deviation of  0.01 g, how imprecise is
the data if the original measurement was 100.00 g? 10.00
grams? 1.00 grams?
We must look at the relative error using a precision
check.
 precision check = average deviaiton x 100
average
Precision Example
 Ex. A student finds the
density of 4 silver samples.
 Average deviation calculation:
 Precision check:
Trial
Density
(g/cm3 )
1
10.35
2
10.55
3
10.50
4
10.49
Average
Reporting the results:
D = ___________  _________g/cm3
Accuracy and Precision practice:
A student calculates the volume of an object using water displacement in 3
separate trials. Given the following results, and the actual volume = 20.4
ml, discuss the data set in terms of accuracy and precision. Calculate %
error and the average deviation for the data set.
 Average volume
 % error =
 Average deviation =
Trial 1
13.5 ml
Trial 2
Trial 3
Average volume
15.6 ml
14.8 ml
 This data set is precise but not accurate. Most likely due to a consistent human or
instrumental error. (maybe water splashed onto the sides of the graduated cylinder
making the volume seem less every time).
Precision continued…
 precision is also expressed when reading an instrument to the
proper number of decimal places. The more decimal places that
can be read, the more precise the measurement.
 Read 1 decimal place past the markings on the instrument – the
last digit is an UNCERTAIN DIGIT. All the digits that can be
read are called
SIGNIFICANT DIGITS or SIGNIFICANT FIGURES.
 Sometimes the last uncertain digit is designated by placing a bar
over that digit.
 Ex.
9800 kg
3000 L or 3000. L
SIGNIFICANT FIGURES
 Used to express precision in a MEASURED quantity
Examples:
Temperature:
Mass:
21.0°C
25.00 g
25
Volume 13.0 ml
 Sig figs eliminate extra digits that the calculator “spits out” and eliminates digits that
we can’t possibly know.
Example: Calculate density given the data above.
D = M/V = 25.00 grams/ 13.0 ml =
D = M/V = 25 g/ 13.0 ml =
How can we possibly know this more precisely than we know our original
measurements?
RULES FOR DETERMINING
SIGNIFICANT FIGURES
1. all non-zero numbers are significant
135.98
2.3 x 103
56.1
189
Exceptions:
a. counting numbers are infinitely significant (ignore
when determining sig figs)
28 students
4 trials
b. exact relationships (like those used in conversion
factors) are infinitely significant (ignore when
determining sig figs)
1 m = 100 cm
1km = 1000 m
2.54 cm = 1 in
ZERO RULES for SF
 Significant figures are less obvious when zeroes are present, because often
times they do not add meaning to the number.
2. Zeros are NOT significant when they are:
a. Place holding zeroes
.00129 .014
b. Alone and before the decimal place 0.45
c. At the end of numbers without a decimal point
67000
560
30
ZERO RULES for SF
3. Zeros are significant when they are:
a. Between two non-zero numbers
50599 40.001
b. In a number with a decimal place after a
non-zero number
55.00
3400.
4.50 x 10-3 .900
SIGNIFICANT FIGURE PRACTICE
Determine the number of sig figs in the
following:
0.13 grams
130.0 g
1.3 x 10-4 m
0.130 mg
1.30 x 104 kg
0.1300 miles
1303 mm
.00130 liter
1300 miles
130. ml
30 students
1300 kg
1 hour/ 60 minutes
CALCULATIONS
WITH SIGNIFICANT FIGURES
RULES
1. When adding or subtracting, the result must be rounded so that the final
digit reflects the fewest number of decimal places in the measured values
Ex.
23.456 g + 20.5 g + 22.0047g
21.0°C – 19°C =
When averaging values, follow this rule. Ex.
13.0 + 12.56 + 12.755
3
2. When multiplying or dividing, the result must be rounded so that it has the same number
of significant digits as the measurement with the fewest number of significant digits.
34.50 g/ 1.23 ml =
89.00 x 20 =
6.05 x 102 / 1.2 x 10-2
3. When multiple calculations occur, perform all calculations
before rounding. Look at individual parts of the calculations
to determine sig figs for each section, then determine for the
overall problem.
14.566 + 12.0 / (23 x 10.00) =
(15.0 + 16.78) / (5.6 x 104)
4. When adding or subtracting numbers in scientific
notation, the exponent must be equal before
determining the number of decimal places to
keep. Change to the higher exponent (less
negative).
1.200 x 10-2 + 1.35 x 10-4 =
3.40 x 10-1 - 4.555 x 10-2 =
Practice
1.
8.91 g – 6.435 g =
2.
4.63cm x .37 cm =
3.
(5.6 g + 4.50 g) / 0.123 ml =
4.
(12 + 13 + 14.5) / 3 trials =
5.
(350.0 / 25) cm + 120.5 cm =
6.
9.815 x 10-3 + 3.21 x 10-2 =
7.
35 km = _____cm
UNITS and CONVERSIONS
UNITS OF MEASUREMENT
•Every quantitative measurement involves a NUMBER and a UNIT.
•All units are based on a STANDARD.
SI MEASUREMENT
•The internationally accepted system of measurement (Le System
International d’Unites) was adopted in 1960.
• Base units of this system are based on a STANDARD measurement.
Ex. The kilogram standard
Multiple copies of a platinum/ iridium cylinder housed in France
Protected in a vacuum, in a vault, underground
Bonus: research another standard unit and determine how it is defined.
SI BASE UNITS
QUANTITY
UNIT
ABBREVIATION
Length
Time
Mass
Amount of a
substance
Temperature
Current
Luminous intensity
•It is not always convenient to measure a quantity using the base unit, so SI
prefixes can be used to modify the unit.
SI PREFIX
ABBREVIATION EXPONENTIAL
NOTATION
MEANING
Giga
Mega
Kilo
Hecto
Deka
BASE UNIT: meters, grams, liters, moles, seconds, etc. 100
Deci
Centi
Milli
Micro
Nano
Pico
Prefix practice:
1.
How many meters are in a kilometer?
2.
How many micrograms are in a gram?
3.
How many liters are in a dekaliter?
4.
How many millimoles are in a mole?
5.
How many nanometers are in a meter?
6.
How many centigrams are in a gram?
7.
How many hertz are in a Megahertz?
CONVERSIONS USING PREFIXES
 While most prefix conversions can be performed by sliding the
decimal, and alternative method is to use conversion factors from
the prefix table.
 Conversion factor = 2 equivalent units combined in a ratio with
numerical equivalent = 1.
Example:
 Determine the conversion factor to be used by unit analysis,
factor label, or dimensional analysis.
Sample SI conversion problems:
1. How many millimeters are in 45.0 meters?
2. How many picograms are present in 3.0 x 10-9 grams?
3. How many moles of atoms are in 0.050 Mmoles?
4. How many centigrams are in 345000 nanograms?
5. How many millimeters are in 23 km?
COMMONLY MEASURED QUANTITIES IN
CHEMISTRY
MASS, LENGTH, VOLUME, DENSITY, and TEMPERATURE
MASS vs. WEIGHT
 Mass – the amount of matter in an object.
 Measured in grams or kilograms.
 Measured using an electronic balance.
MASS vs. WEIGHT
 Weight –is the force on the object due to gravity.
Changes with changes in gravitational fields.
 Measured in pounds or Newtons
 1 N = kgm/s2
 Measured using a scale or spring scale.
MASS CONVERSIONS
 English/ SI conversions
 2.20 lbs = 1 kg
 16 oz. = 1 lb
 1 ton = 2000 lbs
Sample mass conversions:
1.
2.
3.
How many kilograms would a 195 lb person weigh?
A baby weighs 8lbs. 9 oz. at birth, what is this in kg?
A train carries 1.5 ton of coal, ho many kg of coal are present?
LENGTH CONVERSIONS
 English/ SI conversions for length
 2.54 cm = 1 inch
 12 inches = 1 foot
 3 feet = 1 yard
 1 mile = 1.609 km
Sample length conversions:
1. A football players runs 45 yards, how far has he traveled in m?
2. The distance from earth to the moon (on average) is 238,857
miles. What is this in km?
3. 75 km would be equivalent to how many inches?
4. Measure the length and width of your lab bench in inches,
convert these lengths to meters.
5. Find the surface are of the lab bench in m2. This is a derived
unit. Convert the area to cm2.
DERIVED UNITS and
CONVERSIONS
Area, volume, and density
DERIVED UNITS:
 Derived unit- a combination of 2 or more units
 Examples:
 Area
 Volume
 Density
 Molar mass
 Pressure
 Speed
Derived unit conversions
 Ex. Convert 25 miles/ hour to meters/ second
 Ex. Convert 0.027 kg/ L to grams/ cm3
VOLUME
 Defined as: the amount of space matter occupies
 A three dimensional measurement
 Commonly measured using the non-SI unit, liters.
 1L = 1 dm3 (by definition)
 1 L = 1000 cm3 = 1000 ml
VOLUME FORMULAS
 Formulas for volume of regular solids:
 Cube
 Rectangular solid
 Sphere
 Cylinder
 Cone
 Pyramid
Volume conversions: English to SI
 English/ SI conversions for volume
 1L = 1.057 quarts
 4 quarts = 1 gallon
 2 pints = 1 quart
 8 fl. Oz = 1 cup
 4 cups = 1 quart
Sample volume conversions:
1. How many cc are in 2.5 L?
2. How many ml are equivalent to 3.0 cups?
Volume conversions (cont.)
3. A 2.0 L pop bottle contains how many quarts of pop?
4. A cylindrical mug has a height of 4.5 inches and a diameter
of 3.0 inches. What is the volume of the mug in in3? How
many ml of coffee would fit in the mug?
Volume conversions (cont.)
Determine the volume of an ice cream cone if it is 4.0 inches
tall and 2.75 inches at the opening. How many cm3 of ice
cream will fit in the cone?
6. Assume the cone is topped with a hemisphere of ice cream
with the same diameter as the cone, what is the volume of ice
cream on top of the cone (in cups)?
5.
Volume conversions (cont.)
7. You measure 100.0 ml of water in the lab, how many cups of
water are present?
What piece of equipment did you use?
DENSITY
 Defined as: the amount of matter per unit volume
 Calculation: D = Mass
Volume
 Units of density: g/ml or g/cm3
 For gases: g/ L
DENSITY of WATER
Changes with temperature.
D = 1.00 g/ ml (approximately) at room temperature
Density calculations
 Sample density calculations:
1.
2.
3.
4.
Determine the density of a 100.0 gram object that occupies 1.50 in3 of space.
Convert the density to g/ cm3.
What is the mass of 25.0 ml of mercury?
A cube of silver has a mass of 25.00 grams, what are the dimensions of the cube?
A copper metal sphere has a mass of 10.0 grams, determine the volume of the sphere
and the radius of the sphere.
ENERGY
Temperature, heat, and specific heat
What is Energy?
 The ability to move or change
matter.
(Units: Joules)
 All physical and chemical
changes involve energy!
Examples of Energy

Kinetic – energy of motion







KE = ½ mv2
Potential – stored energy/energy of
position
Light
Sound
Electricity
Heat (Thermal)
Chemical
Law of conservation of energy:
 Energy cannot be created or destroyed during any
chemical or physical change.
 Energy may be transferred between the system
and surroundings
 Energy may change forms.
Energy and mass are related
Einstein derived an
equation to show this
relationship in 1905.
 Nuclear reactions can
create energy from
mass.

Energy is transferred during physical and
chemical changes:
 Endothermic – energy is absorbed by the
system
+
 Exothermic – energy is released into the
surroundings
-
What is Heat?
The transfer of energy
between the particles of
two objects due to a
temperature difference
between the two objects.
 Heat always flows from hot
to cold.
 Measured in a calorimeter.
 Units: Joules, Calories, or
calories.

TEMPERATURE
What is temperature?
 Temperature is the measure of
the average kinetic energy of all
the particles within an object.
 Measured with a thermometer.
Heat and temperature
 The transfer of heat does not always result in a
temperature increase. During phase changes,
energy goes directly to changing the phase, not
into increasing the kinetic energy of the particles.
 EX. The heating curve for water.
The heating curve for water shows that temperature does NOT
change during a phase change.
Heating curve points and definitions:
 Melting point/ freezing point of water: 0º C
 Boiling point of water: 100 º C
 Heat of fusion – the amount of energy required to melt a solid
for water: 334.0 J/g
 Heat of crystallization – the amount of energy released when a
solid forms from a liquid
 Heat of vaporization – the amount of energy required to change a
liquid into a gas.
for water: 2257. J/g
Scales to Measure Temperature
 Fahrenheit Scale (U.S.A.)
 Celsius Scale
(everyone else)
 Kelvin Scale
(scientists)
Fahrenheit
Celsius
Kelvin
Proposed by:
Daniel Gabriel
Fahrenheit (1710)
German instrument
maker
Anders Celcius
(1742)
Swedish astronomer
LordWilliam
Thomson Kelvin
(1848)
Scottish physicist
Symbol
F
C
K
-459.0 F
-273.15 C
0K
b.p. water
m.p. water
Difference
Lowest temperature
possible
TEMPERATURE CONVERSIONS
Celsius to Kelvin
Kelvin to Celsius
Fahrenheit to Celcius
Celsius to Fahrenheit
Sample Temperature Conversions
 Convert 212.0F to C and K.
 Convert 300.0 K to C.
How do Thermometers Work?
 Usually contain alcohol or mercury.
 Temperature increase (particles move
faster), liquids expand
 Temperature decreases (particles move
slower), liquids contract
Absolute Zero




The lowest possible temperature
All motion STOPS.
Energy is minimal/absent.
In September 2003, MIT announced a record cold
temperature of 450 pK, or 4.5 × 10-10 K in a BoseEinstein condensate of sodium atoms. This was
performed by Wolfgang Ketterle and colleagues at
MIT.
SPECIFIC HEAT CAPACITY
 Transfer of heat affects substances
differently.
 Measuring heat transferred to and absorbed
by a substance under conditions of constant
pressure yields specific heat capacity.
SPECIFIC HEAT CAPACITY
Specific heat is defined as:
The quantity of heat required to raise
1 gram of a substance 1°C or 1 K.
Symbol: Cp
The p symbolizes that the measurements were taken under
constant pressure.
Units = Joules/ gram °C or J/gK
J/g°C
Sample Cp values
 Metals have low specific heat values which allows them to heat
up with little added energy.
 Iron
0.449 J/g°C
 Copper 0.385 J/g°C
 Platinum
0.133 J/g°C
 Water has a relatively high specific heat
4.184 J/g °C
Questions:
 Which would heat up faster, 5.00
grams of iron or 5.00 grams of
water?
 Which would cool down faster, 5.00
grams of iron or 5.00 grams of
water?
 Which is a better thermal
conductor?
 Which is a better insulator?
MEASURING HEAT
and SPECIFIC HEAT
Must use a calorimeter.
Find the change in
temperature:
 T = (delta T)
change in temperature in °C
 T = T final – T initial
SPECIFIC HEAT CALCULATIONS
q =m x Cp x  T
q=
m=
Cp=
 T=
Rearrange the formula:
m= q/Cp  T
Cp = q/ m  T
 T = q/ m Cp
Sample Heat Calculations
1.Calculate the heat released as an aluminum pan cools from
100.0C to 25.0 C if the pan mass is 0.98kg.
2. What is the mass of iron that absorbs 1200 Joules of heat
when going from 10.0K to 90.0K?
3. What is the specific heat of a 10.0 gram sample of a
substances that gains 20.0 kJ of energy when its temperature
is raised from 25.0 C to 75.0 C?
Sample Heat Calculations
4. What is the final temperature of a 5.00 gram sample of
platinum that has been heated by 450 Joules of energy
starting at an initial temperature of 273K?
Specific Heat Calculations
5. A snickers bar contains 266 calories. Food calories are
actually kilocalories (symbolized C). 1 calorie = 4.184
Joules.
a. determine how much energy would be released by a
snickers bar in a bomb calorimeter.
b. If the calorimeter is filled with 10000.0 grams of water, what
is the temperature change that would result from burning the
snickers bar?
Solving for Tfinal
1. A 50.00 gram piece of copper metal was heated to 100.0C
and then quickly dropped into 100.0 grams of water inside a
calorimeter at 20.0 C. Assuming all the heat lost from the
metal goes into the water, what will be the final temperature
of both materials inside the calorimeter?
Solving for Tfinal
2. What will be the equilibrium temperature of a mixture of
300.0 ml of water at 80.0 C with a 25.0 gram silver spoon
at 20.0 C?