Chapter 2
Measurements
and
Calculations
2.2 Units of Measurement
Nature of Measurement
Measurement - quantitative observation
consisting of 2 parts
Part 1 – number
Part 2 - scale (unit)
Examples:
20 grams
6.63 x 10-34 Joule*seconds
The Fundamental SI Units
(le Système International, SI)
P h y sica l Q u a n tity
N am e
A b b rev ia tio n
k ilo g ram
kg
m eter
m
T im e
seco n d
s
T em p eratu re
K elv in
K
E lectric C u rren t
A m p ere
A
m o le
m ol
can d ela
cd
M ass
L en g th
A m o u n t o f S u b stan ce
L u m in o u s In ten sity
Derived Units
Volume: Liter - non SI unit commonly used
1 liter = 1 cubic decimeter
1 liter = 1000 ml = 1000 cm3
Density: kg/m3 is inconveniently large
commonly used unit is g/cm3 or g/ml
1 g/cm3 = 1 g/ml
gas density is reported in g/liter
Specific Gravity
• A comparison between a substance’s density and
that of water, at a specified temperature and
pressure.
• Since density is divided by density, specific
gravity is a dimensionless quantity, meaning it
has no units.
• It is common to use the density of water at 4 °C
(39 ° F) as reference - at this point the density of
water is at the highest - 1000 kg/m3.
Metric Prefixes
Conversions: Factor-Label Method
(Dimensional Analysis)
Starting Number, Unit X Number, Desired Unit
Number, Starting Unit
Conversion
Factor
Practice
1) What is the density of a sample of ore that has a mass of
74.0 g and occupies 20.3 cm3?
D = 3.65 g/cm3
2) Find the volume of a sample of wood that has a mass of
95.1 g and a density of 0.857 g/cm3.
3
V = 111 cm
3) Express a time period of exactly 1.00 day in terms of
seconds.
t = 86400 s
4) How many centigrams are there in 6.25 kg?
m = 625000 cg
Chapter 2
Measurements
and
Calculations
2.3 Using Scientific Measurement
Uncertainty in Measurement
A digit that must be estimated is called
uncertain. A measurement always has
some degree of 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?
Precision and Accuracy
Accuracy refers to the agreement of a particular
value with the true value.
Precision refers to the degree of agreement
among several measurements made in the same
manner.
Neither
accurate nor
precise
Precise but not
accurate
Precise AND
accurate
Types of Error
Random Error (Indeterminate Error) measurement has an equal probability of
being high or low.
Systematic Error (Determinate Error) Occurs in the same direction each time
(high or low), often resulting from poor
technique or incorrect calibration.
Percent Error
A way to compare the accuracy of an
experimental value with an accepted value.
Percent =
Error
Value accepted – Value experimental x 100
Value accepted
Ex: The actual density of a certain material is 7.44 g/cm3.
A student measures the density of the same material
as 7.30 g/cm3. What is the percentage error of the
measurement?
% Error = 1.9 %
Rules for Counting Significant Figures
Nonzero integers always count as
significant figures.
3456 has
4 sig figs.
Rules for Counting Significant Figures
Zeros
Leading zeros do not count as
significant figures.
0.0486 has
3 sig figs.
Rules for Counting Significant Figures
Zeros
Captive zeros always count as
significant figures.
Also
known as
the
Hugging
Rule!
16.07 has
4 sig figs.
Rules for Counting Significant Figures
Zeros
Trailing zeros are significant only if
the number contains a decimal
point.
9.300 has
4 sig figs.
Rules for Counting Significant Figures
Exact numbers have an infinite
number of significant figures.
1 inch = 2.54 cm, exactly
Sig Fig Practice #1
How 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
Rules for Rounding
Rules for Significant Figures in
Mathematical Operations
Multiplication and Division: # of sig figs in
the result equals the number in the least
precise measurement used in the
calculation.
6.38 x 2.0 =
12.76  13 (2 sig figs)
Sig Fig Practice #2
Calculation
Calculator says:
3.24 m x 7.0 m
22.68 m2
100.0 g ÷ 23.7 cm3
4.219409283 g/cm3
Answer
23 m2
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
0.3588 g/mL
0.359 g/mL
Rules for Significant Figures in
Mathematical Operations
Addition and Subtraction: The number of
decimal places in the result equals the
number of decimal places in the least
precise measurement.
6.8 + 11.934 =
18.734  18.7 (3 sig figs)
Sig Fig Practice #3
Calculation
Calculator says:
Answer
3.24 m + 7.0 m
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
Scientific Notation
A method of representing very large or very
small numbers
M x 10n
• M is a number between 1 and 9
• n is an integer
• all digits in M are significant
Reducing to Scientific Notation
– Move decimal so that M is between 1 and 9
– Determine n by counting the number of places
the decimal point was moved
Moved to the left, n is positive
Moved to the right, n is negative
Mathematical Operations Using
Scientific Notation
Addition and subtraction
– Operations can only be performed if the
exponent on each number is the same
Multiplication
– M factors are multiplied
– Exponents are added
Division
– M factors are divided
– Exponents are subtracted (numerator denominator)
Operations with Units
» Cancellation occurs with the units in the same
way that it occurs with numbers common to
both the numerator and denominator
– Units are handled algebraically, just like
numbers
– Analysis of units can be a clue as to whether
a problem was set up correctly
» Calculations involving units must have the
correct units shown throughout the working of
the problem and attached to the answer
Direct Proportions
 The quotient of two variables is a
constant
k = y/x
 As the value of one variable
increases, the other must also
increase
 As the value of one variable
decreases, the other must also
decrease
 The graph of a direct proportion is a
straight line
Inverse Proportions
 The product of two variables is a
constant
k = xy
 As the value of one variable
increases, the other must decrease
 As the value of one variable
decreases, the other must increase
 The graph of an inverse
proportion is a hyperbola
Practice
1) Calculate the density of a liquid given that 41.4 mL of it
has a mass of 58.24 g.
D = 1.41 g/mL
2) How many kilometers are there in 6.2 x 107 cm?
6.2 x 102 km
3) How m\any hours are there in exactly 3 weeks?
504 hours