Lecture 2
Significant Figures and Dimensional
Analysis
Ch 1.7-1.9
Dr Harris
8/23/12
HW Problems: Ch 1: 31, 33, 37
Significant Figures
• Precision is indicated by the number of significant figures. Significant
figures are those digits required to convey the precision of a result.
• There are two types of numbers: exact and inexact
• Exact numbers have defined values:
* There are 12 eggs in a dozen
* There are 24 hours in a day
* There are 1000 grams in a kilogram
• Inexact number are obtained from measurement. Any number that is
measured has some error because:
• Limitations in equipment
• Human error
Significant Figures
• Example: Laboratory balances are precise
to the nearest cg (.01g). Lets say you
measure the mass of a particular sample
and you find the sample to have a mass of
335.49 g.
• If you choose to report error, you would
give the mass of the sample as
335.49±.01g because there is uncertainty
in the last digit (9).
• The actual mass may be 335.485 g, or
335.494 g. But because the scale is
limited to two decimal places, it
rounds up or down. Hence, we use ±
to include all possibilities.
Significant Figures
• The value 335.49 has 5 significant figures, with the hundredths
place (9) being the uncertain digit.
• Exact numbers have infinite sig. figs because there is no limit
of confidence.
• Other examples of inexact numbers?
•
•
•
Speedometer
Thermometer
Scale
How to Determine if a Digit is Significant
• All non-zeros and zeros between non-zeros are significant
• 457 (3) ; 2.5 (2) ; 101 (3) ; 1005 (4)
• Zeros at the beginning of a number aren’t significant. They
only serve to position the decimal.
• .02 (1) ; .00003 (1) ; 0.00001004 (4)
• For any number with a decimal, zeros to the right of the
decimal are significant
• 2.200 (4) ; 3.0 (2)
Ambiguity
• Zeros at the end of a number with no decimal may or may not be
significant
• 130 (2 or 3), 1000 (1, 2, 3, or 4)
• This is based on scientific notation
• 130 can be written as:
1.3 x 102  2 sig figs
1.30 x 102  3 sig figs
• If we convert 1000 to scientific notation, it can be written as:
1 x 103  1 sig fig
1.0 x 103  2 sig figs
1.00 x 103  3 sig figs
1.000 x 103  4 sig figs
* Numbers that must be treated as significant CAN NOT disappear in
scientific notation
Calculations with Significant Figures
• You can not get exact results using inexact numbers
• Multiplication and division
• Result can only have as many sig figs as the least precise number
6.22 𝑐𝑚 𝑥 𝟓. 𝟐 𝑐𝑚 = 32.3492 𝑐𝑚2 = 32 𝑐𝑚2
3 𝑠. 𝑓.
2 𝑠. 𝑓.
(2 𝑠. 𝑓. )
8 𝑠. 𝑓.
2 𝑠. 𝑓.
105.86643 𝑚
𝑚
𝑚
𝑚
2
= 108.0269694
= 110
𝑜𝑟 1.1 𝑥 10
0. 𝟗𝟖 𝑠
𝑠
𝑠
𝑠
2 𝑠. 𝑓.
𝑚
𝑘𝑔 𝑚
𝑘𝑔 𝑚
𝑘𝑔 𝑚
5
=
173080
=
200000
𝑜𝑟
2
𝑥
10
𝑠2
𝑠2
𝑠2
𝑠2
1 𝑠. 𝑓.
(1 𝑠. 𝑓. )
43270.0 𝑘𝑔 𝑥 𝟒
6 𝑠. 𝑓.
Calculations with Significant Figures
• Addition and Subtraction
• Result must have as many digits to the right of the decimal as the
least precise number
20.4
1.322
83
+ 104.722
211.942
212
Group Problems
• Solve the following. Use proper scientific notation for all answers. Also,
include correct units.
• Using scientific notation, convert 0.000976392 to 3 significant figures
• Using scientific notation, convert 198207.6 to 1 significant figure
H=10.000 cm
W = .50 cm
L = 30.000 cm
•
•
Volume of rectangle (volume = LWH) ?
Surface area (SA = 2WH + 2LH + 2LW) ?
note: the constants in an equation are exact numbers
Dimensional Analysis
• Dimensional analysis is an algebraic method used to convert
between different units
• Conversion factors are required
• Conversion factors are exact numbers (infinite sig figs), that are
equalities between one unit and another.
• For example, we can convert between inches and feet. The
conversion factor can be written as:
12 inches
1 foot
or
1 foot
12 inches
• In other words, there are 12 inches per foot, or 1 foot per 12 inches.
Dimensional Analysis
conversion factor (s)
desired units
given units x
given units
 desired units
• Example. How many feet are there in 56 inches?
• Our given unit of length is inches
• Our desired unit of length is feet
• We will use a conversion factor that equates inches and feet to
obtain units of feet. The conversion factor must be arranged
such that the desired units are ‘on top’
𝟓𝟔 𝑖𝑛𝑐ℎ𝑒𝑠 𝑥
1 𝑓𝑜𝑜𝑡
= 4.6666 𝑓𝑡
12 𝑖𝑛𝑐ℎ𝑒𝑠
4.7 ft
Group Examples
• Answer the following using
dimensional analysis. Consider
significant figures
• 35 minutes to hours
Non-SI to SI conversions
1 in = 2.54 cm
1 ft = 12 in.
1 mile = 5280 ft
1 quart = 946.3 mL
1 gallon = 4 quarts
• Convert 40 weeks to seconds
• Convert 4 gallons to Liters
• 4 gallons to cm3 ??
1 min = 60 sec
60 min = 1 hr
24 hr = 1 day
• 13 lbs to kg
1 lb = 453.59 g
Solutions
ℎ𝑜𝑢𝑟
1. 35 min 𝑥
= .583333 ℎ𝑜𝑢𝑟 =. 𝟓𝟖 𝒉𝒓
60 𝑚𝑖𝑛
2. 40 𝑤𝑒𝑒𝑘𝑠 𝑥
7 𝑑𝑎𝑦𝑠 24 ℎ𝑜𝑢𝑟𝑠 60 𝑚𝑖𝑛 60 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
𝑥
𝑥
𝑥
= 24192000 𝑠𝑒𝑐
𝑤𝑒𝑒𝑘
𝑑𝑎𝑦
ℎ𝑜𝑢𝑟
𝑚𝑖𝑛
= 𝟐. 𝟒 𝐱 𝟏𝟎𝟕 𝒔𝒆𝒄
4 𝑞𝑡 946.3 𝑚𝐿 10−3 𝐿
3𝑎. 4 𝑔𝑎𝑙 𝑥
𝑥
𝑥
= 15.1408 𝐿
𝑔𝑎𝑙
𝑞𝑡
𝑚𝐿
𝑐𝑚3
= 15140.8 𝑐𝑚3
3𝑏. 15140.8 𝑚𝐿 𝑥
𝑚𝐿
4.
13 𝑙𝑏𝑠 𝑥
= 𝟐𝟎 𝑳 𝒐𝒓 𝟐 𝒙 𝟏𝟎𝟏 𝑳
= 𝟐𝟎𝟎𝟎𝟎 𝒄𝒎𝟑 𝒐𝒓 𝟐 𝒙 𝟏𝟎𝟒 𝒄𝒎𝟑
453.59 𝑔
𝑘𝑔
𝑥
= 5.89667 𝑘𝑔
𝑙𝑏
103 𝑔
= 𝟓. 𝟗 𝒌𝒈
Converting Cubic Units
• As we previously learned, the units of volume can be expressed
as cubic lengths, or as capacities. When converting between
the two, it may be necessary to cube the conversion factor
• Ex. How many mL of water can be contained in a cubic
container that is 1 m3
3
1𝑚 𝑥
3
𝑐𝑚
𝒎𝑳
𝑥
10−2 𝑚
𝒄𝒎𝟑
Must use this equivalence to convert
between cubic length to capacity
Cube this conversion factor
𝟑
𝒄𝒎
𝑚𝐿
= 1 𝑚3 𝑥
𝑥
𝟏𝟎−𝟔 𝒎𝟑 𝑐𝑚3
= 𝟏 𝒙 𝟏𝟎𝟔 𝒎𝑳
Group Examples
• Convert 48.3 ft3 to cm3
• Convert 10 mL to m3
1 in = 2.54 cm
1 ft = 12 in.
mL = cm3
• Convert 100 L to µm3
•
**
A certain gasoline tank can hold
12.50 gallons of fuel. Assuming a
gasoline density of 0.797 g/cm3,
calculate the mass of gasoline in a
full tank.
k = 10 3
c = 10 -2
m = 10 -3
μ = 10 -6
1 quart = 946.3 mL
1 gallon = 4 quarts