Transcript Numbers and Conversions - Chemistry at Winthrop University

Introduction to General
Chemistry
Part 3:
Numbers and Conversions
Ch. 1.6 - 1.9
Accuracy and Precision
• Accuracy defines how close to the correct answer you are.
• Precision defines how repeatable your result is.
• Ideally, data should be both accurate and precise, but it may be
one or the other, or neither.
Accurate only
Precise only
Accurate and precise
Neither
Measuring Accuracy
• Accuracy can be described using by percentage error (%E)
%E 
average value  true value
x100
true value
Ex. The table below shows values obtained using three different
methods to determine the mg of sodium in a candy bar, which must be
reported to within 3% error. If the actual quantity of sodium in the
candy bar is 115 mg, which method(s) would you describe as precise,
accurate, or both? What is the % error of each method?
Method 1
Method 2
Method 3
109
110
114
110
115
115
110
120
116
109
116
115
110
113
115
Measuring Precision: “Average ± range
Trial
Mass (g)
1
2.5270
2
2.5271
3
2.5272
4
2.5271
5
2.5269
• The balance to the left can
measure out to four decimal
places, or to the nearest
0.0001g.
• Imagine that we use this
balance to measure the mass of
a penny 5 times, and we obtain
the results shown.
We can express the result as an average value plus or minus some
deviation that includes all observed values. This means that all results
for this measurement are expected to fall somewhere in this range.
𝐴𝑣𝑒𝑟𝑎𝑔𝑒: 2.5271
𝑀𝑖𝑛: 2.5269 = Average −. 𝟎𝟎𝟎𝟐
𝑀𝑎𝑥: 2.5272 = Average + . 𝟎𝟎𝟎𝟏
𝑹𝒆𝒑𝒐𝒓𝒕𝒆𝒅 𝒗𝒂𝒍𝒖𝒆: 𝟐. 𝟓𝟐𝟕𝟏 ± . 𝟎𝟎𝟎𝟐
Reporting Precision: Significant Figures
• The number of significant figures used to report a value
indicates how certain we are of the value. The two balances
below have very different limits of precision. As you can see,
the second balance can only report to two decimal places.
• Let‘s revisit the penny example. While the 1st balance would
report 2.5271g, the 2nd must report this value as 2.53g.
Therefore, the 1st result has 5 significant figures, the 2nd has 3
significant figures. “Significant” means that there is confidence
in the value. There is greater certainty in the 1st reading.
Measuring Precision: Significant Figures
• There are two types of numbers: exact and inexact
– Exact numbers have defined values and possess an infinite
number of significant figures because there is no limit of
confidence:
* There are exactly 12 eggs in a dozen
* There are exactly 24 hours in a day
* There are exactly 1000 grams in a kilogram
– Inexact number are obtained from measurement. Any number
that is measured has error because:
• Limitations in equipment
• Human error
Determining the Number of Significant Figures In a Result
• 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. To prove this, write these
values in scientific notation.
– .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)
Determining the Number of Significant Figures In a Result
• Zeros at the end of an integer may or may not be significant
– 130 (2 or 3), 1000 (1, 2, 3, or 4)
• This is based on scientific notation.
– 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
*This will depend on the precision of the result.
*Numbers that must be treated as significant CAN NOT
disappear in scientific notation
Calculations Involving Significant Figures
• You can not get exact results using inexact numbers
• Multiplication and division
– Result can only have as many significant figures as the
least precise number
6.2251 𝑐𝑚 𝑥 𝟓. 𝟖𝟐 𝑐𝑚 = 36.230082 𝑐𝑚2 = 36.2 𝑐𝑚2
(3 s.f.)
105.86643 𝑚
𝑚
𝑚
𝑚
= 108.0269694
= 1𝟏0
𝑜𝑟 1.1 𝑥 102
0. 𝟗𝟖 𝑠
𝑠
𝑠
𝑠
(2 s.f.)
𝑚
𝑘𝑔 𝑚
𝑘𝑔 𝑚
𝑘𝑔 𝑚
5
43270.0 𝑘𝑔 𝑥 𝟒 2 = 173080
= 200000
𝑜𝑟 2 𝑥 10
𝑠
𝑠2
𝑠2
𝑠2
(1 s.f.)
Calculations Involving Significant Figures
• Addition and Subtraction
– Result is only as precise as the least precise number.
20.4
1.322
83
+ 104.722
211.942
Limit of certain is the ones place
212
Group Work
• Using scientific notation, convert 0.000976392 to 3 sig. figs.
• Using scientific notation, convert 198207.6 to 1 sig. fig.
H=10.000 cm
W = .40 cm
L = 31.00 cm
•
•
Volume of rectangle ?
Surface area (SA = 2WH + 2LH + 2LW) ?
note: constants in an equation are exact numbers
= 2(10.000 𝑐𝑚) .40 𝑐𝑚 + 2(10.000 𝑐𝑚) 31.00𝑐𝑚2 + 2(31.00 𝑐𝑚)(.40 𝑐m)
= 2 4. 𝟎𝑐𝑚2 + 2 310. 𝟎𝑐𝑚2 + 2(1𝟐. 4𝑐𝑚2 )
= 8. 𝟎𝑐𝑚2 + 620. 𝟎𝑐𝑚2 + 2𝟒. 8𝑐𝑚2
= 65𝟐. 8𝑐𝑚2 = 65𝟑𝑐𝑚2
Limit of
certainty is
the ones
place
Dimensional Analysis
• Dimensional analysis is an algebraic method used to
convert between different units
• Conversion factors are required
– Conversion factors are exact numbers which are
equalities between one unit set 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 1 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’
1 𝑓𝑜𝑜𝑡
𝟓𝟔 𝑖𝑛𝑐ℎ𝑒𝑠 𝑥
= 4.6666 𝑓𝑡
12 𝑖𝑛𝑐ℎ𝑒𝑠
4.7 ft
Group Work
• Answer the following using dimensional analysis. Consider
significant figures.
– Convert 35 minutes to hours
– Convert 40 weeks to seconds
– Convert 4 gallons to cm3 (1 gal = 4 quarts, 1 quart = 946.3 mL)
– *35
𝒎𝒊𝒍𝒆𝒔
𝒉𝒓
to
𝒊𝒏𝒄𝒉𝒆𝒔
𝒔𝒆𝒄
(1 mile = 5280 ft and 1 ft = 12 in)
High Order Exponent Unit Conversion (e.g. 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
1 𝑚3
𝑥
𝑐𝑚 3
𝒎𝑳
𝑥
10−2 𝑚
𝒄𝒎𝟑
Must use this equivalence to convert
from cubic length to capacity
Cube this conversion factor
=1
𝑚3
𝒄𝒎𝟑
𝑚𝐿
𝑥
𝑥
𝟏𝟎−𝟔 𝒎𝟑 𝑐𝑚3
= 𝟏 𝒙 𝟏𝟎𝟔 𝒎𝑳
Group Work
• Convert 10 mL to m3 (c = 10-2)
• Air has a density of 1.024 kg/m3. Determine the mass, in
grams, of air in a room that is 15 ft x 18 ft x 11 ft., given that 1
ft = 0.3048 m.