Numbers
What is the length of the rod?
1
• Expressing a number correctly is determined
by the method used in the measurement!
• How many numbers should I include?
Different measurement tools give different numbers:
Which ruler is better?
Significant Digits (Figures)
Consider: the exactness of the measured value
• Short Hand expression translates the
number: Scientific Notation
? cm
4.2 - 4.3cm
? cm
4.24 - 4.25cm
© Copyright 1998-2007 R.J. Rusay
Buret
01_08
Measurement Assignment
mL
0
http://chemconnections.llnl.gov/General/Chem120/volume1.htm
10
22.2 mL
20
30
40
50
What does each line represent?
1 mL
What can be estimated?
O.1
O.1 mL
1
Temperature Scales
Relative to Water
QUESTION
Dr. R. walks into class and claims, “It is very cold in here today. It
feels like 242 K.” If that were the temperature, would you agree
that you would feel cold? What would that be in Celsius degrees?
1.
2.
3.
4.
I agree, that would be 31°C.
I agree, that would be – 31°C.
I do not agree, that would be 31°C.
I agree, that would be –31.15°C.
01_09
Precision & Accuracy
Temperature
(a)
(b)
(c)
QUESTIONS:
1) Rank the images from best to worst precision.
2) Rank the images from best to worst accuracy.
2
Precision & Accuracy
QUESTION
Two Chem 120 students are each drinking a soft drink after class.
The volumes of both containers are respectively listed as 375
milliliters. Philip remarks that the law requires bottlers to be very
precise. Susan correctly responded:
1. If precision were the only requirement, bottlers could claim
any volume as long as it was always very nearly the
same volume.
2. Since precision is a requirement, bottlers have to get exactly
375 mL in every can.
3. Bottlers must have a precise average of all of the containers in a
case of soft drinks equal to 375 mL.
4. If there were a difference of no more than +/- 1 mL between
containers, the bottlers can sell their beverage.
Numerical Data
a)
a)
b)
9.52 9.52
8.36 8.36
7.29 7.29
8.34 8.34
__________
__________
Average
8.378
Average
Round Off
8.38
Average
Round Off
c)
7.99
b)
deviation
-1.14
0.02
1.09
0.04
8.40
8.35
8.42
8.36
__________
0.573
8.383
8.38 +/- 0.57
c)
7.95 7.95
8.00 8.00
8.05 8.05
7.95 7.95
__________
__________
7.988
8.38
a)
9.52
8.36
7.29
8.34
__________
8.378
b)
8.40 8.40
8.35 8.35
8.42 8.42
8.36 8.36
__________
__________
8.383
c)
deviation
-0.02
0.03
-0.04
0.02
7.95
8.00
8.05
7.95
__________
0.028
7.988
8.38 +/- 0.03
7.99
deviation
0.04
-0.01
-0.06
0.04
0.038
+/- 0.04
Absolute value ( all of the - become +)
QUESTION
QUESTION
Rank the relative precision of the three sets of data:
a), b) and c).The accepted value is 8.08.
Rank the relative accuracy of the three sets of data:
a), b) and c).The accepted value is 8.08.
Average
Average
Average
Average
Average
a)
b)
c)
a)
b)
c)
8.38
8.38
7.99
8.38
8.38
7.99
Average
average
average
average
average
average
average
deviation
deviation
deviation
deviation
deviation
deviation
a)
b)
c)
a)
b)
c)
+/- 0.57
+/- 0.03
+/- 0.04
+/- 0.57
+/- 0.03
+/- 0.04
A) Precision: a > c > b
B) Precision: b > c > a
A) Accuracy: a > c > b
B) Accuracy: b > c > a
C) Precision: a = b > c
D) Precision: a > b > c
C) Accuracy: c > a = b
D) Accuracy: a = b > c
3
Reporting Numbers
Reporting Numbers
Rules for Significant Digits (Figures)
Rules for Significant (Digits) Figures
Nonzero integers always count
as significant figures.
3456 g has how many sig figs?
4 sig figs.
• Expressed in scientific notation?
3.456 x 10 3 g
Exact numbers (unit, conversion
or scale factors) can have an infinite
number of significant figures.
1 liter = 1,000. ml, exactly
1 inch = 2.54 cm, exactly
Systematic Problem Solving
Complete the following
Dimensional/Unit Analysis
Units & Conversions
How many mL of milk are in a1/2 gallon carton?
0.50 gal
? mL
Number
13,000,000,000 yrs.
Scientific Notation
________________
___________ mL
______________ mL
____________ kg __8.45
__8.45 x 10 -1 kg___
kg___
1 gal = 4 qt
Named unit
__? gigayears
0.546 Liters
_?
_? grams__
grams__
1 qt = 946 mL
0.50 gal | 1 qt | 946 mL
= ? mL
| 4 gal | 1 qt
4
Zeros

Zeros
Leading zeros do not count as
significant figures.
Captive zeros always count as
significant figures.
0.0486 mL has how many sig figs?
3 sig figs.
• Number expressed in scientific
notation?
4.86 x 10 -2 mL
16.0
16.07 cm has how many sig figs?
4 sig figs.
Number expressed in scientific
1.607 x 10 1 cm
notation?
Zeros
Trailing zeros are significant only if
the number contains a decimal
point.
QUESTION
Which one of the following does NOT represent a result with four
significant digits?
1.
2.
3.
4.
0.07100
0.7100
0.7010
0.0710
9.300 kg has how many sig figs?
4 sig figs.
• Number expressed in scientific
9.300 kg
notation?
5
Mathematics & Arithmetic
1
Computational Rules
• Relative to method(s) of measurement
• Short Hand expression: Scientific Notation
• Numbers : How many to include?
Quantitative vs. Qualitative
• Addition/Subtraction......
• Multiplication/Division.....
• What is “significant”?.....Rounding Off
1
• Addition/Subtraction: Answer expressed
•
to the least number of decimal places of
the figures in the process
Multiplication/Division: Answer
expressed to the least number of
significant figures
• http:dbhs.wvusd.k12.ca.us/SigFigsFable.html
© Copyright 1998-2007 R.J. Rusay
© Copyright 1998-2007 R.J. Rusay
Addition
Four students were each asked to
measure a piece of wire and provide a
total length for the four pieces.
Report the result correctly:
0.05
12.01
1.9
+
2.386
_______
cm
cm
cm
cm
QUESTION
If you were unloading a 23.50 kg box of books from your car and
a “friend” added two more 482 gram chemistry books, how much
in kg and using the rules for significant digits, would you be
lifting?
1.
2.
3.
4.
23.98 kg
24.464 kg
24.46 kg
24.5 kg
6
Mathematical Processes:
Provide correct answers assuming each value (unit
omitted) is written with the correct number of sig figs:
12.01
x
1.90
_______ _______
_______
=
2.386
QUESTION
The average mass of a certain brand of vitamin C tablets is 253 mg.
What is the mass of three such tablets rounded to the proper
number of significant digits?
1.
2.
3.
4.
0.760 grams
0.759 grams
0.7590 grams
0.253 grams
12.01
x
1.90
_______ _______
_______
+ 0.05 =
2.386
Conversion Factor Method
Density
(Dimensional Analysis)
1
• Qualitative Descriptions vs. Quantitative
• Use exact numbers / “scale factor” UNITS
• A Bookkeeping Method: Example
___ ft___in --------> ? m
• (1 ft = 12 in; 2.54 cm = 1 in; 100 cm = 1 m)
• ___ft x 12 in/ft + ___in = ___in
• ___in x 2.54 cm/in x 1 m/100cm = ___m
http://www.density.com/what.htm
• Density = Mass / Volume [g/mL or g/cm ; g/L]
• Least dense man-made solid substance:
3
1
Aerogel, D = 3.025 x 10-3 g/cm3
http://eetd.lbl.gov/ECS/aerogels/aerogels.htm
http://stardust.jpl.nasa.gov/spacecraft/aerogel.html
• D = 1.22 x 10 g/cm (1.22 g/L)
• Densest known substance: a White Dwarf
-3
air
3
http://antwrp.gsfc.nasa.gov/apod/ap961203.html
1.0 teaspoon = 3.0 T; D = ? g/cm3
(1 tsp = 4.93 mL; 1 mL = 1 cm3 )
© Copyright 1998-2007 R.J. Rusay
© Copyright 1998-2007 R.J. Rusay
7
QUESTION
QUESTION
The volume of a sample can be obtained from its density and mass.
If the mass of a sample of acid from a battery were 5.00 grams and
the density was 1.2 g/mL, what would you report in mL and with
the proper number of significant digits, as the sample volume?
Which would provide more grams of NaCl, sample one with a
mass of 2,350 mg, or sample two, a solid with a volume of 2.00
cm 3? (The density of solid salt is 2.16 g/cm 3.) Report your choice
and report the grams of the more massive sample.
1.
2.
3.
4.
1.
2.
3.
4.
6.0 mL
6.00 mL
4.2 mL
4.17 mL
Sample two; 1.08 grams
Sample two; 4.32 grams
Sample one; 2.35 grams
Sample one; 2.350 grams
Percent
1
• A comparison based on normalization to
100.
• George Washington University: 64
unsealed addressed envelopes with $10 in
each were dropped on campus in different
classrooms.
• In economics 18 of 32 were mailed, in
business, history and psychology 10 of 32
were mailed. What is the percent for each
of the 2 groups of students?
© Copyright 1998-2007 R.J. Rusay
Percent Continued
1
• The Professor conducting the study
received 43.75% of the $640 in the
mail. How much did he receive?
• How many of you would mail the
envelop presuming no one knows you
found it?
• One student mailed an empty envelop
with the return address: Mr. IOU, 1013
Indebted Lane, Bankrupt City, MS (WSJ
1/18/95)
© Copyright 1998-2007 R.J. Rusay
8