Chapter 1
Measurement and
Problem Solving
By Dr. Rana Nabil Malhas
Introductory
Chemistry, 6th Edition
Nivaldo Tro
What Is a Measurement?
• Quantitative observation.
• Every measurement has a number
and a unit.
2
1.1 Scientific Notation
• A way of writing large and small numbers.
The sun’s
diameter is
1,392,000,000 m.
An atom’s
average diameter is
0.000 000 000 3 m.
Big and Small Numbers
3
Scientific Notation
• To compare numbers written in scientific notation:
• First compare exponents on 10.
• If exponents are equal, then compare decimal numbers
Exponent
1.23 x
Decimal part
10-8
1.23 x 105 > 4.56 x 102
4.56 x 10-2 > 7.89 x 10-5
7.89 x 1010 > 1.23 x 1010
Exponent part
4
Standard form to Scientific :
move to right, exp. (-)
move to left, exp. (+)
Scientific to Standard form:
move to right, exp. (+)
move to left, exp. (-)
5
Writing a Number in Scientific
Notation, Continued
1. Locate the decimal point.
important
12340
12340.
↑
2. Move the decimal point to obtain a number between 1 and
10.
1.234
3. Multiply the new number by 10n .
• Where n is the number of places you moved the decimal
point.
1.234 x 104
4. If you moved the decimal point to the left, then n is +; if you
moved it to the right, then n is − .
1.234 x 104
⑩@
>
-
1
.
23
-
4: x104
⑭
6
Writing a Number in Scientific Notation, Continued
0.00012340
1.
Locate the decimal point.
0.00012340
2.
Move the decimal point to obtain a number between 1
and 10.
1.2340
3.
Multiply the new number by 10n .
• Where n is the number of places you moved the decimal
point.
1.2340 x 104
4.
If you moved the decimal point to the left, then n is +; if
you moved it to the right, then n is − .
1.2340 x 10-4
7
Writing a Number in Standard Form
1.234 x 10-6
• Since exponent is -6, make the number smaller by
moving the decimal point to the left 6 places.
0
.
000001 234
.
• When you run out of digits to move around, add zeros.
• Add a zero in front of the decimal point for decimal
numbers.
000 001.234
0.000 001 234
8
Practice
-
write the
Notation
1
123 4
.
[1
.
.
234X102
46 000
.
2
4
following in scientific
8 0012
.
E8
.
.
0012 X10
:
.
45 x105
.
5
6
:
[
0
.
00234
2 34 X10
:
3
.
2G
ub
S
1
.
.
O
HE
45
X100
5
.
0
.
0123
[123 x 108
.
0
E
.
000
008306
8 706 x 10
.
-
6
Example 1
• The U.S. population in 2007 was estimated to be
301,786,000 people. Express this number in
scientific notation.
• 301,786,000 people = 3.01786 x 108 people
9
Practice—Write the Following in Scientific
Notation
123.4
8.0012
145000
0.00234
25.25
0.0123
1.45
0.000 008706
10
Practice 1 —Write the Following in Scientific
Notation, Continued
123.4 = 1.234 x 102
8.0012 = 8.0012 x 100
145000 = 1.45 x 105
0.00234 = 2.34 x 10-3
25.25 = 2.525 x 101
0.0123 = 1.23 x 10-2
1.45 = 1.45 x 100
0.000 008706 = 8.706 x 10-6
11
Practice 2 —Write the Following in
Standard Form
2.1 x 103
4.02 x 100
9.66 x 10-4
3.3 x 101
6.04 x 10-2
1.2 x 100
12
Practice 2 answer—Write the Following
in Standard Form, Continued
2.1 x 103 = 2100
4.02 x 100 = 4.02
9.66 x 10-4 = 0.000966
3.3 x 101 = 33
6.04 x 10-2 = 0.0604
1.2 x 100 = 1.2
13
1.2 Significant Figures
Writing numbers to reflect precision.
Significant Figures
Rules:
❖
Zeros on the left of a no is not sig.
❖
Zeros between numbers are sig.
❖
•
•
Zeros on the right of number:
sig. if there is a decimal point
Not sig. if there is no decimal point
15
Significant Figures
• Any non-zero digits is significant
• 1.5 has 2 significant figures.
• Zeros on the left of any number are not significant
•
0.001050 has 4 significant figures.
• Zeros on the right of a number is only significant if
there is only decimal point other wise its not
significant
• Example:
• 12.0 has 3 Significant Figures
• 12.30 has 4 Significant Figures
• 120 has 2 Significant Figures
16
Example—Determining the Number of
Significant Figures in a Number, Continued
• How many significant figures are in each of the
following numbers?
0.0035
1.080
2371
2.97 × 105
2 significant figures—leading zeros are
not significant.
4 significant figures—trailing and interior
zeros are significant.
4 significant figures—All digits are
significant.
3 significant figures—Only decimal parts
count as significant.
17
Determine the Number of Significant Figures, the
Expected Range of Precision, and Indicate the
Last Significant Figure
• 12000
↳2
• 120.
Es
• 12.00
↳
H
• 1.20 x 103
↳
• 0.0012
↳
z
① I-9
② Lero
significant
left
not
significant
• 0.00120
↳3
• 1201
E
H
• 1201000
↳
4
18
Determine the Number of Significant Figures, the
Expected Range of Precision, and Indicate the
Last Significant Figure, Continued
• 12000
2
•*
0.0012
2
12
d
the
• 120. 3
•&
0.00120
3
• 12.00
4
• 1201
4
• 1.20 x 103
3
• 1201000
4
3 Zeros are
.
counted
19
not
Rounding
•
•
•
When rounding to the correct number of significant figures, if
the number after the place of the last significant figure is:
0 to 4 (under 5), stay the same
5 to 9, round it up.
Examples:
round to 2 significant figures.
• 2.34 rounds to 2.3.
• 2.37 rounds to 2.4.
• 2.349865 rounds to 2.3.
20
More Rounding examples
Round to 2 significant figures
• 0.0234 rounds to 0.023 or 2.3 × 10-2.
• 0.0237 rounds to 0.024 or 2.4 × 10-2.
• 0.02349865 rounds to 0.023 or 2.3 × 10-2.
• 234 rounds to 230 or 2.3 × 102 .
• 237 rounds to 240 or 2.4 × 102 .
• 234.9865 rounds to 230 or 2.3 × 102 .
21
1.3: Significant Figures in Calculations
1.3.1: Multiplication and Division with Significant
Figures
• When multiplying or dividing measurements with
significant figures, the result has the same number
of significant figures as the measurement with the
fewest number of significant figures.
5.02 × 89,665 × 0.10 = 45.0118 = 45
3 sig. figs.
5 sig. figs.
5.892 ÷
4 sig. figs.
2 sig. figs.
2 sig. figs.
6.10 = 0.96590 = 0.966
3 sig. figs.
3 sig. figs.
22
0v82u4
Determine the Correct Number of Significant Figures for Each
Calculation and Round and Report the Result
1. 1.01 × 0.12 × 53.51 ÷ 96 = 0.067556
2. 56.55 × 0.920 ÷ 34.2585 = 1.51863
23
Determine the Correct Number of Significant Figures for Each
Calculation and Round and Report the Result, Continued
1. 1.01 × 0.12 × 53.51 ÷ 96 = 0.067556 = 0.068
3 sf
2 sf
4 sf
2 sf
Result should 7 is in place
have 2 sf. of last sig. fig.,
2. 56.55 × 0.920 ÷ 34.2585 = 1.51863 = 1.52
4 sf
3 sf
6 sf
number after
is 5 or greater,
so round up.
Result should 1 is in place
have 3 sf. of last sig. fig.,
number after
is 5 or greater,
so round up.
24
1.3.2: Addition and Subtraction with Significant Figures
• When adding or subtracting measurements with
significant figures, the result has the same number
of decimal places as the measurement with the
fewest number of decimal places.
5.74 + 0.823 +
2.651 = 9.214 = 9.21
2 dec. pl.
4.8 1 dec. pl
3 dec. pl.
3.965
3 dec. pl.
3 dec. pl.
=
0.835 =
2 dec. pl.
0.8
1 dec. pl.
25
Determine the Correct Number of
Significant Figures for Each Calculation and
Round and Report the Result
1. 0.987 + 125.1 – 1.22 = 124.867
2. 0.764 – 3.449 – 5.98 = -8.664
26
Determine the Correct Number of
Significant Figures for Each Calculation and
Round and Report the Result, Continued
1. 0.987 + 125.1 – 1.22 = 124.867 = 124.9
3 dp
1 dp
2 dp
2. 0.764 – 3.449 – 5.98 = -8.664
3 dp
3 dp
2 dp
8 is in place
of last sig. fig.,
number after
is 5 or greater,
-8.66so round up.
Result should
have 1 dp.
=
Result should
have 2 dp.
6 is in place
of last sig. fig.,
number after
is 4 or less,
so round down.
27
Both Multiplication/Division and Addition/Subtraction
with Significant Figures
• When doing different kinds of operations with
measurements with significant figures, evaluate the
significant figures in the intermediate answer, then
do the remaining steps.
• Follow the standard order of operations.
• Please Excuse My Dear Aunt Sally.
( )→
n
→→ +-
3.489 × (5.67 – 2.3) = 12
2 dp
1 dp
3.489 ×
3.4 =
4 sf
2 sf
28
Example —Perform the Following Calculations to
the Correct Number of Significant Figures
a ) 1 .1 0  0 .5 1 2 0  4 .0 0 1 5  3 .4 5 5 5
b)
0.355
+ 105.1
− 100.5820
c)
4 .5 6 2  3 .9 9 8 7 0  (4 5 2 .6 7 5 5 − 4 5 2 .3 3 )
d)
(1 4 . 8 4  0 . 5 5 ) − 8 . 0 2
29
Example—Perform the Following Calculations to the
Correct Number of Significant Figures, Continued
a ) 1 .1 0  0 .5 1 2 0  4 .0 0 1 5  3 .4 5 5 5 = 0 .6 5 2 1 9 = 0 .6 5 2
b)
0.355
+ 105.1
− 100.5820
4.8730 = 4.9
c)
4 .5 6 2  3 .9 9 8 7 0  (4 5 2 .6 7 5 5 − 4 5 2 .3 3 ) = 5 2 .7 9 9 0 4 = 5 3
d)
(1 4 . 8 4  0 . 5 5 ) − 8 . 0 2 =
0 .1 4 2 = 0 .1
30
1.4 Basic Units of Measure
The Standard Units
• Scientists generally report results in an agreed
upon International System.
• The SI System
• Aka Système International
Quantity
Length
Mass
Time
Temperature
Unit
meter
kilogram
second
kelvin
Symbol
m
kg
s
K
32
Some Standard Units in the
Metric System
Quantity
Measured
Name of
Unit
Abbreviation
Mass
gram
g
Length
meter
m
Volume
liter
L
Time
seconds
s
Temperature
Kelvin
K
33
Common Prefixes in the
SI System
Prefix
Symbol
Power of 10
mega-
M
Base x 106
kilo-
k
Base x 103
deci-
d
Base x 10-1
centi-
c
Base x 10-2
milli-
m
Base x 10-3
micro-
m or mc
Base x 10-6
nano-
n
Base x 10-9
34
Common Units and Their Equivalents, Continued
Mass
1 kilogram (km) = 2.205 pounds (lb)
1 pound (lb) = 453.59 grams (g)
1 ounce (oz) = 28.35 (g)
Volume
1 liter (L)
1 liter (L)
1 liter (L)
1 U.S. gallon (gal)
=
=
=
=
1000 milliliters (mL)
1000 cubic centimeters (cm3)
1.057 quarts (qt)
3.785 liters (L)
35
Units
• Always include units in your calculations.
• You can do the same kind of operations on units as you
can with numbers.
• cm × cm = cm2
• cm + cm = cm
• cm ÷ cm = 1
• Using units as a guide to problem solving is called
dimensional analysis.
36
1.5 Problem Solving and
Dimensional Analysis, Continued
desired unit
start unit
= desired unit
start unit 
related unit desired unit
start unit 

= desired unit
start unit
related unit
37
1.6 Density
40
Density: Ratio of mass:volume.
• Its value depends on the kind of material, not the
amount.
• Solids = g/cm3
M ass
• 1 cm3 = 1 mL
Density =
• Liquids = g/mL
• Gases = g/L
• Density : solids > liquids > gases
Volume
• Except ice is less dense than liquid water!
• D α M ; if D increase , M increase M
• D α 1/ V; if D increase , V decrease
41
Using Density in Calculations
Solution Maps:
Mass
Density =
Volume
m, V
D
Mass
Volume =
Density
m, D
V
V, D
m
M ass = D ensity  V olume
42
She places the ring on a balance and finds it has a mass of 5.84 grams.
She then finds that the ring displaces 0.556 cm3 of water. Is the ring
made of platinum? (Density Pt = 21.4 g/cm3)
Given: Mass = 5.84 grams
Volume = 0.556 cm3
Find: Density in grams/cm3
Equation: m
V
=d
Solution Map:
m and V → d
m, V
m
=d
V
d
43
She places the ring on a balance and finds it has a mass of 5.84 grams.
She then finds that the ring displaces 0.556 cm3 of water. Is the ring
made of platinum? (Density Pt = 21.4 g/cm3)
Apply the Solution Map:
m
=d
V
m, V
m
=d
V
d
5.84 g
g
= 10.5
3
3
cm
0.556 cm
Since 10.5 g/cm3  21.4 g/cm3, the ring cannot be platinum.
44
Example:
A 55.9 kg person displaces
57.2 L of water when
submerged in a water tank.
What is the density of the
person in g/cm3?
Information:
Given: m = 55.9 kg
V = 57.2 L
Find: density, g/cm3
Solution Map: m,V→ d
Equation: d = m
V
Density =
(55.9 x 103)g / (57.2 x103 cm3)
= 0.9772727 g/cm3
= 0.977 g/cm3
45
Example:
A 55.9 kg person displaces
57.2 L of water when
submerged in a water tank.
What is the density of the
person in g/cm3?
Information:
Given: m = 5.59 x 104 g
V = 5.72 x 104 cm3
Find: density, g/cm3
Solution Map: m,V→ d
Equation: d = m
V
• Apply the solution maps—equation.
m
5.59 x 104 g
d =
=
V 5.72 x 104 cm3
= 0.9772727 g/cm3
= 0.977 g/cm3
46
1.7: Temperature conversion Temperature Scales
• Fahrenheit scale, °F.
• Celsius scale, °C.
• Kelvin scale, K.
47
Fahrenheit vs. Celsius
(
 F - 32 )
C =
1.8
F = 1.8 * C + 32
Kelvin vs. Celsius
K = C + 273
48
Example —Convert –25 °C to Kelvins
K = (-25 ºC) +273 = 248 K
Example B) —Convert 55° F to Celsius
C =
( 55  F - 32 )
1.8
= 12 . 778  C
Example C—Convert 310 K to Fahrenheit
C = 310 − 273 = 37 C
 F = 1 .8 (3 7  C ) + 3 2 = 9 8 .6  F
49
Thank You
College of Engineering
#