Transcript Solution

Math Session:
- Measurement
- Dimensional Analysis
SC155: Introduction to Chemistry
Freddie Arocho-Perez
Numbers in Science
• Integer Numbers:
2
6
• Signed Numbers:
-2
+4
• Irrational/Decimal Numbers:
2.3
0.35
-2.68
+9.87
English vs. Metric System
Physical Quantity
Metric Unit
English Unit
Mass
Gram (g)
Pound (lb)
Volume
Liter (L)
Gallon (gal)
Length
Meter (m)
Inch (in)
Time
Second (s)
Minute (min)
Temperature
Celsius (°C)
Kelvin (K)
Fahrenheit (°F)
Metric System
• Length
– Measurement of distance or dimension.
– The base unit: meter.
– It is a little over 1 yard long, more precisely 39.4 inches long. Here
are some other conversions:
1 meter (m) = 39.4 inches = 1.094 yards (about one big step)
1 meter (m) = 100 centimeters (cm)
1 kilometer (km) = 1000 meters = 0.62 miles
• Mass
– Amount of matter or material in an object.
– The base unit: gram.
– Here are some other conversions:
1 gram (g) = 0.0353 ounce
1 pound (lb) = 453.6 g
1 ounce (oz) = 28.35 grams
1 kilogram (kg) = 1000 grams
Metric System
• Volume
– Amount of space occupied
by an object.
– The base unit:
• liter (L)
• milliliter (mL)
– 1 L = 1,000 mL
– A milliliter is a cube 1 cm
long on each side (1 cm3).
– 1 mL = 1 cm3 = 1 cc
Temperature
Temperature
• In scientific measurements, the Celsius (C) and Kelvin (K)
scales are most often used.
• The Celsius scale is based on the properties of water.
– 0 C is the freezing point of water
– 100 C is the boiling point of water
Temperature
• Kelvin is one of the standard units of
temperature:
K = C + 273.15
• Celsius is the other standard unit.
• Fahrenheit is not used in scientific
measurements.
• Other Formulas:
F = (1.8 x C) + 32
C = (F - 32) x 0.555
Temperature
• If a weather forecaster predicts that the
temperature for the day will reach 31 C,
what is the predicted temperature:
(a) in K ?
(b) in F ?
• Solution:
– (a) Using Kelvin Equation, we have
K = C + 273.15
= 31 + 273.15
= 304.15 K ~ 304 K
Temperature
• Temperature: 31 C
• Solution:
– (b) Using Fahrenheit Equation, we have
F = (1.8 x C) + 32
= (1.8 x 31) + 32
= 55.8 + 32
= 87.8 F
~ 88 F
Temperature
• 85.0 F is approximately the same as?
• Solution: Use the Celsius Equation
C = (F - 32) x 0.555
= (85.0 - 32) x 0.555
= 53 x 0.555
= 29.4 C
Density
• Physical property of a substance
• Relation between mass and volume
mass
m
d

volume v
Density
• Calculate the density of mercury if 100 g
occupies a volume of 7.36 mL.
• Solution:
d=m/v
d = 100 g / 7.36 mL
d = 13.6 g/mL
mass
Density 
volume
Dimensional Analysis
• Dimensional Analysis
– Also called Factor-Label Method or
the Unit Factor Method
• This a problem-solving method that
uses the fact that any number or
expression can be multiplied by
one without changing its value.
Dimensional Analysis
• Unit factors may be made from any two
terms that describe the same or
equivalent “amounts” of what we are
interested in.
• For example, we know that:
1 inch = 2.54 centimeters
1 dozen = 12 items
Dimensional Analysis
• We can make two unit factors from this
information:
1 dozen = 12 items
OR
12 items = 1 dozen
• Arrangement:
desired unit  given unit 
desired unit
given unit
Dimensional Analysis
• How many items are in 2 dozens?
• Conversion Factor: 1 dozen = 12 items
• Solution:
? items  2 dozens 
12 items
1 dozen
 24 items
Dimensional Analysis
• How many dozens are in 6 items?
• Conversion Factor: 1 dozen = 12 items
• Solution:
? dozens  6 items 
1 dozen
12 items
 0.5 dozen
Dimensional Analysis
• How many centimeters are in 6.0
inches?
• Conversion Factor: 1 in = 2.54 cm
• Solution:
? cm  6.0 inches 
2.54 cm
1 inch
 15.2 cm
Dimensional Analysis
• How many inches are 24.0
centimeters?
• Conversion Factor: 1 in = 2.54 cm
• Solution:
? inches  24.0 cm 
1 inch
2.54 cm
 9.4 inches
Dimensional Analysis
• Convert 5.0 L to milliliters (mL).
• Conversion Factor: 1 L = 1,000 mL
• Solution:
? mL  5.0 L 
1,000 mL
1 L
 5,000 mL
Dimensional Analysis
• Convert 50.0 mL to liters (L).
• Conversion Factor: 1 L = 1,000 mL
• Solution:
? L  50.0 mL 
1 L
1,000 mL
 0.05 L
Dimensional Analysis
• If a lady has a mass of 115 lb, what is
her mass in grams?
Solution: Because we want to change from lb to g, we look for
a relationship between these units of mass. We have that
1 lb = 453.6 g. In order to cancel pounds and leave grams, we
write the conversion factor with grams in the numerator and
pounds in the denominator:
• Answer: 52,164 g
Dimensional Analysis
• You can also string many unit factors together.
• How many minutes are in 2.0 years?
? min  2.0 yr 
365 days
1 yr
= 1,051,200 minutes

24 hours
1 day

60 min
1 hour
Dimensional Analysis
• Units are a critical part of describing every
measurement.
• Before you can work with units
mathematically, you frequently must convert
from one unit to another.
• Dimensional analysis does not do your math
for you, but it makes sure you get your
multiplications and divisions straight.
• After that, all you have to do is find the
conversion factors and plug into a calculator.
Significant Figures
• The term significant figures refers to
digits that were measured.
• When rounding calculated numbers, we
pay attention to significant figures so we
do not overstate the accuracy of our
answers.
Significant Figures
1. All nonzero digits are significant.
2. Zeroes between two significant figures
are themselves significant.
3. Zeroes at the beginning of a number
are never significant.
4. Zeroes at the end of a number are
significant if a decimal point is written
in the number.
Significant Figures
• Examples: How many significant figures
are present in the following numbers?
Number
48,923
3.967
900.06
0.0004
8.1000
# Significant Figures
5
4
5
1
5
Rule(s)
1
1
1, 2
1, 3
1, 4
Significant Figures
• When math operations are performed, answers are
rounded to the number of digits that corresponds to
the least number of significant figures in any of the
numbers used in the calculation.
• Example: How many significant figures should be
shown for the following calculation?
1.25  0.45
2.734
• Answer: 0.621799561 = 0.62 (2 significant figures)
Powers of Ten
• Scientific Notation
• Way to deal with large and small
numbers: abbreviate them.
• Examples:
0.00001 = 1 x 10-5
0.005 = 5 x 10-3
3,000 = 3 x 103
100,000 = 1.0 x 105
6,000,000 = 6.0 x 106
Powers of Ten
• For numbers larger than 10, the power of 10 is a positive value and
negative for numbers less than 1.
• For numbers between 0 and 10, the power is a positive fraction.
• In the examples that follow, notice what happens to the decimal point:
100 = 1.
101 = 10.
102 = 100.
106 = 1000000.
= 1. with the decimal point moved 0 places
= 1. with the decimal point moved 1 place to the right
= 1. with the decimal point moved 2 places to the right
= 1. with the decimal point moved 6 places to the right
And
10-1 = 0.1
= 1. with the decimal point moved 1 place to the left
10-2 = 0.01
= 1. with the decimal point moved 2 places to the left
10-6 = 0.000001 = 1. with the decimal point moved 6 places to the left