Transcript Converting Between the Fahrenheit and Celsius Scale

Chapter 2
Measurements and
Calculations
2.1 Scientific Notation
 Scientific
notation is a method for making
very large or very small numbers more
compact and easy to write.
 Scientific notation simply expresses a
number as a product of a number between
1 and 10, and the appropriate power of 10.
 Example: 93,000,000 = 9.3 x 10,000,000 =
9.3 x 107
 To
determine the power of ten you must
determine how many places the decimal
must be moved to make the first number a
number between 1 and 10. Example:
0.000167 = 1.67 x 10-4
 In the above example the decimal point is
moved four times, so the exponent is four.
 When the number is smaller than one the
exponent is negative.
Practice
Represent the following numbers in scientific
notation
1. 238,000
2. 1,500,000
3. 0.00043
4. 0.089
5. 357
6. 0.0055
Calculations in Scientific Notation

To multiply numbers in scientific notation multiply
the coefficients and add the exponents
 To divide numbers in scientific notation divide
the coefficients and subtract the exponent in the
denominator from the exponent in the numerator
 Before adding or subtracting numbers in
scientific notation you must make the exponents
the same by moving the decimal point. Once
the exponents are the same align the decimal
points and add the numbers.
Examples

Multiplication
(3.0 x 104) x (2.0 x 102) = (3.0 x 2.0) x 104+2
= 6.0 x 106
 Division
3.0 x 104 / 2.0 x 102 = (3.0 / 2.0) x 104-2 =
1.5 x 102
 Addition / Subtraction
5.40 x 103 + 6.0 x 102 = 54.0 x 102 + 6.0 x 102 =
60.0 x 102 = 6.0 x 103
Examples
1.
2.
3.
4.
Multiply 3.9 x 107 by 1.2 x 102
Divide 8.4 x 10-3 by 4.7 x 104
Add 2.3 x 102 and 1.9 x 101
Subtract 3.1 x 10-3 from 8.5 x 10-2
2.2 Units
 The
units part of a measurement tells what
scale or standard is being used to
represent the results of the measurement.
 There are many different systems of
measurement, but the two most common
are the English system and the metric
system.
 The metric system is used in science.
 In
1960 scientists had a conference to
agree on which units to use. The system
adopted at the conference is the
International System (le Systeme
International), or the SI system.
 The SI system uses the units of the metric
system. In the SI system uses prefixes to
change the size of the unit.
2.3 Measurements of Length,
Volume, and Mass
 In
the SI system there are seven base
units from which all other units are
derived.
 Derived units are used for measurements
such as volume, which come from the
units for length.
Base Units
Quantity
Measured
Length
Mass
Temperature
Time
Luminous
Intensity
Current
Amount of
substance
Unit Electric
Abbreviation
meter
kilogram
Kelvin
seconds
candella
M
Kg
K
S
Cd
ampere
mole
A
Mol
Units of Volume







The space occupied by a sample of matter is called its
volume.
To calculate the volume of any cubic or rectangular
object you multiply length, width, and height.
The unit for volume therefore comes from the meter
which is used to measure length, width, and height.
The liter is the most common unit used to measure
volume.
A liter is one cubic decimeter (1 L = 1 dm3). A decimeter
is equal to 10 centimeters.
A smaller unit of volume is the milliliter. A milliliter is
equal to one cubic centimeter (1 mL = 1 cm3).
The volume of any substance will change with
temperature.
Units of Mass
 Weight
is a force that measures the pull of
gravity on an object.
 Mass is different from weight. Mass
measures the amount of matter in an
object.
 The weight of an object can change with
location, but the mass does not.
 An object can become weightless, but not
massless.
Prefix
Unit
abbreviation
Meaning
Example
Giga
G
1,000,000,
000
1 Gm =
1,000,000,
000 m
Mega
M
1,000,000
1 Mm =
1,000,000
Kilo
K
1,000
1 km =
1,000 m
Hecto
h
100
1 hm = 100
m
2.4 Uncertainty in Measurement
 Whenever
a measurement is made an
estimate is required.
 The numbers that are known because the
measuring device has graduations are
called certain numbers.
 The last digit is estimated, and is called an
uncertain number.





In lab, when measurements are made, they
should be made with all of the numbers known
with certainty plus one digit that is estimated.
A measurement always has some degree of
uncertainty.
The amount of uncertainty depends on the
device used to make the measurement.
The numbers recorded in a measurement are
called significant figures.
The number of significant figures is determined
by the uncertainty in the measuring device.
2.5 Significant Figures
 Chemistry
often requires many types of
calculations.
 Uncertainty accumulates as calculations
are performed on numbers with
estimation, for this reason there are rules
for working with significant figures.
Rules for Counting Significant
Figures
1.
2.
3.
Nonzero integers. Nonzero integers always count as significant
figures.
Zeros

Leading zeros. Leading zeros are zeros that precede all of the
nonzero digits. They never count as significant figures.

Captive zeros. Captive zeros are zeros that fall between
nonzero digits. Captive zeros always count as significant
figures.

Trailing zeros. Trailing zeros are zeros are the right end of a
number. They are significant only if the number is written with
a decimal point.
Exact numbers. Often calculations involve numbers that were not
obtained using measuring devices but were determined by
counting. They can be assumed to have an unlimited number of
significant figures. Exact number can also arise from definitions.
Sig Figs & Scientific Notation
 Rules
for counting significant figures also
apply to numbers in scientific notation.
 Scientific notation offers two major
advantages: the number of significant
figures can be easily determined and
fewer zeros are needed to write a very
large or very small number.
Example 2.3
1. Give the number of significant figures for each
of the following measurements.
a. A sample of orange juice contains 0.0108 g of
vitamin C.
b. A forensic chemist in a crime lab weighs a
single hair and records its mass as 0.0050060
g.
c. The distance between two points was found to
be 5.030 x 103 ft.
d. In yesterdays bicycle race, 110 riders started
but only 60 finished
Self-Check Exercise 2.2
2. Give the number of significant figures for
each of the following measurements.
a. 0.00100 m
b. b. 2.0800 x 102 L
c. 480 Corvettes
Rounding Off Numbers
 When
you perform a calculation on your
calculator, the number of digits displayed
is usually greater than the number of
significant figures that the result should
possess.
 Therefore you must “round off” your
answer.
 The rules for rounding off are on the next
slide.
Rules for Rounding Off
If the digit to be removed
1.


Is less than five, the preceding digit stays the same.
Is equal to or greater than 5, the preceding digit is
increased by 1.
In a series of calculations, carry the extra digits
through to the final result and then round off.
When rounding off, use only the first number to the
right of the last significant digit.
2.
Rules for Using Significant
Figures in Calculations
1.
2.


For multiplication and division, the number of
significant figures in the result is the same as
that in the measurement with the smallest
number of significant figures.
For addition or subtraction, the limiting term is
the one with the smallest number of decimal
places.
Note that for multiplication and division
significant figures are counted.
For addition and subtractions, the decimal
places are counted.
Example 2.4

a.
b.
c.
d.
Without performing the calculations, tell
how many significant figures each
answer should contain.
5.19 + 1.9 +0.842
1081 – 7.25
2.3 x 3.14
the total cost of 3 boxes of candy as
$2.50 a box
Example 2.5

a.
b.
c.
d.
e.
Carry out the following mathematical
operations and give each result to the
correct number of significant figures.
5.18 x 0.0208
(3.60 x 10-3) x (8.123) / 4.3
21 +13.8 +130.36
116.8 – 0.33
(1.33 x 2.8) + 8.41
Self- Check Exercise 2.3

a.
b.
c.
Give the answer for each calculation to
the correct number of significant figures.
12.6 x 0.53
(12.6 x 0.53) – 4.59
(25.36 – 4.15) / 2.317
2.6 Problem Solving and
Dimensional Analysis
 Converting
from one unit to another is
encountered frequently in daily life.
 Dimensional analysis can be used to help
convert from one unit to another.
 To change from one unit to another a
conversion factor is used.
 A conversion factor is a ratio that relates
two units to each other.

An equivalence statement is a statement that
gives two different units that are equal to each
other. Example: 100 cm = 1.00 m
 When using dimensional analysis units cancel.
Choose a conversion factor that cancels the
units you want to discard and leaves the units
we want in the result.
 Always check to make sure your answer makes
sense. This is a good rule for any problem you
are completing.
 When
using dimensional analysis units
cancel. Choose a conversion factor that
cancels the units you want to discard and
leaves the units we want in the result.
 Always check to make sure your answer
makes sense. This is a good rule for any
problem you are completing
Example 2.6
 An
Italian bicycle has its frame size given
as 62 cm. What is the frame size in
inches?
Example 2.7
 The
length of a marathon race is
approximately 26.2 mi. What is the
distance in kilometers?
Self-Check Exercise 2.5
 Racing
cars at the Indianapolis Motor
Speedway now routinely travel around the
track at an average speed of 225 mi/h.
What is this speed in kilometers per hour?
2.7 Temperature Conversions:
An Approach to Problem Solving
 The
Fahrenheit scale is widely used in the
United States and Great Britain, and is the
scale employed in the engineering
sciences.
 The Celsius scale is a second temperature
scale used in Canada, Europe and in the
physical and life sciences.
 On
both the Fahrenheit and Celsius scale
the unit of temperature is called the
degree, and the symbol for it is followed by
the capital letter representing the scale on
which the units are measured.
 A third temperature scale is the absolute
scale, or the Kelvin scale. The unit of
temperature is called the Kelvin and is
symbolized by K.
Some important facts about the
three temperature scales



The size of each temperature unit is the
same on the Kelvin and Celsius scales.
The Fahrenheit degree is smaller that the
Celsius and Kelvin unit.
The zero points are different on all three
scales.
Converting between the Kelvin
and Celsius Scales
 Converting
from Celsius to Kelvin involves
add 273 to the Celsius temperature.
 In order to convert from Kelvin to Celsius,
simply subtract 273 from the Kelvin
temperature.
Example 2.8
 The
boiling point of water at the top of Mt.
Everest is 70.ºC. Convert the temperature
to the Kevin scale
Example 2.9
 Liquid
nitrogen boils at 77 K. What is the
boiling point of nitrogen on the Celsius
scale?
Self-Check Exercise 2.6
 Which
75ºC?
temperature is colder, 172 K or -
Converting Between the
Fahrenheit and Celsius Scale
 The
conversion between Fahrenheit and
Celsius requires adjusting for the different
size units and the different zero points.
 The factor 1.80 is multiplied by the Celsius
temperature to convert between the
different sized units, and then 32 is added
to the account for the different zeros.
Example 2.10
 On
a summer day the temperature in the
laboratory is 28ºC. Express this using the
Fahrenheit scale.
Example 2.11
 Express
the temperature -40. ºC on the
Fahrenheit scale.
Self-Check Exercise 2.7
 Hot
tubs are often maintained at 41ºC.
What is this temperature in Fahrenheit?
Example 2.12
 On
of the body’s responses to injuries is to
elevate its temperature. A certain flu
victim has a body temperature of 101ºF.
What is this temperature on the Celsius
scale?
Self-Check
anti-freeze solution in a car’s radiator
boils at 239ºF. What is this temperature
on the Celsius scale?
 An