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Fundamentals of General, Organic,
and Biological Chemistry
5th Edition
Chapter Two
Measurements in
Chemistry
James E. Mayhugh
Oklahoma City University
2007 Prentice Hall, Inc.
Outline
► 2.1 Physical Quantities, Metric System
► 2.2 Measuring Mass
► 2.3 Measuring Length and Volume
► 2.4 Measurement and Significant Figures
► 2.5 Scientific Notation
► 2.6 Rounding Off Numbers
► 2.7 Converting a Quantity from One Unit to Another
► 2.8 Problem Solving: Estimating Answers
► 2.9 Measuring Temperature
► 2.10 Energy and Heat
► 2.11 Density
► 2.12 Specific Gravity
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2.1 Physical Quantities
Physical properties such as height, volume, and
temperature that can be measured are called physical
quantities. Both a number and a unit of defined size
is required to describe physical quantity.
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► A number without a unit is meaningless.
► To avoid confusion, scientists have agreed on a standard set of
units.
► Scientists use SI or the closely related metric units.
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► Scientists work with both very large and very
small numbers.
► Prefixes are applied to units to make saying and
writing measurements much easier.
► The prefix pico (p) means “a trillionth of.”
► The radius of a lithium atom is 0.000000000152
meter (m). Try to say it.
► The radius of a lithium atom is 152 picometers
(pm). Try to say it.
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Frequently used prefixes are shown below.
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2.2 Measuring Mass
► Mass is a measure of the amount of matter in an
object. Mass does not depend on location.
► Weight is a measure of the gravitational force
acting on an object. Weight depends on location.
► A scale responds to weight.
► At the same location, two objects with identical
masses have identical weights.
► The mass of an object can be determined by
comparing the weight of the object to the weight
of a reference standard of known mass.
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a) The single-pan balance with sliding
counterweights. (b) A modern electronic balance.
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Relationships between metric units of mass and the
mass units commonly used in the United States are
shown below.
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Problem
►Express 3.27 mg in units of grams.
►Express 19.3 kg in units of mg.
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2.3 Measuring Length and Volume
► The meter (m) is the standard measure of length or
distance in both the SI and the metric system.
► Volume is the amount of space occupied by an
object. A volume can be described as a length3.
► The SI unit for volume is the cubic meter (m3).
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Relationships between metric units of length and
volume and the length and volume units commonly
used in the United States are shown below and on the
next slide.
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A m3 is the volume of a cube 1 m or 10 dm on edge.
Each m3 contains (10 dm)3 = 1000 dm3 or liters. Each
liter or dm3 = (10cm)3 =1000 cm3 or milliliters. Thus,
there are 1000 mL in a liter and 1000 L in a m3.
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The metric system is based on factors of 10 and is
much easier to use than common U.S. units. Does
anyone know how many teaspoons are in a gallon?
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2.4 Measurement and Significant
Figures
► Every experimental
measurement has a
degree of uncertainty.
► The volume, V, at right
is certain in the 10’s
place, 10mL<V<20mL
► The 1’s digit is also
certain, 17mL<V<18mL
► A best guess is needed
for the tenths place.
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► To indicate the precision of a measurement, the
value recorded should use all the digits known
with certainty, plus one additional estimated digit
that usually is considered uncertain by plus or
minus 1.
► No further insignificant digits should be
recorded.
► The total number of digits used to express such a
measurement is called the number of significant
figures.
► All but one of the significant figures are known
with certainty. The last significant figure is only
the best possible estimate.
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Below are two measurements of the mass of the
same object. The same quantity is being described
at two different levels of precision or certainty.
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► When reading a measured value, all nonzero digits
should be counted as significant. There is a set of
rules for determining if a zero in a measurement is
significant or not.
► RULE 1. Zeros in the middle of a number are like
any other digit; they are always significant. Thus,
94.072 g has five significant figures.
► RULE 2. Zeros at the beginning of a number are
not significant; they act only to locate the decimal
point. Thus, 0.0834 cm has three significant
figures, and 0.029 07 mL has four.
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► RULE 3. Zeros at the end of a number and after
the decimal point are significant. It is assumed
that these zeros would not be shown unless they
were significant. 138.200 m has six significant
figures. If the value were known to only four
significant figures, we would write 138.2 m.
► RULE 4. Zeros at the end of a number and before
an implied decimal point may or may not be
significant. We cannot tell whether they are part
of the measurement or whether they act only to
locate the unwritten but implied decimal point.
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2.5 Scientific Notation
► Scientific notation is a convenient way to
write a very small or a very large number.
► Numbers are written as a product of a number
between 1 and 10, times the number 10 raised
to power.
► 215 is written in scientific notation as:
215 = 2.15 x 100 = 2.15 x (10 x 10) = 2.15 x 102
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Two examples of converting scientific notation back to
standard notation are shown below.
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► The distance from the Earth to the Sun is
150,000,000 km. Written in standard notation this
number could have anywhere from 2 to 9 significant
figures.
► Scientific notation can indicate how many digits are
significant. Writing 150,000,000 as 1.5 x 108
indicates 2 and writing it as 1.500 x 108 indicates 4.
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2.6 Rounding Off Numbers
► Often when doing arithmetic on a pocket
calculator, the answer is displayed with more
significant figures than are really justified.
► How do you decide how many digits to keep?
► Simple rules exist to tell you how.
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RULE 1. In carrying out a multiplication or division,
the answer cannot have more significant figures than
either of the original numbers.
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►RULE 2. In carrying out an addition or
subtraction, the answer cannot have more digits
after the decimal point than either of the original
numbers.
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Rounding
► RULE 1. If the first digit you remove is 4 or less,
drop it and all following digits.
2.4271 becomes 2.4 when rounded off to two
significant figures.
► RULE 2. If the first digit removed is 5 or greater,
round up by adding 1 to the last digit kept.
4.5832 is 4.6 when rounded off to 2 significant
figures.
► If a calculation has several steps, it is best to round
off at the end.
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Problems
►Express 1,066,298 to 5 significant figures using
scientific notation.
►Express 2/3 to 4 significant figures.
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Problem
►Determine the answer to:
6.023 x 1023 x 2.32 x 10-7
______________________________
Calculator gives too many significant figures
1.397336 x 1017 ??
1.40 x 1017 (3 sig. figures)
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2.7 Problem Solving: Converting a
Quantity from One Unit to Another
► Factor-Label Method: A quantity in one unit is
converted to an equivalent quantity in a different
unit by using a conversion factor that expresses the
relationship between units.
(Starting quantity) x (Conversion factor) = Equivalent quantity
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Writing 1 km = 0.6214 mi as a fraction restates it in
the form of a conversion factor. This and all other
conversion factors are numerically equal to 1.
The numerator is equal to the denominator.
Multiplying by a conversion factor is equivalent to
multiplying by 1 and so causes no change in value.
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When solving a problem, the idea is to set up an
equation so that all unwanted units cancel, leaving
only the desired units. How many km are there in
26.22 mi (a marathon)?
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Problem
►An infant weighs 9.2 lb at birth. How many kg does
the infant weigh?
1kg = 2.205 lb is the conversion factor
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Problem
►The directions on the bottle of cough syrup say to
give 4.0 fl oz to the child, but your measuring spoon
is in units of mL. How many mL should you
administer?
1 L = 0.264 gal
1 gal = 4 qt
1 qt = 32 fl oz
1L = 1000 mL
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Problem 2.15
from McMurry:
►One international nautical mile is defined as exactly
6076.1155 ft, and a speed of 1 knot is defined as one
nautical mile per hour. What is the speed in m/sec of
a boat traveling at 14.3 knots?
What we know:
1 knot = 1 naut. mi/hr = 6076.1155 ft/hour
1 ft = 0.3048 m
1 hr = 60 min
1 min = 60 sec
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Problem Solved !
naut. mi
14.3
x
hr
units
6.073x10 3 ft
x
1 naut. mi
0.3048 m
x
1 ft
ft/hr
m/hr
1 hr
60 min
m/min
x
1 min
60 sec
m/sec
= 7.35 m/sec
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2.9 Measuring Temperature
► Temperature is commonly reported either in
degrees Fahrenheit (oF) or degrees Celsius (oC).
► The SI unit of temperature is the Kelvin (K).
► 1 Kelvin, no degree, is the same size as 1 oC.
► 0 K is the lowest possible temperature, 0 oC =
273.15 K is the normal freezing point of water.
To convert, adjust for the zero offset.
► Temperature in K = Temperature in oC + 273.15
► Temperature in oC = Temperature in K - 273.15
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Freezing point of H2O
32oF
0oC
Boiling point of H2O
212oF
100oC
212oF - 32oF = 180oF covers the same range of
temperature as 100oC - 0oC = 100oC covers.
Therefore, a Celsius degree is exactly 180/100 = 1.8
times as large as a Fahrenheit degree. The zeros on
the two scales are separated by 32oF.
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Fahrenheit, Celsius, and Kelvin temperature scales.
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► Converting between Fahrenheit and Celsius scales
is similar to converting between different units of
length or volume, but is a little more complex.
The different size of the degree and the zero offset
must both be accounted for.
►
►
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oF
= (1.8 x oC) + 32
oC = (oF – 32)/1.8
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Problem
Problem
►A patient has a temperature of 38.8oC. What is her
temperature in oF?
We know 2 equations:
oF = (1.8 x oC) + 32
oC = (oF – 32)/1.8
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2.10 Energy and Heat
► Energy: The capacity to do work or supply heat.
► Energy is measured in SI units by the Joule (J); the
calorie is another unit often used to measure energy.
► One calorie (cal) is the amount of heat necessary to
raise the temperature of 1 g of water by 1°C.
► A kilocalorie (kcal) = 1000 cal. A Calorie, with a
capital C, used by nutritionists, equals 1000 cal.
► An important energy conversion factor is:
1 cal = 4.184 J
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► Not all substances have their temperatures raised to
the same extent when equal amounts of heat energy
are added.
► One calorie raises the temperature of 1 g of water by
1°C but raises the temperature of 1 g of iron by
10°C.
► The amount of heat needed to raise the temperature
of 1 g of a substance by 1°C is called the specific
heat of the substance.
► Specific heat is measured in units of cal/gC
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► Knowing the mass and specific heat of a
substance makes it possible to calculate how
much heat must be added or removed to
accomplish a given temperature change.
► (Heat Change) = (Mass) x (Specific Heat) x
(Temperature Change)
► H in cal; mass in g; C in cal/goC; Temp in oC
► Using the symbols D for change, H for heat, m
for mass, C for specific heat, and T for
temperature, a more compact form is:
DH = mCDT
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Problem
►What is the specific heat of Aluminum if it takes 161
cal to raise the temperature of 75 g of Al by 10.0oC?
DH = mCDT
solve for C (specific heat, cal/g.oC)
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2.11 Density
Density relates the mass of an object to its volume.
Density is usually expressed in units of grams per cubic
centimeter (g/cm3) for solids, and grams per milliliter
(g/mL) for liquids.
Density =
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Mass (g)
Volume (mL or cm3)
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Problem
►What volume in mL should you measure out if you
need 16 g of ethanol which has a density of 0.79 g/mL?
(You only have a graduated cylinder, no balance.)
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►Which is heavier, a ton
of feathers or a ton of
bricks?
►Which is larger?
►If two objects have the
same mass, the one with
the higher density will
be smaller.
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2. 12 Specific Gravity
Specific gravity (sp gr): density of a substance
divided by the density of water at the same
temperature. Specific gravity is unitless. The
density of water is so close to 1 g/mL that the
specific gravity of a substance at normal
temperature is numerically equal to the density.
Density of substance (g/ml)
Specific gravity =
Density of water at the same temperature (g/ml)
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The specific gravity of a
liquid can be measured using
an instrument called a
hydrometer, which consists of
a weighted bulb on the end of
a calibrated glass tube. The
depth to which the hydrometer
sinks when placed in a fluid
indicates the fluid’s specific
gravity.
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Chapter Summary
► Physical quantities require a number and a unit.
► Preferred units are either SI units or metric units.
► Mass, the amount of matter an object contains, is
measured in kilograms (kg) or grams (g).
► Length is measured in meters (m). Volume is
measured in cubic meters in the SI system and in
liters (L) or milliliters (mL) in the metric system.
► Temperature is measured in Kelvin (K) in the SI
system and in degrees Celsius (°C) in the metric
system.
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Chapter Summary Cont.
► The exactness of a measurement is indicated by
using the correct number of significant figures.
► Significant figures in a number are all known with
certainty except for the final estimated digit.
► Small and large quantities are usually written in
scientific notation as the product of a number
between 1 and 10, times a power of 10.
► A measurement in one unit can be converted to
another unit by multiplying by a conversion factor
that expresses the exact relationship between the
units.
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Chapter Summary Cont.
► Problems are solved by the factor-label method.
► Units can be multiplied and divided like numbers.
► Temperature measures how hot or cold an object is.
► Specific heat is the amount of heat necessary to
raise the temperature of 1 g of a substance by 1°C.
► Density relates mass to volume in units of g/mL for a
liquid or g/cm3 for a solid.
► Specific gravity is density of a substance divided by
the density of water at the same temperature.
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