Transcript 2 or 3 significant figures

2 Measurement
Contents
2-1 Measurement of Matter: SI (Metric) Units
2-2 Converting Units
2-3 Uncertainty in Measurements
2-4 Significant Figures in Calculations
2-5 Density
2-6 Measuring Temperature
2-7 Atomic Masses
2-8 Formula Masses
2-9 Amount of Substance
The rulers
The TEM
The SEM
The speedometer
2-1
Measurement of Matter: SI(Metric) Units.
SI Units
• There are two types of units:
– fundamental (or base) units;
– derived units.
• There are 7 base units in the SI system.
The scientific community uses SI units ( the units of
International System of units) for measurement properties of
matters.
There are seven SI base units from which all other
necessary units (derived units) are derived.
Length
Although the meter is the base SI unit used for length, it
may not be convenient to report the length of an
extremely small object or an extremely large object in
units of meters.
Decimal prefixes allow us to choose a unit that is
appropriate to the quantity being measured.
a very small
object
a very large
object
measured in
millimeters (1
mm = 0.001 m)
measured in
kilometers (1 km
= 1000 m).
2-2
Converting Units
Decimal prefixes allow us to choose a unit that is
appropriate to the quantity being measured.
Selected Prefixes used in SI System
Derived Units
• Derived units are obtained from the 7 base SI units.
• Example:
units of distance
Units of velocity 
units of time
meters

seconds
 m/s
Derived Units
Note that there are no units of volume in SI system. For
measurements of volume, density, and other properties,
we must derive the desired units from SI base units.
The SI unit of mass and
volume are gram(g) and
meter (m)
Deriv
e
Density has units of grams per
cubic centimeter, g/cm3.
the basic SI
unit for
density is
Derived Units
The SI unit of
length is meter (m)
Deriv
e
The unit of
volume is cubic
meter (m3)
the basic SI
unit for
volume is
milliliters, mL
(1 mL = 1 cm3).
liters, L (1 cubic
decimeter, or 1 dm3)
Volume
• The units for volume
are given by (units of
length)3.
– SI unit for volume is 1
m3.
• We usually use 1 mL
= 1 cm3.
• Other volume units:
– 1 L = 1 dm3 = 1000 cm3
= 1000 mL.
Volume
Temperature
The SI unit of temperature is the Kelvin, although the
Celsius scale is also commonly used. The Kelvin scale is
known as the absolute temperature scale, with 0 K being the
lowest theoretically attainable temperature.
Kelvin from
Celsius
Celsius from
Fahrenheit
T (K) = t(ºC ) + 273.15
t (ºC) = 5/9 [t (º F)-32]
Definition
Kelvin scale: The absolute temperature scale; the SI unit
for temperature is the Kelvin. Zero on the Kelvin scale
corresponds to -273.15°C; therefore, K = °C + 273.15.
SI units: The preferred metric units for use in science.
Celsius scale: A temperature scale on which
water freezes at 0° and boils at 100° at sea level.
Figure 1.12
shows a
comparison
of the Kelvin,
Celsius, and
Fahrenheit
scales.
2-3
Uncertainty in Measurements
Two kinds of numbers are encountered in scientific
work:
Exact Numbers (those whose values are known
exactly)
Inexact Numbers (those whose values have some
uncertainty).
Key points
Numbers obtained by measurement are always
inexact.
2-3
Uncertainty in Measurements
Even the most carefully taken measurements are
always inexact.
This inexactness can be a
consequence of
inaccurately
calibrated
instruments
human error
any number of
other factors
• All scientific measures are subject to error.
• These errors are reflected in the number of figures
reported for the measurement.
• These errors are also reflected in the observation
that two successive measures of the same quantity
are different.
Two terms are used to describe the
quality of measurements.
precision
accuracy
The repeatability of
measurements is called precision,
which refers to how closely two or
more measurements of the same
property agree with one another.
The correctness of measurements is
called accuracy, which refers to how
close a measurement is to the true
value of a property.
The difference between the accuracy and precision
The analogy of darts stuck in a dartboard pictured in Figure 2.1
illustrates the difference between the two terms.
The relationship between the accuracy and precision
In general, the more precise a
measurement, the more accurate it is.
We again confide in the accuracy of a measurement if we
obtain nearly the same value in many different experiments.
It is possible, however, for a precise
value to be inaccurate.
If a very sensitive balance is poorly calibrated, for example, the
masses measured will be inaccurate even if they are precise.
Measurement = a number + a unit
Measurement = quantity + units + uncertainty
2-4
Significant Figures in Calculations
Significant figures
In order to convey the appropriate uncertainty in a
reported number, we must report it to the correct number
of significant figures.
Example
has three digits. All three digits are
significant. The 8 and the 3 are
"certain digits" while the 4 is the
The number 83.4 "uncertain digit."
Figure 1.14 Number of Significant figures
this number implies uncertainty
of plus or minus 0.1, or error of
1 part in 834.
The number 83.4
Thus, measured quantities are generally reported
in such a way that only the last digit is uncertain.
All digits, including the uncertain one, are called
significant figures.
Guideline
s
1. Nonzero digits are
always significant.
457 cm (3 significant figures);
2.5 g (2 significant figures).
2. Zeros between nonzero
digits are always
significant.
1005 kg (4 significant figures);
1.03 cm (3 significant figures).
0.02 g (one significant figure);
0.0026 cm (2 significant figures).
3. Zeros at the
beginning of a number
are never significant.
4. Zeros that fall at the end of a
number and after the decimal point
are always significant
0.0200 g (3 significant figures);
3.0 cm (2 significant figures).
5. When a number ends in zeros but
contains no decimal point, the zeros
may or may not be significant
130 cm (2 or 3 significant figures);
10,300 g (3, 4, or 5 significant figures).
Scientific (or exponential) notation
To avoid ambiguity with regard to the number of significant
figures in a number with tailing zeros but no decimal point,
we use scientific (or exponential) notation to express the
number.
Example
We can express it as
700 or 7.00 × 102
If we are reporting the number
700 to three significant figures,
we can express this
number as 7.0 × 102
if there really should be only
two significant figures
we can write 7 × 102.
if there should be only
one significant figure,
Scientific notation is convenient for expressing the
appropriate number of significant figures. It is also useful to
report extremely large and extremely small numbers.
the number 1.91 × 10-24
we can express the number
0.00000000000000000000000191.
Significant figures in calculation
When measured numbers are used in a calculation, the
precision of the result is limited by the precision of the
measurements used to obtain that result.
If we measure the length of one
side of a cube to be 1.35 cm,
Original number
had three
significant
figures
calculate the volume of the
cube to be 2.460375 cm3.
If we report the volume to seven significant
figures, we are implying an uncertainty of 1
part in over two million! We can't do that.
Guideline
s
In order to report results of calculations so as to imply a realistic
degree of uncertainty, we must follow the following rules.
1. the C must have the same number of significant
figures as the A or B with the fewest significant
figures.
If A x B or A / B = C
If A + B or A - B = C
2. the C can have only as many places to the right of the
decimal point as the A or B with the smallest number of
places to the right of the decimal point.
Using above rules,
If we measure the length of one
side of a cube to be 1.35 cm,
The volume of the cube
should to be 2.46 cm3.
Original number
had three
significant
figures
The significant figures of volume
should not be more than three.
Rounding off Numbers
In rounding off numbers, look at the leftmost digit to be
removed:
1.If the leftmost digit removed is less than 5, the
preceding number is left unchanged.
Thus, rounding 7.248 to two significant figures gives
7.2.
2. If the leftmost digit removed is 5 or greater, the
preceding number is increased by 1.
Rounding 4.735 to three significant figures gives 4.74,
and rounding 2.376 to two significant figures gives 2.4.
Question
1. What is the answer to the following
problem, reported to the correct
number of significant figures.
0.11807
0.1181
0.118
0.12
0.1
2. How many significant figures
are there in the number
0.0012?
1
2
3
4
5
3. How many significant figures
are there in the number
1020.5?
2
3
4
5
6
2-5
Density and percent Composition:
Their Use in Problem Solving
What weighs more, a
ton of stones or a ton
of cottons?
Density
stone
cotton
If you answer that
they weigh the
same, you
demonstrate a clear
understanding of the
meaning of mass---a measure of a
quantity of matter.
WHAT IS THE DIFFERENCE BETWEEN MASS AND WEIGHT?
Mass is defined as the amount of matter an object
has. One of the qualities of mass is that it has inertia.
Mass is a measure of how much inertia an object
shows.
The weight of an object on earth depends on the
force of attraction (gravity) between the object and
earth. Weight will change according to the force of
attraction.
Since the moon has 1/6 the mass of earth, it would
exert a force on an object that is 1/6 that on earth.
Density
• Used to characterize substances.
• Defined as mass divided by volume:
mass
Density 
volume
• Units: g/cm3.
• Originally based on mass (the density was defined
as the mass of 1.00 g of pure water).
Density in conversion Pathways
If we measure the mass of an
object and its volume, simple
division gives us its density.
Density (d) = mass(m) / volume(V)
mass(m) = Density (d) x volume(V)
volume(V) = mass(m) / Density (d)
Density is the
ratio of mass
to volume.
For example
What is the volume of a
5.25-gram sample of a liquid
with density 1.23 g/ml?
solution
Using:
volume(V) = mass(m) / Density (d)
Volume in mL = 5.25g x 1mL/ 1.23 g = 4.27 mL
Percent as a conversion factor
A common way of referring to composition is
through percentages.
definition
Percent
is the number of parts of a
constituent in 100 parts of the
whole.
a seawater sample contains 3.5%
sodium chloride by mass means
there is 3.5 g of sodium chloride
in every 100 g of the seawater
a seawater sample contains 3.5%
sodium chloride by mass means
We can express this percent by writing the following
ratios and we can use this type of ratio as a
conversion factor.
3.5 g of sodium chloride /100 g of the seawater
100 g of the seawater/ 3.5 g of sodium chloride
Questions
choice
1. A square metal sheet measures 12.3 cm on a side. It
is 3.6 mm thick and has a mass of 121.35 g. What is
the density of the metal?
a.
b.
c.
d.
e.
2.22806 g/cm3,
0.22 g/cm3,
2.2 g/cm3,
0.76 g/cm3,
1.27 g/cm3.
Questions
2. How many cubic
millimeters are there
in a cubic centimeter?
a. 10,
b.
100,
d. 0.1,
e.
1 x 103
c. 1000,
choice
Questions
3.
Which sample has the greater volume? 16.2 g
of a liquid with density 1.045 g/cm3 or 52.0 g of a
solid with density 3.354 g/cm3
choice
a. The solid
b. The liquid
c. They both have
The same volume
d.
2-6
Measuring Temperature
Temperature
There are three temperature scales:
•Kelvin Scale
–Used in science.
–Same temperature increment as Celsius scale.
–Lowest temperature possible (absolute zero) is zero
Kelvin.
–Absolute zero: 0 K = -273.15 oC.
Temperature
• Celsius Scale
– Also used in science.
– Water freezes at 0 oC and boils at 100 oC.
– To convert: K = oC + 273.15.
• Fahrenheit Scale
– Not generally used in science.
– Water freezes at 32 oF and boils at 212 oF.
– To convert:
5
C  F - 32 
9
9
F  C   32
5
Temperature
The SI unit of temperature is the Kelvin, although the
Celsius scale is also commonly used. The Kelvin scale is
known as the absolute temperature scale, with 0 K being
the lowest theoretically attainable temperature.
Kelvin from
Celsius
Celsius from
Fahrenheit
T (K) = t(ºC ) + 273.15
t (ºC) = 5/9 [t (º F)-32]
Figure 2.2
shows a
comparison
of the Kelvin,
Celsius, and
Fahrenheit
scales.
Class Practice Example
• Make the following temperature conversions: (a)
68 oF to oC; (b) -36.7 oC to oF
5
C  F - 32 
9
9
F  C   32
5
2-7
Atomic Masses
Proton: Small particles with a unit positive charge in the
nucleus of an atom.
Neutron: Particles with no charge and are present in the nuclei
of all atoms expect one isotope of hydrogen.
Atomic number: The number of protons in the nucleus.
Neutron number: The number of neutrons in the nucleus.
Isotopes: Atoms of an element that differ in mass.
Mass Number: The sum of the number of protons and
neutrons in the nucleus of an atom.
2-8
Formula Masses
Formula Masses of compounds are equal to the sum of atomic
masses of the atoms forming the compound.
Example: Na2SO4
Formula Masses of Na2SO4
=22.99×2+32.07+16.00×4=142.05
2-9
Amount of Substance
One mole is defined as the amount of substance in a sample
that contains as many units as there are atoms in exactly 12 g
of carbon-12.
Avogadro’s number: The number of units in a mole is
6.022137×1023
Molar mass: The mass of one mole in grams.
Homework
All the questions are from the textbook:
Genaral Chemistry.
P28: 1.41
P66: 2.45
P67: 2.70, 2.75