Download 3 significant figures

Scientific Method: review chapter 2, section 1 (this is also a review of the
scientific method as you have talked about it in every science class since you
have been in school). You should already know the main points of this
discussion.
Nature of Measurement
• Measurement - quantitative observation consisting of 2 parts
• Part 1 – number
• Part 2 – scale (unit)
A number without a unit is NOT a measurement!
• Examples:
• 20 grams
What quantity is this a measurement of?
• 6.63 × 104 meters per second (m/s)
What quantity is this a
measurement of?
Can you think of other examples?
A base unit is not made up of other units.
gram is an example of a base unit.
A derived unit is made up of two or more other units.
m/s is an example of a derived unit.
The 7 Fundamental SI Units (Base Units)
Quantity (symbol)
Mass
(symbol is “m”)
Name of Unit
Abbreviation
kilogram
kg
Length (symbol is “l”)
meter
m
Time
second
s
Temperature (symbol is “T”)
kelvin
K
Amount of
substance
mole
mol
Electric current (symbol is “I”)
ampere
A
Luminous
intensity
candela
cd
(symbol is “t”)
(symbol is “n”)
(symbol is “IV”)
Why is volume not listed as a quantity?
Volume is a derived unit (length x length x length = volume)!
Any unit not listed above is a derived unit.
Derived Units
Derived units are units that are defined by a combination of
two or more base units.
Volume = (length x width x height)
If distances are in meters, what would the units for
volume be? (m)x(m)x(m) = m3
(cubic meters)
A definition: 1 cm3 = 1 mL
(one cubic centimeter is one milliliter)
Density = mass ÷ volume = m/V
If mass is in grams and volume is in mL, what
would the units for density be?
(g)÷(mL) = g/mL
(gram per milliliter)
Definitions of SI Prefixes:
Big
terra (T)
giga (G)
mega (M)
kilo (k)
hecto (h)
deka (da)
means
means
means
means
means
means
Middle
Small
deci (d)
centi (c)
milli (m)
micro (m)
nano (n)
pico (p)
femto (f)
atto (a)
means
means
means
means
means
means
means
means
1X1012
1X109
1X106
1X103
1X102
1X101
1000 000 000 000
1000 000 000
1000 000
1000
100
10
1X100
1
1X10-1
1X10-2
1X10-3
1X10-6
1X10-9
1X10-12
1X10-15
1X10-18
0.1
0.01
0.001
0.000001
0.000000001
0.000000000001
0.000000000000001
0.000000000000000001
SI Prefixes written in Equivalence Statement format
1 TL
1 Gs
1 Mg
1 km
=
=
=
=
1X1012 L
1X109 s
1X106 g
1X103 m
Notice the 3, 6, 9, 12 pattern!
1g
1L
1m
1s
=
=
=
=
1X103 mg
1X106 mL
1X109 nm
1X1012 ps
Notice that if we write 1 large unit on the left side of the equal sign, then there
must be a larger number of the smaller units on the right side to be equal.
If we look back at the previous slide we will see negative signs in the definitions
for milli, micro, nano, and pico. Why is that?
Problem Solving Strategy Illustration
A solid object is found to have a mass of 84.241 g and a volume of 28.53 mL.
What is the density of the object?
First Step: Highlight key concepts or quantities in the word problem
Second Step: Assign an appropriate symbol for all key quantities
mass = m = 84.241 g
volume = V = 28.53 mL
density = d = ?
Third Step: Use the list of symbols to identify any useful equations
d =
m
V
m = 84.241 g
V = 28.53 mL
d =
d=?
m
V
Fourth and Fifth Steps: Arrange the symbols in the equation so that the
unknown variable is by itself on one side and then substitute quantities into
the mathematical equation and complete the indicated mathematics
d =
84.241 g
28.53 mL
Sixth Step: Check significant figures and units and write the correct answer
You have 14.3 mL of an object that has a density of 7.932 g/mL.
What is the mass of the object?
You have 435.3 g of a liquid that has a density of 0.8325 g/mL.
What is the volume of the liquid?
What if you could only remember one of the two temperature
conversion equations? Can you change one into the other?
o
C=
5 (oF - 32.00)
9
To change from oC to K:
o
F=
9 o
( C)
5
+ 32.00
K = oC + 237.15
Practice
Convert -15 oF
into
Convert 45 oC
into K
Convert 245 K into
o
o
C
F
Mathematics with Scientific Notation:
Let your calculator handle the exponents!
Let’s do an example: 3.4X106
2.8X105
+
On your calculator (Texas Instruments), type the following in order:
3.4
2nd
EE
6
+
2.8
2nd
EE
On your calculator (Casio), type the following in order:
3.4
exp
6
+
2.8
exp
5
=
5
=
How many Gm are in 1.5X1013 meters?
We need the following equivalence statement:
1 Gm
=
1X109 m
Now we use the equivalence statement so that the “m” units cancel out
and are replaced by the units “Gm”.
(
1.5X1013
m
)(
)
1 Gm
1X109 m
= 1.5X104 Gm
How many mg are in 3.42X10-4 g?
(
3.42X10-4 g
)(
)
Equivalence Statements and Conversion Factors
Any statement that says that one quantity is equal to another.
12 things = 1 dozen
1 inch = 2.54 cm
1 km = 1000 m
Each of these equivalence statements can be used to create
conversion factors.
Example:
1 inch
1 inch
2.54 cm
=
2.54 cm
2.54 cm
1 inch
1 inch
=
2.54 cm
This is a conversion factor that converts
from “in” into “cm”
This is a conversion factor that converts
from “cm” into “in”
How many dozen apples do you have if you have 270 apples?
We need the following equivalence statement: 12 apples = 1 dozen apples
(
)(
270 apples
1 dozen apples
12 apples
)
= 22.5 dozen apples
Conversion Factor
(created from the equivalence statement)
We could have tried to remember that 1/12 is 0.0833 and then used the
value 0.0833 as a conversion factor. However, in the long run it is more
efficient to learn the equivalence statements and then use them to create
conversion factors as needed.
How many apples do you have if you have 13.5 dozen apples?
Dimensional Analysis
Using units to guide your use of conversion factors to solve problems.
If you know that your car has a mileage rating of 23.5 miles per gallon and you
assume that gas costs $3.60 per gallon, how much will it cost you to travel
545 miles?
What were you given?
Important Equivalence statements:
1 gal = 23.5 miles
Travel 545 miles
What were you asked to find?
$3.60 = 1 gal
Cost for traveling 545 miles
Start with what you were given and convert the units into what you were asked
to find (using the equivalence statements you know).
(545 miles)(
1 gal
23.5 miles
= $ 83.5
)(
$ 3.60
1 gal
)
a) Not Precise and not Accurate
b) Not Accurate but Precise
c) Accurate and Precise
The Difference Between Precision and Accuracy can be more
difficult to see when numbers are given instead of the darts.
Trial
Volume (mL)
1
3.6
2
3.5
3
3.7
Average
3.6
The values are only changing in the last
decimal place, so they are precise.
If the TRUE value for the volume is 3.6, then
the data is also accurate.
However, if the TRUE value for the volume
was 4.2, then the data would only be precise
and would not be accurate.
The Difference Between Precision and Accuracy can be more
difficult to see when numbers are given instead of the darts.
Trial
Volume (mL)
1
3.8
2
2.8
3
4.2
Average
3.6
Trial
Volume (mL)
1
2.99
2
3.01
3
2.97
Average
2.99
The values are changing both decimal places,
so they are not precise.
Once the data is not precise, it really can not
be called accurate even if by some chance the
average value is close to the accepted value.
What about this set of data?
Uncertainty in Measurement
• A digit that must be estimated is called an
uncertain digit.
• All measurements include all the digits we are
certain of plus one guess digit.
• A measurement always has some degree of
uncertainty because we can always make a
guess about the last digit.
Generally, the more digits a measurement has, the more
precise it is considered to be. Between two numbers,
the number with uncertainty in the smallest decimal
place is the more precise number.
3.28 g
3.2764 g
Let’s use an illustration of measuring the length of a nail that is about 6.3 cm long to
discus uncertainty in measurements.
Since the nail is longer than 6.3 cm but is not
longer than 6.4 cm, we are certain of the digits 6.3
Since there are no marks between 6.3 cm and 6.4 cm, we must
guess how far between the marks we think the length is-this
guess is an uncertain digit.
Significant Figures are those digits in a measurement that we are
certain of plus one guess digit at the end.
All non-digital devices have precisions that are one place smaller than the
smallest marking on the device. In the case of the ruler, the smallest marks are
at the 0.1 cm scale. Therefore, the precision would be at the 0.01 cm scale.
We would say the measurement was 6.36 cm +/- 0.01 cm (or +/- 0.05 cm
depending upon how well we can estimate our guess). This measurement
would have 3 significant figures!
However, if you are not making the measurement, and the measurement is
given to you, you must use different rules to determine significant figures.
Rules for Significant Figures in Measurements given to you by an
outside source
• Nonzero integers always count as significant figures.
– 3456 has 4 sig figs.
• Leading zeros do not count as significant figures.
– 0.048 has 2 sig figs.
• Captive zeros always count as significant figures.
– 16.07 has 4 sig figs.
• Trailing zeros are significant only if the number contains a decimal point.
– 9.300 has 4 sig figs
– 150 has 2 sig figs.
• Exact numbers have an infinite number of significant figures.
– 1 inch = 2.54 cm, exactly
Exact numbers are definitions or simple counting: 12 is 1 dozen and 4 cars
How many significant figures are in each of the following numbers?
These measurements are given to you by someone else-they are not
numbers that you obtained from a measurement. This means that we
must apply the arbitrary rules for significant figures.
450 g
0.029 m
20.3 s
0.00300 g
$45,700,000
13 people
6.2X10-2 mL
1.300X108 m
Significant Figure Rules for Mathematical Operations:
Multiplication and Division: the number with the fewest
significant figures in the calculation determines how many
significant figures the answer will have.
Examples:
(4.53 m)*(0.28 m)*(1.342 m) = 1.7021928 m3
(from the calculator)
1.7 m3
(correct answer)
(678.3 m)÷(18.4 s) =
36.86413043 m/s
(from the calculator)
36.9 m/s
(correct)
Significant Figure Rules for Mathematical Operations:
Addition and Subtraction: The largest position “guess”
number determines the position of the last significant figure
in the answer.
a)
4,300
+ 298
4,598
4,600
b)
the “3” in 4,300 is a guess number and is in the hundred’s position
m
the “8” in 298 is a guess number and is in the one’s position
m
m (calculator answer)
m (correct answer)
Since the hundred’s position is larger
than the one’s position, the answer must
have its guess number in the hundreds
position.
321.4 m
- 298 m
23.4 m (calculator answer)
23 m (correct answer)
Error (or absolute error) is the difference between the accepted value for a
measurement and the experimental value for a measurement.
Error = Accepted Value – Experimental Value
The Accepted Value for a measurement is usually determined by completing
many trials. If the measurements are in close agreement, then the average
value from those trials will be taken as the accepted value.
The Experimental Value is the value obtained in a single experiment.
Note that Error can be positive or negative.
Percent Error is the Error expressed as a percentage!
% Error =
(
Accepted Value − Experimental Value
Accepted Value
)
* 100
Example: The accepted density for chloroform is 2.97 g/mL;
In an experiment, a student obtained a value of 2.85 g/mL. Find the error and
percent error in the students measurement.
Given:
Accepted Value = 2.97 g/mL
Experimental Value = 2.85
Error = Accepted Value – Experimental Value Click to see Solution
Error = 2.97 g/mL – 2.85g/mL = 0.12 g/mL
% Error =
% Error =
(
(
Accepted Value − Experimental Value
Accepted Value
g
g
2.97mL − 2.85 mL
g
)
mL
(2.97
)
*100 = 4.0%
)
* 100
Examples of Two Types of Graphs
Bar Graph
Circle Graph
25
Seniors
20
Juniors
15
Sophomors
10
Freshmen
5
0
Seniors
Juniors
Sophomors
Freshmen
A third type of graph:
X-Y scatter plot
7
What happens to “y”
as “x” increases?
6
Y values
5
4
Since “y” gets
smaller as “x” gets
larger, the slope will
be negative.
3
2
1
0
0
1
2
3
4
5
6
X values
Line will have the form: y = mx + b
where “m” is slope and “b” is y intercept
7
This data represents directly proportional data (y/x = constant).
45
40
35
30
25
20
15
10
5
0
0
20
40
60
80
100
120
This data represents inversely proportional data (y*x = constant).
600
500
400
300
200
100
0
0
100
200
300
400
500