Download Unit 6 – Scientific Notation and Significant Figures

UNIT 6 - SCIENTIFIC NOTATION
AND SIGNIFICANT FIGURES
CH1030
Mark Stacey
SCIENTIFIC NOTATION
Science often has to use very large or very
small numbers. This would be very difficult
to quickly and neatly write out or to
communicate. Scientific Notation is a form
that allows very large or very small
numbers to be expressed easily.
SCIENTIFIC NOTATION
Scientific notation
uses an exponent of
10 along with a
base number to
communicate a
number.
12000.00
Becomes
1.230000 x 10 4
SCIENTIFIC NOTATION
An easy way to
convert to scientific
notation is to count
the number of times
the decimal has to
be moved to place it
at the ones position.
12000.00
4 places left
1.230000 x 10 4
SCIENTIFIC NOTATION
For large numbers,
this will mean
moving the decimal
to the left.
This results in the 10
in scientific notation
having a positive
exponent.
12000.00
4 places left
1.230000 x 10 4
SCIENTIFIC NOTATION
For small numbers
(numbers less than
1), the decimal is
moved to the right.
In scientific notation,
this results in a
negative exponent.
0.00000123
6 places right
1.230000 x 10 -6
SI UNITS
Chemists worldwide have agreed on a
single system of units, the SI Units (Système
International d'Unités) so that measurements
are understood and shared accurately.
However, for many everyday calculations,
other units are occasionally used. Published
research will generally use SI Units.
SI UNITS
Most of these units
should be familiar
from everyday life.
Note that the kilogram
(kg) is used as the base
unit, not the gram (g).
This is because a single
gram is a very small
amount of material,
making kilograms more
useful in most situations.
(1 kg = 1000g)
SI UNITS
When larger or
smaller amounts
of things need to
be measured,
the SI system
uses the
standard metric
prefix system.
SI UNITS
Units for
“derived”
measurements
have direct and
simple
translations to
their SI
components.
SI UNITS – CELSIUS/KELVIN
The most commonly used non-SI unit is Celsius for
temperature in place of Kelvin.
Celsius is avoided as its negative values make using
them in formulas difficult.
However, as the difference in one degree Celsius is
equal to one Kelvin, they are equivalent when
talking about temperature change (eg: a
temperature increase of 5˚C is the same as 5K)
SI UNITS – CELSIUS/KELVIN
Converting from Celsius and Kelvin is easy:
˚C = K - 273.15
K = ˚C + 273.15
ACCURACY AND PRECISION
Accuracy and precision are often used
interchangeably in everyday speech.
In science, they have specific meanings.
Accuracy refers to how close a measured
value is to the true value.
Precision refers to how consistent a device is
at measuring things.
ACCURACY AND PRECISION
An accurate device obtains values close to
the true value.
Example: If a liquid is at 25˚C and a
thermometer records values between 24 and 26,
it is fairly accurate. An inaccurate thermometer
could record numbers between 27 and 28, or
between 23 and 24.
ACCURACY AND PRECISION
A precise device obtains consistent results
(regardless of how accurate they are to the
true value).
 If a pH meter was used on an acid and returned results
of 3.0, 3.1, and 2.9 – it is fairly accurate.
 If another meter recorded results of 2.5, 3.0 and 4.0, it
is much less accurate.
 Again, accuracy does not reflect how close the values
are to the true value (that is accuracy) but is only concern
with how consistent the values are.
ACCURACY AND PRECISION
ACCURACY AND PRECISION
Naturally, scientists aim for experimental
tools and methods that are both accurate
and precise.
Scientists state their level of accuracy and
precision when reporting their work.
This can be shown mathematically or
graphically.
ACCURACY AND PRECISION
Graphs like this
show the
potential error in
their data using
“I”-shaped bars.
This suggests that
the true value is
within that range.
SIGNIFICANT DIGITS
In any measurement,
some of the digits of
a number are absolutely known, while others
are a best estimate.
In this example, the value “1” is known for
sure, but the next number is estimated by the
observer.
SIGNIFICANT DIGITS
If an observer
estimates this ruler to
read 1.2m, the “1” is known for certain, while
the “2” is uncertain.
Numbers we know for certain are called
significant digits or figures.
SIGNIFICANT DIGITS
It is important to take into account how
many digits in a measurement are
significant.
More significant digits in a measurement
mean that our answers and conclusions will
have more significant digits, and thus be
more detailed, nuanced and
accurate/precise.
SIGNIFICANT DIGITS
There are two major methods for indicating
significant digits.
20.345
20.345
Drop all non-significant digits
down lower
Underlining (or over-lining) the
last significant digit.
Both are acceptable, but please stick to
one system or the other.
SIGNIFICANT DIGITS
All measured values should have a specific
number of significant digits. Generally,
when using a device we record all
significant digits and one non-significant
digit.
However, there are two types of numbers
that have infinite significant digits.
SIGNIFICANT DIGITS
An exact number, or whole number, has
infinite significant digits.
For example, if a number of people were
to be recorded to be 41, this could be
treated as 41.000000…. with infinite
zeroes. This is because a person is exactly
one person. This is known to absolute
accuracy.
SIGNIFICANT DIGITS
Conversion Factors also have infinite
significant digits.
This is because 1 cm is equal to exactly
10mm. So we can write the conversion
factor 10 cm/mm with infinite significant
digits.
Simply put, converting from one unit to
another should not make your measurement
any more or less accurate/precise.
SIGNIFICANT DIGITS - RULES
When using measurements in calculations,
we often need to know how many
significant digits a number contains.
There are 5 basic rules for determining the
number of significant digits:
SIGNIFICANT DIGITS - RULES
1) Absolute numbers have infinite significant
figures.
2) All non-zero numbers are considered
significant.
 1.05
0.0110
SIGNIFICANT DIGITS - RULES
3) Interior zeros (zeros surrounded by nonzero numbers) are always significant.
 1.05
50.012
0.01010
4) Leading zeros (zeros with no non-zero
number to the left of them) are not
significant.
0.01010
0.00000012
SIGNIFICANT DIGITS - RULES
5) Trailing zeroes (zeroes to the right of a
number) are considered significant.
0.0100
2.30030
134.00
If you have a non-significant zero to the right of
your number, you should remove them, or rewrite
in scientific notation to remove them.
Instead of 240 (where the 0 is nonsignificant), rewrite to 2.4 x 102
SIGNIFICANT DIGITS – ADD/SUBTRACT
When adding or subtracting. The final answer
contains the same number of decimal places as
the input number that has the fewest decimal
places.
6.01 + 0.334 = 6.344 = 6.34
(2 dp) (3 dp)
(2 dp)
SIGNIFICANT DIGITS – MULTIPLY/DIVIDE
When multiplying or dividing, the answer will
have the same number of significant figures as
the component number with the fewest significant
figures.
5.5 x 10.4 = 57.2 = 57
(2 sd) (3sd)
(2sd)
SIGNIFICANT DIGITS –
MULTI-STEP PROBLEMS
In multiple-step problems, retain one nonsignificant digit (if any) between steps and only
round off at the very end. This avoids making the
answer more inaccurate by rounding after every
single step.
SIGNIFICANT DIGITS –
MULTI-STEP PROBLEMS
Example 1:
3.489 x (5.67 – 2.3)
3.489 x (3.37)
11.758
12
[2 dp – 1 dp]
[4 sd x 2 sd]
SIGNIFICANT DIGITS –
MULTI-STEP PROBLEMS
Example 2:
19.667 – (5.4 x 0.916)
19.667 – (4.95)
14.7206
14.7
[2 sd x 3 sd]
[3 dp – 1 dp]
DERIVED UNIT - DENSITY
Density is a useful unit in that it
compares an object’s mass with
volume.
D=m/V
D = desnity, m = mass, V = volume
DERIVED UNIT - DENSITY
Density does not have a set unit to
describe it. In SI units, density will
typically be given in kg/L (kilograms per
litre), but any unit of mass and volume
can be used (kg/cm3, g/mL, etc).
However, you cannot compare densities
accurately unless they are in the same
units.
DERIVED UNIT - DENSITY
Example 1:
What is the density of a 5.0g mas
that is 10 cm3 in size?
d = m /V = 5.0 g / 10 cm3
d = 0.5 g/cm3
DERIVED UNIT - DENSITY
Example 2:
What is the mass of 5.0 mL of a liquid
with a density of 3.0 g/mL?
d=m/V
m=d*V
m = 3.0 g/mL * 5.0 mL
m = 15 g
DERIVED UNIT - DENSITY
Example 3:
If a liquid has a density of 0.50 g/cm3,
what volume of it has a mass of 17 g?
d=m/V
V=m/d
V = 17 g / 0.50 g/cm3
V = 34 cm3
DERIVED UNIT - DENSITY
The SI Units are based on water.
1 g of pure water has a volume of
1 mL or 1 cm3
This means that the density of pure
water is 1.0 g/mL