Transcript Scientific Measurement

Chapter 3
Scientific Measurement
Measurement
In chemistry, #’s are either very small or very
large
1 gram of hydrogen =
602,000,000,000,000,000,000,000 atoms
Mass of an atom of gold =
0.000 000 000 000 000 000 000 327 gram
Scientific Notation
• Condensed form of writing large or small numbers
• When a given number is written as the
product of 2 numbers
• M x 10n
 M must be:
• greater than or equal to 1
• less than 10
 n must be:
• whole number
• positive or negative
 Find M by moving the decimal point
over in the original number to the
left or right so that only one non-zero
number is to the left of the decimal.
 Find n by counting the number of
places you moved the decimal:
To the left (+) or
To the right (-)
Scientific Notation Examples
 20 = 2.0 x 101
 200 = 2.0 x 102
 501 = 5.01 x 102
 2000 = 2.000 x 103
More examples…
 0.3 = 3 x 10-1
 0.21 = 2.1 x 10-1
 0.06 = 6 x 10-2
 0.0002 = 2 x 10-4
 0.000314 = 3.14 x 10-4
Rule:
If a number starts out
as < 1, the exponent
is always negative.
Scientific Notation
Adding & Subtracting:
• if they have the same n, just add or subtract the
M values and keep same n
• if they don’t have the same n, change them so
they do
Scientific Notation
Multiplying:
• the M values are multiplied
• the n values are added
Scientific Notation
 Division:
• the M values are divided
• the n values are subtracted
Accuracy & Precision
‘How close you are really
counts!’
Accuracy
• Accuracy – a measure of how close a
measurement comes to the actual or true
value of what is measured
To evaluate… the measured
value must be compared
to the correct value
Precision
• Precision – a measure of how close a series
of measurements are to one another
To evaluate… you must compare the values of
2 or more repeated measurements
Accuracy vs. Precision
Errors are Unavoidable
• Measuring instruments have limitations
• Hence, there will always be errors in
measurement.
Not All Errors are Equal
• Consider the following two errors:
• You fly from NY to
• You are an eye surgeon
San Francisco
• Your plane is blown off • Your scalpel misses the
course by 3cm
mark by 3cm
The errors sound equal… but are they?
Absolute Error
• The error in each of the previous examples is 3cm
• But the error in each is not equivalent!
• This type of error is the absolute error.
Absolute error = | measured value – accepted value |
 Accepted value is the most probable value or the
value based on references
 Only the size of the error matters, not the sign
Significance of an Error
• The absolute error tells you how far you are from
the accepted value
• It does not tell you how significant the error is.
o
o
Being 3cm off course on a trip to San Francisco
is insignificant because the city of San
Francisco is very large.
Being 3cm off if you are an eye surgeon means
your operating on the wrong eye!
• It is necessary to compare the size of the error to
the size of what is being measured to understand
the significance of the error.
Percentage Error
• The percentage error compares the absolute
error to the size of what is being measured.
% error = |measured value – accepted value| x 100%
accepted value
Sample Problem
• Example: Measuring the boiling point of H2O
Thermometer reads – 99.1OC
You know it should read – 100OC
Error = measured value – accepted value
% error =
|error|
accepted value
x 100%
|99.1oC – 100.0oC|
100oC
% error =
=
0.9o C x 100%
100o C
= 0.009 x 100%
= 0.9%
x 100%
Significant Figures
• Used as a way to express which numbers
are known with certainty and which are
estimated
What are significant figures?
Significant Figures –
all the digits that are known, plus
a last digit that is estimated
Rules
1) All digits 1-9 are significant
Example: 129
3 sig figs
2) Embedded zeros between significant digits
are always significant
4 sig figs
Example: 5,007
3) Trailing zeros in a number are significant only if
the number contains a decimal point
Example: 100.0 4 sig figs
3600
2 sig figs
4) Leading zeros at the beginning of a number
are never significant
Example: 0.0025
2 sig figs
5) Zeros following a decimal significant
figure are always significant
3 sig figs
Example: 0.000470
0.47000
5 sig figs
6) Exceptions to the rule are numbers with an
unlimited number of sig figs
Example = Counting – 25 students
Exact quantities – 1hr = 60min, 100cm = 1m
Significant Figure Examples









123m = 3
9.8000 x 104m = 5
0.070 80 = 4
40, 506 = 5
22 meter sticks = unlimited
98, 000 = 2
143 grams = 3
0.000 73m = 2
8.750 x 10-2g = 4
Calculations Using
Significant Figures
• Rounding
1st determine the number of sig figs
Then, count from the left, & round
If the digit < 5, the value remains the same.
If the digit is ≥ 5, the value of the last sig fig
is increased by 1.
Try your hand at rounding…
 Round each measurement to 3 sig figs.






87.073 meters = 87.1m
4.3621 x 108 meters = 4.36 x 108 m
0.01552 meter = 0.0155m or 1.55 x 10-2m
9009 meters = 9010m
1.7777 x 10-3 meter = 1.78 x 10-3m
629.55 meters = 630. m or 6.30 x 102m
• Multiplying and Dividing
Limit and round to the least number of
significant figures in any of the factors.
23.0cm x 432cm x 19cm = 188,784cm3
Answer = 190,000cm3 or 1.9 x 103cm3
Because 19 only has 2 sig figs
• Addition and Subtraction
Limit and round your answer to least
number of decimal places in any of the
numbers that make up your answer.
123.25mL + 46.0mL + 86.257mL = 255.507mL
Answer = 255.5mL
Because 46.0 has only 1 decimal place
The International
System of Units
• Based on the #10
• Makes conversions easier
• Old name = metric system
Units and Quantities
• Length – the distance between 2 points or
objects
Base unit = meter
• Volume – the space occupied by any sample of matter
V = length x width x height
Base unit = liter
Based on a 10cm cube
(10cm x 10cm x 10cm = 1000cm3)
1 liter = 1000cm3
• Mass – the amount of matter contained in
an object
Base unit = gram
Different than weight…
Weight - a force that
measures the pull of gravity
Metric Conversion Chart
Move Decimal Right OR Multiply
1000
100
10
1
10-1
10-2
10-3
10-6
10-9
KILO – HECTA – DEKA – [BASE] – DECI – CENTI – MILLI – MICRO - NANO
Meter
Liter
Gram
Move Decimal Left OR Divide
IF YOU ARE MOVING THE DECIMAL POINT:
1. Start with the unit given to you
2. Count how many times you need to move to get to the new unit
3. Move the decimal in the number that many spaces and
in the same direction.
4. Re-write the number with the new units.
IF YOU ARE MULTIPLYING OR DIVIDING:
1. Start with the unit given to you.
2. If moving to the Right Multiply: x10 for 1 jump,
x100 for 2 jumps, x1000 for 3 jumps, etc.
3. If moving to the Left Divide: /10 for 1 jump,
/100 for 2 jumps, /1000 for 3 jumps, etc.
4. Rewrite the number with the new units.
• Temperature – a measure of the energy of
motion
How fast are the molecules moving?
When 2 objects are at different
temperatures heat is always
transferred from the warmer
→ the colder object
Temperature Scales
• Celsius scale –
Freezing point of H2O = 0oC
Boiling point of H2O = 100oC
• Kelvin scale –
Freezing point of H2O = 273.15K
Boiling point of H2O = 373.15K
K = C + 273
C = K - 273
Temperature Scale Conversions
Conversion Factors
and
Unit Cancellation
A physical quantity must include:
Number + Unit
1 foot = 12 inches
1 foot = 12 inches
1 foot
12 inches
=
1
1 foot = 12 inches
1 foot
12 inches
12 inches
1 foot
=
1
=
1
1 foot
12 inches
12 inches
1 foot
“Conversion factors”
1 foot
12 inches
1 foot
12 inches
“Conversion factors”
How many inches are in 3 feet?
(
3 feet
)(
12 inches
1 foot
)
=
36 inches
How many cm are in 1.32 meters?
equality: 1 m = 100 cm
conversion factors:
______
1m
100 cm
X cm = 1.32 m
or
(
100 cm
______
1m
100 cm
______
1m
)
= 132 cm
We use the idea of unit cancellation
to decide upon which one of the two
conversion factors we choose.
How many meters is 8.72 cm?
equality: 1 m = 100 cm
conversion factors:
______
1m
100 cm
X m = 8.72 cm
or
(
1m
______
100 cm
100 cm
______
1m
)
= 0.0872 m
Again, the units must cancel.
How many feet is 39.37 inches?
equality: 1 ft = 12 in
conversion factors:
______
1 ft
12 in
X ft = 39.37 in
or
( )
____
1 ft
12 in
______
12 in
1 ft
= 3.28 ft
Again, the units must cancel.
How many kilometers is
15,000 decimeters?
X km = 15,000 dm
( )(
1m
____
1 km
______
10 dm
1,000 m
)
= 1.5 km
How many seconds
is 4.38 days?
( )(
24 h
X s = 4.38 d ____
1d
)( )
60
min
_____
1h
60 s
____
1 min
= 378,432 s
If we are accounting for significant
figures, we would change this to…
3.78 x 105 s
Why do some objects float in
water while others sink?
• Need to know the ratio of the mass of an
object to it’s volume
• Pure H2O at 4oC = 1.000g/cm3
• If an object has a lower ratio it will float
• If an object has a greater ratio it will sink
Density
• The ratio of an object’s mass to it’s volume
Density =
mass
volume
Example: A 10.0cm3 piece of lead has a mass of
114g. What is the density of lead?
114g
= 11.4g/cm3
10.0cm3
Recall…
What type of property is density?
Does the density of a material change in
relation to the sample size?
NO… density is an Intensive property
it depends only on the composition of
the material
What might affect a substance’s
density?
• Temperature
 The volume of most substances ↑ with an ↑ in
temperature
 the mass remains the same
If the volume increases… what affect does it have on
a substance’s density?
The density decreases
*Exception – H2O
Water’s volume ↑ with a ↓ in temperature
Its density decreases & ice floats
Calculating Density
What is the volume of a pure silver coin that has
a mass of 14g, and a density of 10.5g/cm3?
D = 10.5g/cm3
M = 14g
V=?
M
Rearrange the density
formula to solve for V
D
V
V=
V=
M
D
14g
= 14 g x 1 cm3 = 1.3cm3
10.5g/cm3
10.5 g
What is the mass of mercury that has a density
of 13.5g/cm3 and a volume of 0.324cm3?
Once again, rearrange the
density formula… and solve
for M.
M=DxV
M
D
M = 13.5g x 0.342 cm3 = 4.62g
cm
3
V