Scientific Measurement
Chapter 3
Measurement & Uncertainty
• Making measurements and performing
calculations with measurements is very
important in science and many other fields
• Any measurement has a number with a
unit
• How do you know if a measurement is
true?
• Are there limits to measurement?
Scientific Notation
A convenient way of writing very large and very small
numbers
A way to indicate significant figures
Standard (Decimal) notation
0.00000000000030 m (radius of H atom)
Scientific notation
3.0 x 10-13 m
coefficient x 10 power
first digit must be from 1 to 9
1.65 x 10 4
Correct format?
0.053 x 10 -2
Correct format?
12.63 x 10 15
Correct format?
Calculations with Scientific Notation
• Review scientific notation in your text
– Read pages R56-57, Appendix C
• Your calculator uses a special key to enter
scientific notation
• EE, E, Exp, Sci
• These keys mean “x 10exp” to your
calculator
• Do not use 10^
Calculations with Scientific
Notation
•
•
•
•
•
How to enter 6.022 x 1023
6.022
2nd EE
23
Your calculator screen should show
6.022E23
Calculations with Scientific
Notation
•
•
•
•
•
•
•
•
•
Calculate 6.52 x 1018 ÷ 4.91 x 10-5
6.52
2nd EE
18
÷
4.91
2nd EE
-5
ENTER = 1.33…..E23
Accuracy, Precision, & Error
Accuracy and precision are not the same thing
Accuracy
how close a measurement is to the true value (actual or
accepted value)
Precision
how close measurements agree
how exact a measurement is
Example: a centigram balance (0.01g) is more precise
than a decigram balance (0.1g)
Error
difference between actual and experimental value
Accuracy & Precision
Accuracy & Precision
To evaluate accuracy of a measurement:
compare measurement to true value
To evaluate precision of a measurement:
compare values of two or more repeated
measurements
Uncertainty in Measurement
• All measurements are approximations
• All measurements contain error, so we can
only report numbers that we know for sure
(certain)
• The certainty of a measurement is
determined by the precision of the
measurement
• Significant figures are used to reflect
certainty of measured value
Uncertainty in Measurement
• Digital instruments (like our electronic
scales) estimate the final digit
• Example: 5.67 g
• In this measurement, the 7 is estimated by
the scale
• The uncertainty of the scale is the smallest
division reported by the scale (0.01 g)
• Recording the measurement with its
uncertainty: 5.67 ± 0.01 g
Significant Figures
• All digits that are known, plus one last
estimated digit
• Represent certainty of a measurement
• Must be handled properly in calculations to
prevent overstating precision
• Review rules to determine significant
figures (p. 66-67)
Significant Figures in Measurement
Rules for Determining Significant
Figures
•
•
•
•
•
•
All non-zeros
Zeros between non-zeros
Zeros at the beginning of a #
Zeros at the end, to right of “.”
Final zeros without “.”
Final zeros with “.”
YES
YES
NO
YES
NO
YES
Significant Figures in
Calculations
• Multiplication & Division
– Result must have the same # of s.f. as the
measurement with the fewest s.f.
– 6.221 cm x 5.2 cm = 32.3492 cm2 → 32 cm2
• Addition & Subtraction
– Result may not have more decimal places
than the number with the fewest decimal
places
– 20.4 + 1.322 + 83 = 104.722 → 105
Uncertainty in Measurement
• An error due to limitations of the
instrument
• For a digital instrument
– +/- the smallest digit
– 62.56g +/- 0.01 g
• For an analog instrument
– +/- the estimated digit
– See example
Determining Error
Error:
the difference between the accepted and
experimental measurement
Example:
Water was measured to boil at 101.5ºC
The known bp of water is 100.0ºC
Calculate the error in the measurement
Error  experiment al value - accepted value
Error  101.5 C - 100.0 C
Error  1.5 C
Percent Error
Error is often better understood as a percent of
the true value
% error 
experiment al - accepted
% error 
101.5 - 100.0
accepted
100.0
 1.5%
x 100
x 100
Note that the numerator is absolute value!
3.2 International System of Units
• SI units (System International) used to be called
the metric system
• Standard units used in science
Metric Prefixes*
*Memorize these prefixes and their factors
Common Units of Volume
Mass vs. Weight
• Mass is a measure of matter
• Anything that occupies space has mass
• Weight is a force
– The force of gravity acting on a mass
Temperature Scales Used in
Science
• Kelvin (Absolute
Temperature)
• Absolute zero
0º K = -273.15º C
no negative temps
• Celsius
0 C = +273.15 K
• A Kelvin degree and a
Celsius degree have
the same size
Conversions Between the Celsius
and Kelvin Scales
We will not use the Farenheit scale!
Energy
Units of Energy
• Energy is the capacity to do work or to produce
heat.
• The joule (J) is the SI unit of energy.
• One calorie (cal) is the quantity of heat that raises
the temperature of 1 g of pure water by 1°C.
The Joule
Pronunciation Guide
NO
NO
YES!
• Energy can be converted into other
forms, but the units are still joules (J)
• This house is equipped with solar panels.
The solar panels convert the radiant energy
from the sun into electrical energy that can
be used to heat water and power
appliances.
3.3 Conversion Problems
• Conversion Factors
• Ratio of two
equivalent
measurements
• 1 dozen = 12 items
1 dozen
12 items
or
12 items
1 dozen
Dimensional Analysis
• When solving problems, units must be
consistent
• Unit conversion are often necessary
• Use conversion factors
• Problem: Determine how many centimeters
are in 1 yd.
• 1 yd x 36.0 in x 2.54 cm = 91.44 cm
1 yd
1 in
3.4 Density
• Density is the ratio of mass
to volume
• Density is an intensive
property
• Density of a pure
substance is constant at a
given temperature
m
d
V
Density
• Depends on temperature
temp  density
What if temp decreased?
• Units
g/cm3 or g/mL for solids & liquids
g/L for gases