Measurement

1
Measurement

• Observations can be qualitative or quantitative
• Qualitative observations are non-numerical, they
  ask what
• Quantitative observations are numerical, they ask
  how much
• Quantitative observations are also called
  measurements

2

• Measurements
• Always involve a comparison
• Require units
• Involve numbers that are inexact (numbers in
  mathematics are exact)
• Include uncertainty due to the inherent physical
  limitations of the observer and the instruments
  used (to make the measurement)
• Uncertainty is also called error

3
Measurements

• Use the appropriate mass scale for the size
  object.
• A dump truck is measured in tons
• A person is measured in kg or pounds
• A paperclip is measured in g or ounces
• An atom?
• For atoms, we use the atomic mass unit (amu)
• 1 amu 1.661 x 10-24 g

4

• Chemists use SI units for measurements
• All SI units are based on a set of seven measured
  base units

Measurement Unit
Symbol Length meter
m Mass gram
g Time second
s Electrical current ampere
A Temperature kelvin
K Amount of substance mole
mol Luminous intensity candela cd
5

• Derived units involve a combination of base
  units, including

Measurement Formula
SI Units Area length x
width m2 Volume
length x width x height m3 Velocity
distance/time
m/s Acceleration velocity/time
m/s2 Density
mass/volume kg/m3
Note Many other derived units exist
6

• Base units are frequently to large or small for a
  measurement
• Decimal multipliers are used to adjust the size
  of base units, including

Prefix Symbol Factor Power of 10 kilo
k 1000 103 deci
d 0.1 10-1 centi
c 0.01 10-2 milli
m 0.001 10-3
See Table 3.3 for a more complete list
7

• You may encounter non-SI metric system units,
  including

Measurement Name Symbol Value
Length angstrom
Å 10-10 m Mass
amu u 1.66054 x 10-27
kg metric ton
t 103 kg Time
minute min 60 s
hour h
3600 s Volume liter
L 1000 cm3
You probably use English system units for your
everyday measurements
8

• English and Metric units are related using
  conversion factors

Measurement English to Metric Metric to English
Length 1 in. 2.54 cm 1 m 39.37 in. 1 yd
0.9144 m 1 km 0.6215 mile 1 mile 1.609
km Mass 1 lb 453.6 g 1 kg 2.205 lb 1 oz
28.35 g Volume 1 gal 3.785 L 1 L 1.057
qt 1 pt 946.4 mL 1 oz (fluid) 29.6 mL
It is also useful to know that 1 m 100 cm
1000 mm, 1 cm 10 mm, and 1 L 1000 mL 1000
cm3
9

• ENGLISH AND METRIC UNITS
• English system - a collection of measures
  accumulated throughout English history.
• no systematic correlation between measurements.
• 1 gal 4 quarts 8 pints
• Metric System - composed of a set of units that
  are related to each other decimally.
• That is, by powers of tens

11
10
Table 1.2 Some Common Metric Prefixes
Prefix Multiple Decimal Equivalent
mega (M) 106 1,000,000. kilo (k) 103
1,000. deka (da) 101 10. deci
(d) 10-1 0.1 centi (c) 10-2
0.01 milli (m) 10-3
0.001 micro (m) 10-6 0.000001 nano
(n) 10-9 0.000000001
1.4 Measurement in Chemistry
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Metric System

• This table has Giga!

1.4 Measurements
12
Metric English Systems

• Table no need to memorize

13

• To measure volumes in the laboratory, one might
  use one of these

14
The milliliter and the cubic centimeter are
equivalent so 1 mL 1 cm3
Measurements
15
Mass and Weight

• Mass the quantity of matter in an object
• mass is independent of location
• Weight the result of mass acted upon by gravity
• weight depends on location depends on the force
  of gravity at the particular location
• Mass is determined by weighing the object using a
  balance
• Common metric units
• 1 kg 1000g
• 1 mg 0.001g

Measurements
16

• Temperature is measured in degrees Celsius or
  Fahrenheit using a thermometer

Relationship between the kelvin (SI), Celsius,
and Fahrenheit temperature scales. Kelvin
temperature is also called the absolute
temperature scale.
17
Temperature

• Fahrenheit (F) defined by setting the freezing
  point of water at 32F and the boiling point of
  water at 212F
• Celsius (C) defined by setting freezing point of
  water at 0C and boiling point of water at 100C
• Will be given the conversions on tests

1.4 E. Measurements
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• The Kelvin scale is another temperature scale.
• It is of particular importance because it is
  directly related to molecular motion.
• As molecular speed increases, the Kelvin
  temperature proportionately increases.

1.4 Measurements
K oC 273
19
Temperature

• Kelvin (K) zero is the lowest possible
  temperature also called the absolute scale
• degree is the same size as Celsius degree
• K C 273

1.4 E. Measurements
20
Temperature
1.4 E. Measurements
21
Time

• Units are the same for all systems (yeah!)
• 60 s 1 min
• 60 min 1 h

1.4 D. Measurements
22

• The difference between a measurement and the
  true value we are attempting to measure is
  called the error
• Errors are due to limitations inherent in the
  measurement procedure
• In science, all digits in a measurement up to and
  including the first estimated digit are recorded
• These digits are called significant digits or
  significant figures

23

• The number of significant digits in a measurement
  may be increased by using a more precise
  instrument

Using the first thermometer, the temperature is
24.3 ºC (3 significant digits). Using the more
precise (second) thermometer, the temperature is
24.32 ºC (4 significant digits)
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• Errors arise from a number of sources including
• Reading scales incorrectly
• Using the measuring device incorrectly
• Due to thermal expansion or contraction
  (temperature changes)
• Errors can often be detected by making repeated
  measurements
• The central value can be estimated by reporting
  the average or mean

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• Accuracy and precision are terms used to describe
  a collection of repeated measurements
• An accurate measurement is close to the true or
  correct value
• A precise measurement is close to the average of
  a series of repeated measurements
• When calibrated instruments are used properly,
  the greater the number of significant figures,
  the greater is the degree of precision for a
  given measurement

26

• Confusion can be avoided by representing
  measurement in scientific or exponential notation
• Scientific notation is reviewed on the web site
  at www.wiley.com/college/brady
• When measurements are expressed in scientific
  notation to the correct number of significant
  digits, the number of digits written is the same
  regardless of the units used to express the
  measurement

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• Measurements limit the precision of the results
  calculated from them
• Rules for combining measurements depend on the
  type of operation performed
• Multiplication and division
• The number of significant figures in the answer
  should not be greater than the number of
  significant figures in the least precise
  measurement.

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• Addition and Subtraction
• The answer should have the same number of decimal
  places as the quantity with the fewest number of
  decimal places

3.247 ? 3 decimal places 41.36 ? 2
decimal places 125.2 ? 1 decimal place
169.8 ? answer rounded to 1 decimal place
Note Remember that numbers are exact. Numbers
that come from definitions or direct counts have
no uncertainty and can be assumed to contain an
infinite number of significant figures.
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RECOGNITION OF SIGNIFICANT FIGURES

• All nonzero digits are significant.
• 3.51 has 3 sig figs
• The number of significant digits is independent
  of the position of the decimal point
• Zeros located between nonzero digits are
  significant
• 4055 has 4 sig figs

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• Zeros at the end of a number (trailing zeros) are
  significant if the number contains a decimal
  point.
• 5.7000 has 5 sig figs
• Trailing zeros are ambiguous if the number does
  not contain a decimal point
• 2000. versus 2000
• Zeros to the left of the first nonzero integer
  are not significant.
• 0.00045 (note 4.5 x 10-4)

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Examples of Significant Figures
How many significant figures are in the
following? 7.500 2009 600. 0.003050 80.0330
4 4 3 4 6
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Examples of Significant Figures
2.30900 0.00040 30.07 300 0.033
6 2 4 1,2,or 3 2
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SCIENTIFIC NOTATION Sig Figs

• Often used to clarify the number of significant
  figures in a number.
• Example
• 4,300 4.3 x 1,000 4.3 x 103
• 0.070 7.0 x 0.01 7.0 x 10-2

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SIGNIFICANT FIGURES IN CALCULATION OF RESULTS

• I. Rules for Addition and Subtraction
• The answer in a calculation cannot have greater
  significance than any of the quantities that
  produced the answer.
• example 54.4 cm 2.02 cm
• 54.4 cm
• 2.02 cm
• 56.42 cm

correct answer 56.4 cm
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• II. Rules for Multiplication and Division
• The answer can be no more precise than the least
  precise number from which the answer is derived.
• The least precise number is the one with the
  fewest sig figs.

Which number has the fewest sig figs?
The answer is therefore, 3.0 x 10-8
36
Rules for Rounding Off Numbers

• When the number to be dropped is less than 5 the
  preceding number is not changed.
• When the number to be dropped is 5 or larger, the
  preceding number is increased by one unit.
• Round the following number to 3 sig figs 3.34966
  x 104

3.35 x 104
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• Length - the distance between two points
• long distances are measured in km
• distances between atoms are measured in nm. 1 nm
  10-9 m
• Volume - the space occupied by an object.
• the liter is the volume occupied by 1000 grams of
  water at 4 degrees Celsius (oC)
• 1 mL 1/1000 L 1 cm3

1.4 Measurements
38

• UNIT CONVERSION
• You need to be able to convert between units
• within the metric system
• between the English and metric system
• The method used for conversion is called the
  Factor-Label Method or Dimensional Analysis

Unit Conversions
!!!!!!!!!!! VERY IMPORTANT !!!!!!!!!!!
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• The factor-label method, or dimensional analysis,
  can be used to help perform the correct
  arithmetic to solve a problem
• This involves treating a numerical problem as one
  involving a conversion from one kind of units to
  another
• This is done using one or more conversion factors
  to change the units of the given quantity to the
  units of the answer

40

• A conversion factor is a fraction formed from a
  valid relationship or equality between units
• Conversion factors are used to switch from one
  system of measure to another
• Example Convert 72.0 in. to cm using the
  equality 1 in. 2.54 cm (exactly).

72.0 in. x 2.54 cm 183 cm 1 1
in.
Note Since the units cancel correctly the
arithmetic is probably setup correctly.
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• In order to rely on measured properties of
  substances, reliable measurements must be made
• The accuracy and precision of measured results
  allow us to estimate their reliability
• To trust conclusions drawn from measurements, the
  measurements must be accurate and of sufficient
  precision
• This is a key consideration when designing
  experiments

42

• For example, if you measured the length, width,
  and height of a block you could calculate the
  volume of a block
• Length 0.11 cm
• Width 3.47 cm
• Height 22.70 cm
• Volume 0.11cm x 3.47cm x 22.70cm
• 8.66459 cm3
• Where do you round off?
• 8.66? 8.7? 8.66459?