Units of

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Units of Measurement
Courtesy www.lab-initio.com
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• Quantitative observation measurement

Two Parts
Number
Unit

• Both must be present for measurement to have
  meaning!!
• Two major systems English system (used in the
  US) and the metric system (used by most of rest
  of the world).
• Scientists worldwide use the metric system.

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• In 1960, the International System of units was
  created and is known as the SI system.

Fundamental SI Units
Physical Quantity Name of Unit Abbreviation
Mass kilogram kg
Length meter m
Time second s
Temperature kelvin K
Electric current ampere A
Amount of substance mole mol
Luminous intensity candela cd
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• Prefixes are used to change the size of
  fundamental SI units.

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• Volume has the dimensions of (distance)3.
• Example length x width x height
• Derived SI unit cubic meter (m3).

Volume

• Traditional metric unit of volume is the liter
  (L).
• In SI terms, 1 L 1 dm3.
• Normally we measure volumes in lab in units of
  milliliters (mL).
• 1000 mL 1L

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Common types of laboratory equipment to measure
volume.
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Mass versus Weight

• Mass the quantity of matter in an object
• Measured using a balance.
• Weight the force exerted on an object by
  gravity.
• Measured using a spring scale.

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Uncertainty in Measurement

• When a person uses a measuring device such as a
  buret or a ruler, there is always uncertainty in
  the measurement.
• In other words, the last digit in a measurement
  is always estimated.
• The uncertainty of a measurement depends on the
  precision of the measuring device.

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• Consider the measurement of volume from the
  buret.
• Suppose 5 different people read this buret and
  the following measurements are obtained.
• Person Measurement
• 1 20.15 mL
• 2 20.14 mL
• 3 20.16 mL
• 4 20.17 mL
• 5 20.16 mL
• The first 3 digits are all the same these are
  digits read with certainty.
• The last digit is estimated and is called the
  uncertain digit.

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• Consider the two centimeter rulers at the left.
• Each ruler is measuring the same pencil.

• The best measurement obtained from the first
  ruler would be 9.5 cm.
• The best measurement obtained from the second
  ruler would be 9.51 cm.
• Why two different measurements?
• The second measurement is more precise, because
  you used a smaller unit to measure with.
• Digits that result from measurement such that
  only the digit farthest to the right is not known
  with certainty are called significant figures.

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Precision and Accuracy

• Precision the degree of agreement among general
  measurements of the same quantity (how
  reproducible your measurements are).
• Accuracy the agreement of a particular value
  with the true value.

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• Random errors mean that a measurement has an
  equal probability of being high or low. Occurs
  in estimating the value of the uncertain digit.
• Systematic errors occur in the same direction
  each time they are always high or always low.
• Figure (a) indicates large random errors (poor
  technique).
• Figure (b) indicates small random errors but a
  large systematic error.
• Figure (c) indicates small random errors and no
  systematic error.

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Example Precision and Accuracy

• Each of 4 general chemistry students measured the
  mass of a chemistry textbook. They each weighed
  the book 4 times. Knowing that the true mass is
  2.31 kg, which student weighed the book
• a. accurately and precisely
• b. inaccurately but precisely
• c. accurately but imprecisely
• d. inaccurately and imprecisely
• Weighing student 1 student 2 student 3 student
  4
• 1 2.38 kg 2.06 kg 2.32 kg 2.71 kg
• 2 2.23 kg 1.94 kg 2.30 kg 2.63 kg
• 3 2.07 kg 2.09 kg 2.31 kg 2.66 kg
• 4 2.55 kg 2.40 kg 2.32 kg 2.68 kg
• Average 2.310.16 kg 2.120.14 kg 2.310.01
  kg 2.670.03kg

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• Solution
• You have to ask yourself two questions about each
  data set
• Is the average close to the accepted (true)
  value? If it is, then the result is accurate.
• Is the average deviation small relative to the
  actual value? If it is, then the result is
  precise.

Student 1 2.38 kg 2.23 kg 2.07 kg 2.55 kg
2.310.16 kg
Student 2 2.06 kg 1.94 kg 2.09 kg 2.40 kg
2.120.14 kg
Student 3 2.32 kg 2.30 kg 2.31 kg 2.32 kg
2.310.01 kg
Student 4 2.71 kg 2.63 kg 2.66 kg 2.68 kg
2.670.03 kg
Student 1 accurate but imprecise
Student 2 inaccurate and imprecise
Student 3 accurate and precise
Student 4 inaccurate but precise
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Significant Figures

• When taking measurements all certain digits plus
  the uncertain digit are significant.
• Example Your bathroom scale weighs in 10 Newton
  increments and when you step onto it, the pointer
  stops between 550 and 560. You look at the scale
  and determine your weight to 557 N. You are
  certain of the first two places, 55, but not the
  last place 7. The last place is a guess and if
  it is your best guess, it also is significant.

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• When given measurements, the numbers that are
  significant are the digits 1 9, and the 0 when
  it is not merely a place holder.
• When 0s are between significant figures, 0s are
  always significant.
• Example 101 has 3 sig. fig. and 34055 has 5
  sig. fig.
• When the measurement is a whole number ending
  with 0s, the 0s are never significant.
• Example 210 has 2 sig. fig. and 71,000,000
  also has 2 sig. fig.
• Removal of the 0s DO change the value (size) of
  the measurement, but the 0s are place holders
  and are thus not significant.

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• When the measurement is less than a whole number,
  the 0s between the decimal and other significant
  figures are never significant (they are place
  holders).
• Example 0.0021 has 2 sig. fig. and 0.0000332
  has 3 sig. fig.
• Removal of the 0s DO change the value (size) of
  the measurement, the 0s are place holders and
  are thus not significant.
• When the measurement is less than a whole number
  and the 0s fall after the other significant
  numbers, the 0s are always significant.
• Example 0.310 has 3 sig. fig. and 0.3400 has 4
  sig. fig.
• The 0s have no effect on the value (size) of
  the measurement. Therefore, these 0s must have
  been included for another reason and that reason
  is to show precision of the measurement. Since
  these 0s show precision they must be significant.

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• When the measurement is less than a whole number,
  and there is a 0 to the left of the decimal, the
  0 is not significant.
• Example 0.02 has only 1 sig. fig. and 0.110
  has 3 sig. fig.
• The 0 to the left of the decimal is only for
  clarity, it is neither a place holder nor adds to
  the accuracy of the measurement.
• When the measurement is a whole number but ends
  with 0s to the right of the decimal, the 0s are
  significant.
• Example 20.0 has 3 sig. fig. and 18876.000 has
  8 sig. fig.
• The 0s have no effect on the value (size) of
  the measurement. Therefore, these 0s must have
  been included for another reason and that reason
  is to show precision of the measurement. Since
  these 0s show precision they must therefore be
  significant.

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Rules for Significant Figures in
Mathematical Calculations

• For addition or subtraction, the result has the
  same number of decimal places as the least
  precise measurement (the measurement with the
  least number of decimal places) used in the
  calculation.
• Example
• 12.011
• 18.0
• 1.013
• ________
• 31.123

Limiting term has one decimal place
Corrected
31.1
One decimal place
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• When adding or subtracting numbers written with
  the notation, always add the uncertainties
  and then round off the value to the largest
  significant digit. Round off the answer to
  match.
• Example (22.4 0.5) (14.76 0.25) 37.16
  0.75 37.2 0.8
• The uncertainty begins in the tenths place it
  is the last significant digit.

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• For multiplication or division, the number of
  significant figures in the result is the same as
  the number in the least precise measurement (the
  measurement with the least number of significant
  figures) used in the calculation.
• Example
• 4.56 x 1.4 6.38

Corrected
6.4
Two significant figures
Limiting term has two significant figures
Example How should the result of the following
calculation be expressed? 322.45 x 12.75 x
3.92 16116.051 16100 (3 sig. fig.)
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Rules for Rounding

• In a series of calculations, carry the extra
  digits through to the final result, then round.
• If the digit to be removed
• is less than 5, the preceding digit stays the
  same. For example, 1.33 rounds to 1.3.
• is equal to or greater than 5, the preceding
  digit is increased by 1. For example, 1.36
  rounds to 1.4.

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Order of Operations

• You will often have to solve problems where there
  is a combination of mathematical operations. To
  get reasonable answers, you need to recall your
  order of operations, given the following table
• Order of Operations
• parentheses
• exponents and logs
• multiplication and division
• addition and subtraction

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Exponential Notation (or Scientific Notation)

• Review Appendix A1.1 and handout.

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Dimensional Analysis

• Dimensional analysis (or unit factor method)
  used to convert from one system of units to
  another.
• In dimensional analysis we treat a numerical
  problem as one involving a conversion of units
  from one kind to another.
• To do this we need one or more conversion factors
  (unit factors) to change the units of the given
  quantity to the units of the answer.
• A conversion factor is a fraction formed from a
  valid relationship or equality between units that
  is used to switch from one system of measurement
  and units to another.

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• For example, suppose we want to express a
  persons height of 72.0 in. in centimeters.
• We need relationship or equality between inches
  and centimeters.
• 2.54 cm 1 in. (exactly)

• If we divide both sides by 1 in., we obtain a
  conversion factor.
• Notice we have canceled the units from both the
  numerator and the denominator of the center
  fraction.
• Units behave just as numbers do in mathematical
  operations, which is a key part of dimensional
  analysis.

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• Now lets multiply 72.0 in. by this fraction

3 sig. fig.
3 sig. fig.

• Because we have multiplied 72.0 in. by something
  that is equal to 1, we know we havent changed
  the magnitude of the persons height.
• We have changed the units (notice that inches
  cancel leaving centimeters which is what we
  wanted).
• Notice that our given quantity and desired
  quantity have the same number of significant
  figures (conversion factors are considered exact
  numbers and do not dictate significant figures).

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• One of the benefits of dimensional analysis is
  that it lets you know when you have done the
  wrong arithmetic.
• From the relationship
• 2.54 cm 1 in.
• we can construct two conversion factors.

• In the previous problem what if we used the
  incorrect conversion factor? Would it make a
  difference in our answer?
• None of our units cancel.
• In this case we get the wrong units!!!!

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• Can we use more than one conversion factor in a
  single problem?
• Example
• In 1975, the world record for the long jump was
  29.21 ft. What is this distance in meters?
• Solution We can state this problem as
• 29.21 ft ? m
• One of several sets of relationships we can use
  is
• 1 ft 12 in.
• 1 in. 2.54 cm
• 100 cm 1 m
• Notice they take us from inches to centimeters to
  meters.

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• Notice if we had stopped after the first
  conversion factor, the units of the answer would
  be inches if we stop after the second, the units
  would be centimeters, and after the third we get
  meters the units we want.
• Note this is not the only way we could have
  solved this problem. Other conversion factors
  could have been chosen.
• Important you should be able to reason your way
  through a problem and find some set of
  relationships that can take you from the given
  information to the answer.

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Temperature
Three systems
Celsius scale
Kelvin scale
Fahrenheit scale
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• Note the size of the temperature unit (the
  degree) is the same for the Kelvin and Celsius
  scales.
• The difference is in their zero points.

• To convert between Kelvin and Celsius scales add
  or subtract 273.15.

Temperature (Kelvin) temperature (Celsius)
273.15 Temperature (Celsius) temperature
(Kelvin) 273.15 Unit for Celsius oC Unit
for Kelvin K
No degree symbol
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• To convert between the Celsius and Fahrenheit
  scales two adjustments must be made
• degree size
• zero point

Zero Point
Degree size
32o F 0oC Subtract or add 32 when converting
 
180o F 100oC
Conversion factor or the reciprocal
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Density

• Property of matter used by chemists as an
  identification tag.
• Density mass of substance per unit volume of
  the substance
• Each pure substance has a characteristic density.
• Density can also be used to convert between mass
  and volume.

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• Be able to manipulate the formula for density.

• If given the density and mass of a substance how
  could you determine volume?

• If given the density and volume of a substance
  how could you determine mass?

• Remember!! A material will float on the surface
  of a liquid if the material has a density less
  than that of the liquid.