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Measurement
Learning Objectives
A checklist of concepts to learn and skills to master in this section.
Lecture Slides
Lecture Notes
Links
Internet sites and paper references for further exploration.
Frequently Asked Questions
Find an answer, or ask a question.
Glossary
Terms and definitions from the glossary are marked with an asterisk (*).
Quiz: Significant Figures
A general quiz on significant figures.
/chem/senese/101/measurement/sigfig-quiz.shtml (01/15/00)

Tutorial: Arithmetic with units
Focuses on units during addition, subtraction, multiplication, and division of measurements. This tutorial is an essential preliminary for unit conversion; it does not cover the arithmetic of significant figures.
/chem/senese/101/measurement/index.shtml#arithmetic with units (3/24/98)

Tutorial: Introduction to unit conversion
Introduction to unit conversion using dimensional analysis. Covers the 5-step approach and includes several examples of serial conversions.
/chem/senese/101/measurement/index.shtml#intro to unit conversion (3/29/98)

Tutorial: Uncertainty in measurement
How uncertainty arises from length, temperature, and volume measurements. How to count significant figures for a single measurement and for a series of measurements. How to round measurements to the correct number of significant figures. A table of contents is available.
/cgi-bin/senese/tutorials/sigfig/index.cgi

Learning objectives

  • Use the SI* system.
    • Know the SI base units*.
    • State rough equivalents for the SI base units in the English system.
    • Read and write the symbols for SI units.
    • Recognize unit prefixes and their abbreviations.
    • Build derived units* from the basic units for mass, length, temperature, and time.
    • Convert measurements from SI units to English, and from one prefixed unit to another.
    • Use derived units like density* and speed as conversion factors.
    • Use percentages, parts per thousand, and parts per million as conversion factors.
  • Use and report measurements carefully.

Lecture outline

Measurement is the collection of quantitative data. The proper handling and interpretation of measurements are essential in chemistry - and in any scientific endeavour. To use measurements correctly, you must recognize that measurements are not numbers. They always contain a unit and some inherent error. The second lecture focuses on an international system of units (the SI system) and introduces unit conversion. In the third lecture, we'll discuss ways to recognize, estimate and report the errors that are always present in measurements.

Measurement

  • quantitative observations
  • include 3 pieces of information
    • magnitude
    • unit
    • uncertainty
  • measurements are not numbers
    • numbers are obtained by counting or by definition; measurements are obtained by comparing an object with a standard "unit"
    • numbers are exact; measurements are inexact
    • mathematics is based on numbers; science is based on measurement
The National Institute of Standards and Technology (NIST) has published several online guides for users of the SI system.

The SI System

  • Le Systéme Internationale (SI) is a set of units and notations that are standard in science.

    Four important SI base units (there are others)
    Quantity SI
    Base Unit
    English
    Equivalent
    length meter (m) 1 m = 39.36 in
    mass kilogram (kg) 1 kg = 2.2 lbs
    time second (s)
    temperature kelvin (K) °F = 1.8(oC)+32
    K = °C + 273.15

  • derived units are built from base units

    Some SI derived units
    Quantity Dimensions SI units Common name
    area length × length m2 square meter
    velocity length/time m/s
    density mass/volume kg/m3
    frequency cycles/time s-1 hertz (Hz)
    acceleration velocity/time m/s2
    force mass × acceleration kg m/s2 Newton (N)
    work, energy, heat force × distance kg m2/s2 Joule (J)

  • Prefixes are used to adjust the size of base units

    Commonly used SI prefixes (there are others).
    Prefix Meaning Abbreviation Exponential
    Notation
    Giga- billion G 109
    Mega- million M 106
    kilo- thousand k 103
    centi- hundredths of c 10-2
    milli- thousandths of m 10-3
    micro- millionths of µ 10-6
    nano- billionths of n 10-9
    pico- trillionths of p 10-12

  • several non-SI units are encountered in chemistry

    Non SI unit Unit type SI conversion Notes
    liter (L) volume 1 L = 1000 cm3 1 quart = 0.946 L
    Angstrom (Å) length 1 Å = 10-10 m typical radius of an atom
    atomic mass unit (u) mass 1 u = 1.66054×10-27 kg about the mass of a proton or neutron; also known as a 'dalton' or 'amu'

Arithmetic with units

  • addition and subtraction: units don't change
    2 kg + 3 kg = 5 kg
    412 m - 12 m = 400 m
    • consequence: units must be the same before adding or subtracting!
      3.001 kg + 112 g = 3.001 kg + 0.112 kg = 3.113 kg
      4.314 Gm - 2 Mm = 4.314 Gm - 0.002 Gm = 4.312 Gm
  • multiplication and division: units multiply & divide too
    3 m × 3 m = 9 m2
    10 kg × 9.8 m/s2 = 98 kg m/s2
    • consequence: units may cancel
      5 g / 10 g = 0.5 (no units!)
      10.00 m/s × 39.37 in/m = 393.7 in/s

Converting Units

  • 5 step plan for converting units
    1. identify the unknown, including units
    2. choose a starting point
    3. list the connecting conversion factors
    4. multiply starting measurement by conversion factors
    5. check the result: does the answer make sense?
  • Common variations
    • series of conversions
      • example: Americium (Am) is extremely toxic; 0.02 micrograms is the allowable body burden in bone. How many ounces of Am is this?
    • converting powers of units
    • converting compound units
    • starting point must be constructed
  • using derived units as conversion factors
    • mass fractions (percent, ppt, ppm) convert mass of sample into mass of component
    • density converts mass of a substance to volume
    • velocity converts distance traveled to time required
    • concentration converts volume of solution to mass of solute

Uncertainty in Measurements

  • making a measurement usually involves comparison with a unit or a scale of units
    • always read between the lines!
    • the digit read between the lines is always uncertain
    • convention: read to 1/10 of the distance between the smallest scale divisions
  • significant digits
    • definition: all digits up to and including the first uncertain digit.
    • the more significant digits, the more reproducible the measurement is.
    • counts and defined numbers are exact- they have no uncertain digits!
Tutorial: Uncertainty in Measurement
  • counting significant digits in a series of measurements
    • compute the average
    • identify the first uncertain digit
    • round the average so the last digit is the first uncertain digit
  • counting significant digits in a single measurement
    • convert to exponential notation
    • disappearing zeros just hold the decimal point- they aren't significant.
    • exception: zeros at the end of a whole number might be significant
  • Precision of Calculated Results
    • calculated results are never more reliable than the measurements they are built from
    • multistep calculations: never round intermediate results!
    • sums and differences: round result to the same number of fraction digits as the poorest measurement
    • products and quotients: round result to the same number of significant digits as the poorest measurement.
Quiz Quiz
Using Significant Figures
  • Precision vs. Accuracy

    good precision & good accuracy

    poor accuracy but good precision

    Accuracy vs. Precision good accuracy but poor precision

    poor precision & poor accuracy


    Precision Accuracy
    reproducibility correctness
    check by repeating measurements check by using a different method
    poor precision results from poor technique poor accuracy results from procedural or equipment flaws
    poor precision is associated with 'random errors' - error has random sign and varying magnitude. Small errors more likely than large errors. poor accuracy is associated with 'systematic errors' - error has a reproducible sign and magnitude.

  • Estimating Precision
    • Consider these two methods for computing scores in archery competitions. Which is fairer?
      1-2-3Score by distance from bullseye
      1-4-9Score by area or target
    • The standard deviation*, s, is a precision estimate based on the area score: s=sqrt((x_i-xbar)^2/(N-1)) where
      xi is the i-th measurement
      is the average measurement
      N is the number of measurements.
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General Chemistry Online! Measurement

Copyright © 1997-2005 by Fred Senese
Comments & questions to fsenese@frostburg.edu
Last Revised 06/16/05.URL: http://antoine.frostburg.edu/chem/senese/101/measurement/index.shtml