Metrics

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.
Consider the reliability of a measurement in decisions based on measurements.
Clearly distinguish between
precision and accuracy
exact numbers and measurements
systematic error and random error
Count the number of significant figures in a recorded measurement. Record measurements to the correct number of digits.
Estimate the number of significant digits in a calculated result.
Estimate the precision of a measurement by computing a standard deviation.
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
identify the unknown, including units
choose a starting point
list the connecting conversion factors
multiply starting measurement by conversion factors
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
Using Significant Figures
Precision vs. Accuracy
good precision & good accuracy
poor accuracy but good 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?
Score by distance from bullseye
Score by area or target
The standard deviation, s, is a precision estimate based on the area score: where
xi is the i-th measurement
is the average measurement
N is the number of measurements.
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