dimensional analysis (unit conversions). conversion factors
TRANSCRIPT
Conversion Factors
• A conversion factor is a fraction relating 2 different units without changing the actual amount• E.g. 1 min = 60 s can be written as the conversion factor of
or
• Remember that if two numbers are equal to each other then when they are divided, the result must be 1
• Therefore when you convert between units, no amounts are changed, only the way of expressing that amount changes
Exact/Defined/Counting Numbers• Exact/defined/counting numbers have infinite sig
figs and are not considered when performing calculations• E.g. is an exact conversion factor because there are
exactly 60 s in 1 min, not 60.000001 or 59.999999999 but 60.00000000000000000000…(zeroes forever)• There is no rounding and there is no uncertainty
• Other examples: 12 in a dozen, 29 students in the classroom, 1 m = 100 cm, 1 ft = 12 in
Exact/Defined/Counting Numbers• Warning: not all conversion factors are exact
• Can you think of an example of an inexact conversion factor?
Exact/Defined/Counting Numbers• Warning: not all conversion factors are exact
• Can you think of an example of an inexact conversion factor?
• Conversions between metric & imperial units • E.g. 1 m = 3.28084… • This number is rounded to 6 sig figs but goes on forever• There is always uncertainty so it counts towards sig figs
Solving Unit Conversion Problems• Step 1: identify the initial amount – what info are you
given?
• Step 2: identify the unknown amount – what are you looking for?
• Step 3: identify the conversion factor – how are 1 & 2 related?
• Overall: unknown amount = initial amount x conversion factor
Example 1
• How many min are there in 3480 s?
• Step 1: initial = 3480 s
• Step 2: unknown = min
• Step 3:
Example 1
• How many min are there in 3480 s?
• Step 1: initial = 3480 s
• Step 2: unknown = min
• Step 3: conversion factor between min & s = or (which one do we use?)
Example 1
• How many min are there in 3480 s?
• Step 1: initial = 3480 s
• Step 2: unknown = min
• Step 3: conversion factor between min & s = • use the one that cancels out with the initial units (s in
this case)
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• ? = 3480 s x conversion factor
• Want to cancel out “s” so use the c.f. that has “s” as the denominator
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• ? = 3480 s x
• Want to cancel out “s” so use the c.f. that has “s” as the denominator
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• ? = 3480 s x
• Want to cancel out “s” so use the c.f. that has “s” as the denominator
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• ? min = 3480 s x
• Want to cancel out “s” so use the c.f. that has “s” as the denominator
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• 58 min = 3480 s x
• Done?
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• 58 min = 3480 s x
• Done? NO! Sig figs!
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• 58 min = 3480 s x
• ? sig figs = ? sig figs x ? sig figs
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• 58 min = 3480 s x
• ? sig figs = 3 sig figs x ? sig figs
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• 58 min = 3480 s x
• ? sig figs = 3 sig figs x ∞ sig figs
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• 58 min = 3480 s x
• 3 sig figs = 3 sig figs x ∞ sig figs
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• 58.0 min = 3480 s x
• 3 sig figs = 3 sig figs x ∞ sig figs
Example 1
• How many min are there in 3480 s?
• Overall: unknown = initial x conversion factor
• 58.0 min = 3480 s x
• Now are we done?
Example 2
The automobile gas tank of a Canadian tourist holds
39.5 L of gas. If 1 L of gas is equal to 0.264 gal in the
US (“gal” is the symbol for “gallon”), and gas is
$1.26/gal in Dallas, Texas, how much will it cost the
tourist to fill his gas tank in Dallas?
Example 2
• $ ? = 39.5 L x c.f. x c.f.
• First c.f. must cancel out litres
• Must have L in the denominator
Example 2
• $ ? = 39.5 L x x c.f.
• First c.f. must cancel out litres
• Must have L in the denominator
Example 2
• $ ? = 39.5 L x x c.f.
• First c.f. must cancel out litres
• Must have L in the denominator
Example 2
• $ ? = 39.5 x x c.f.
• Second c.f. must cancel out gallons
• Must have gal in the denominator
Example 2
• $ ? = 39.5 x x
• Do we have the units we want for our unknown?
• Yes we don’t need anymore conversion factors
• No we need more conversion factors
Example 2
• $ ? = 39.50 L x x
• Finally: use calculator and express in correct sig figs
• ? s.f. = ? s.f. x ? s.f. x ? s.f.
• Are these conversion units exact?
Example 2
• $ ? = 39.50 L x x
• Finally: use calculator and express in correct sig figs
• ? s.f. = ? s.f. x ? s.f. x ? s.f.
• Are the c.f.’s exact? 1st one no, 2nd one yes
Example 2
• $ ? = 39.50 L x x
• Finally: use calculator and express in correct sig figs
• ? s.f. = 4 s.f. x 3 s.f. x ∞ s.f.
• Are the c.f.’s exact? 1st one no, 2nd one yes
Example 2
• $ ? = 39.50 L x x
• Finally: use calculator and express in correct sig figs
• 3 s.f. = 4 s.f. x 3 s.f. x ∞ s.f.
• Are the c.f.’s exact? 1st one no, 2nd one yes
Example 2
• $ ? = 39.50 L x x
• Finally: use calculator and express in correct sig figs
• 3 s.f. = 4 s.f. x 3 s.f. x ∞ s.f.
• $13.1 = 39.5 L x x
Tips to Avoid Rounding Errors• Write only one equation for the entire question
• Do not round before you get the final answer
• Instead, write down as many digits as you can or use the memory function on your calculator (M+)
• This is the difference b/t right and wrong answers!
SI Units
• The International System of Units (Le Système International d’Unités)
• Modernized version of the metric system used in
science
• Any SI prefix can be used with any SI unit
• SI Base Units • SI Prefixes
Quantity Unit name
Unit Symbol
Length metre m
Mass kilogram kg
Volume litre L
Time second s
Temperature Kelvin K
Amount ofSubstance mole mol
Written Prefix
Prefix Symbol
Equivalent Exponential
mega M 106
kilo k 103
hecto h 102
deka da 101
- - 100
deci d 10-1
centi c 10-2
milli m 10-3
micro μ 10-6
Other Units & Equivalences
• 1 t = 1 tonne = 103 kg
• 1 mL = 1 cm3 (cubic centimetres, cc)
• 103 L = 1 m3
Derived Units
• A unit made by combining two or more other units
• Speed: km/h (kilometres per hour)• Density: g/L (grams per litre)