Dimensional Analysis

(Unit Conversions)

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:

• Step 2:

• Step 3:

Example 1

• How many min are there in 3480 s?

• Step 1: initial = 3480 s

• Step 2:

• Step 3:

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

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

Initial =Unknown =Conversion factors:

Example 2

Initial = 39.5 L Unknown =Conversion factors:

Example 2

Initial = 39.5 L Unknown = $ (cost)Conversion factors:

Example 2

Initial = 39.5 L Unknown = $ (cost)Conversion factors:

L gal:

gal $:

Example 2

Initial = 39.5 L Unknown = $ (cost)Conversion factors:

L gal: or

gal $:

Example 2

Initial = 39.5 L Unknown = $ (cost)Conversion factors:

L gal: or

gal $: or

Example 2

• Unknown = initial x c.f. x c.f.

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

• Second c.f. must cancel out gallons

• Must have gal in the denominator

Example 2

• $ ? = 39.5 x x

• 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!

Practice

• Hebden p.14 # 2

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)

From a 3rd year physical chem course at UBC…

Practice

• Hebden p.14-26 #2-40