Transcript SKILLS- Basic Dimensional Analysis

Skills Project
Basic Dimensional Analysis
What is dimensional analysis?
• Dimensional analysis, DA, is a
mathematical tool which uses the labels (or
dimensions) of individual measurements to
calculate solutions to problems.
• In many ways, a DA is simply a massive
fraction which is useful for making
conversions between different values in
chemistry.
Anatomy of a DA:
• Like a fraction, a DA consists of a top and
bottom half. To solve a DA
– Multiply the numbers in the numerator or top
half of the DA, then…..
– Divide by all the numbers in the denominator
or bottom half of the DA.
Example 1: Seconds in a Year
• Convert 1 year into seconds:
____
____
____
____
or hour(s)
3.16e7
60 min x
day(s)
60 sec
x
24
365.2531,557,600
1 year(s)
x
x
___________________________________
31,557,600 or 3.16e7 seconds (s)
1
year(s)
x
x
1 day(s)
x
1 hour(s)
1
min
day = 60
24 seconds
hours
year
365.25
days
111hour
minutes
1 minute
To
solve
the
DA,
multiply
all
the
on
topfor
and
Now,
we
may
begin
converting
11numbers
year
into
seconds.
To
Check
your
work.
All
ofwith
the
(labels)
should
Close
off
this
“bracket”
adimensions
vertical
The
first
According
toconversion
our
conversion
factor,
weline.
can
put
“1”
Tothe
use
this
factor,
we
must
itthe
up
so
that
All
DAs
begin
with
a were
straight
horizontal
line
to
separate
In
problem,
you
given
year
to
begin
with.
Knowing
this,
we
can
continue
adding
conversion
bottom.
Then,
once
you
have
each
product,
divide.
do
we
will
need
conversion
factors.
Ayears
conversion
cancel
out
except
the
one
we
were
solving
for
which,
bracket
will
usually
only
have
something
the
top
years
on
the
365.25
days
on
top.
thethis,
units
(dimensions)
cancel
out.
Since
arewith
onin
the
numerator
from
the
denominator
We’ll
place
this
atbottom
head
ofarranged
our
DA
as
our
starting
factors
in
the
right
order
and
in at
the
correct
factor
simply
states
that
something
is next
equal
to
this
case,
is seconds.
an
1
ofis
the
conversion
factors
top,“invisible”
they
will
beunderneath.
ononthe
bottom
of
the
point.
From
here
out,
ourAll
job
to
carrybracket.
out
direction.
something
else.
For
example:
1 dozen
= 12
that
follow will
have
a top
and
bottom,
however.
conversions
until
weboth
get to
seconds.
DA Mechanics:
• In order to setup a proper DA, you will need to
find conversion factors.
• Typically, many DAs are constructed using the
metric system. These relationships provide
many of our conversion factors.
• For example:
 100 cm = 1.00 m
 1.0 L = 1000 ml
 1.0 g = 1000 mg
 1.609 km = 1 mile
 454 g = 1.0 lb
Example 2: Convert $ to cents
• Convert $5.89 into cents.
____
589xcents 100 cents
$ 5.89
___________________________________
1
$ 1
Check
yourchecks
work
and
cancel
out units.
If you’ve
set up
Everything
out,
so multiply
the top
and divide
bythis
the
To complete
this
problem,
you
know
that:
DAstart
correctly,
we
should
be “1
leftdollar”
with “cents”
at the end.of the
To
thisfind
problem,
place
at the beginning
bottom
to
your
answer.
$1 = 100
cents
DA. We will use a conversion
factor
to find cents.
589 cents
Example 3: Converting Distances
• How many inches are in 1 mile?
____
____
12 in
5280
63360ftin x
x
1 mi
___________________________________
1 mi
1
x
1 ft
Check
yourchecks
work
and
cancel
out units.
If you’ve
set up
Everything
out,
so multiply
the top
and divide
bythis
the
To complete
this
problem,
you
need
to
know
that:
DAstart
correctly,
we
should
be the
left with
inches
the end.
To
thisfind
problem,
place
1 mile
at theatbeginning.
bottom
to
your
answer.
milefeet
= 5280
ft
We’ll turn miles into feet1and
into inches.
63360 in
1 ft = 12 in.
Example 4: Calculating Gas Costs
• You are getting ready to travel 3,500 miles across
the country. Your car’s fuel economy is about 18
miles/gallon and gas will cost $2.30/gallon. How
much will gas cost on this trip?
____
____
$2.30
1$8050
gal
x
x
3500 mi
___________________________________
1818
mi
x
1 gal
From
the
we
are
trying
to
cost
Check
work
and
cancel
out units.
If the
you’ve
set
up
Everything
checks
out,
so
multiply
thefind
top
andtotal
divide
byofthis
the
Fromyour
theproblem,
problem:
gas
onto
this
trip,
inanswer.
dollars.
Wewith
know:
DA
correctly,
we
should
bewant
left
dollars
at the
end. of
To
start
this
problem,
we’ll
toalso
use
the total
number
bottom
find
your
18other
milesvalues
= 1 gallon
miles, 3,500 mi. All the
are conversion factors.
18 miles = 1 gallon
We can convert miles to1 gallons
gallons to dollars.
gallon =and
$2.30
1 gallon = $2.30
$447.22
Example 5: Travel times
• A jet plane travels at 800 mph and consumes
200 gallons of fuel per hour. How far will the
jet travel on 3000 gallons of fuel?
____
____
800 mi
1 hr mi x
2400000
x
3000 gal
___________________________________
200
gal
200
x
1 hr
Check
yourchecks
work
and
cancel
out units.
If you’ve
set up
Everything
out,
so multiply
the top
and divide
bythis
the
To complete
this
problem,
you
know
that:
DAstart
correctly,
we
should
be the
left with
mi,ofatfuel
theatend.
To
thisfind
problem,
place
3000miles,
gallons
the
bottom
to
your
answer.
200 galinto
= 1hours
hour and hours into
beginning. We’ll turn gallons
miles.
1 hour = 800 mi
12000 mi
Example 6: Protons in Silver
• How many protons are in 1 mole of silver
(Ag) atoms?
____
____
23 atoms Ag
25
6.022
x
10
2.830
x
10
protons
x 47 protons
x
1
mol
Ag
___________________________________
1 mol
1 Ag
x
1 atom Ag
Check
yourchecks
work
and
cancel
out units.
If you’ve
set up
Everything
out,
so multiply
the top
and divide
bythis
the
To complete
this
problem,
you
know
that:
25
DAstart
correctly,
we
should
be 1left
with
at the
end.at the
To
thisfind
problem,
place
mol
of protons
silver (Ag)
atoms
bottom
to
your
answer.
23 atoms (Avogadro’s number)
1
mol
=
6.022
x
10
beginning. We’ll turn moles (mol) into atoms and atoms into
protons.
1 atom Ag = 47 protons (Periodic table)
2.830 x 10
protons
(2.830e25 protons)
Example 7: Copper atoms in Cu2O
• How many copper atoms are in 2.5e-10
moles of copper (I) oxide, Cu2O?
____
____
2.5e-10 mol Cu O x 6.022e10
3.011 x 10
atoms Cu
molecules
Cu O x 2 atoms Cu
___________________________________
2
23
14
1 mol
1 Cu2O
2
x
1 molecule Cu2O
Check
yourchecks
work
and
cancel
out units.
If you’ve
set up
Everything
out,
so multiply
the top
and divide
bythis
the
To complete
this
problem,
you
know
that:
14
DAstart
correctly,
we
should
be 2.5e-10
left with mol
atoms
thethe
end.
To
thisfind
problem,
place
of Cu2at
O at
bottom
to
your
answer.
23 molecules (Avogadro’s number)
1
mol
=
6.022
x
10
beginning. We’ll turn moles (mol) into molecules and
molecules into
atoms. Cu O = 2 atoms Cu (formula)
1 molecule
3.011 x 10
atoms Cu
(3.011e14 atoms Cu)
2
Example 8: Moles to Neutrons
• How many neutrons would be found in the
oxygen atoms in 7.4e34 molecules of
phosphate?
____
____
x 2.4e104 atoms
neutrons
O
x 8 neutrons
___________________________________
7.4e34 molecules PO43-
36
1 molecule
PO431
x
1 atom O
Everything
out,
so multiply
the top
and divide
bythis
the
To complete
this
problem,
you
know
that:
Check
yourchecks
work
and
cancel
out units.
If you’ve
set 3up
36
To
start to
thisfind
problem,
place 7.4e34 molecules of PO4 at the
bottom
your
answer.
DA correctly,
we shouldPO
be 3-left
with
neutrons
at the end.
1
molecule
atoms
O (formula)
4 = 4 into
beginning. We’ll turn molecules
atoms
and atoms into
neutrons.
1 atom O = 8 neutrons
2.4 x 10 neutrons
(2.4e36 neutrons)
Example 9: Converting Atoms to Moles
• How many moles of Vanadium (V) carbonate
can be made from 6.5e23 atoms of oxygen?
____
____
6.5e23 atoms O x 1 molecule
(CO
)
1 mol V (CO )
6.5e23 V
mol
V (CO
) x
___________________________________
2
23 5
x
15 atoms
O
9.033e24
3 5
2
3 5
6.022e23 molecules V2(CO3)5
Everything
checks
so multiply
the top
and divide by the
To complete
thisout,
problem,
you know
that:
To
start your
thisfind
problem,
place
6.5e23
atoms
of O (oxygen)
at
Check
work
and
cancel
out units.
If you’ve
set up this
bottom
to
your
answer.
2Vmoles
5 and
15 atoms
O turn
= 1 be
molecule
(CO33
)5V(Formula)
2molecules
the
We’ll
atoms
into
DA beginning.
correctly,
we
should
left with
2(CO
3)5 at the
molecules
moles.
end.1 mol into
= 6.022
x 1023 molecules (Avogadro’s number)
0.072 mol V (CO )
Example 10: Atoms from Moles
• How many atoms of hydrogen will be found
in 2.62e-3 mol of propane, C3H8?
____
____
2.62e-3 mol C H
x 6.022e10
1.26e22 atoms
H
molecules
C H x 8 atoms H
___________________________________
3
8
23
3
1 mol
1 C 3H 8
x
8
1 molecule C3H8
Check
yourchecks
work
and
cancel
out units.
If you’ve
set up
Everything
out,
so multiply
the top
and divide
bythis
the
To complete
this
problem,
you
know
that:
22
DAstart
correctly,
we
should
be 2.62e-3
left with mol
atoms
H3at
end.
To
thisfind
problem,
place
of C
H8the
at the
bottom
to
your
answer.
23 molecules (Avogadro’s number)
1
mol
=
6.022
x
10
beginning. We’ll turn moles (mol) into molecules and
molecules into
atoms. C H = 8 atoms H (formula)
1 molecule
1.26 x 10
atoms H
(1.26e22 atoms H)
3
8
Practice on Your Own:
• Make the following conversions (you
may need a conversion “cheat sheet”):
– 1.45 km  miles 0.901 mi
– 181 oz  grams 5140 g
– 13 decades  centuries 1.3 centuries
– 9.4e4 ml  gallons 25 gal
– 7.32 days  minutes 10500 minutes
Practice on Your Own (2):
• Given 2.5 mol of H2O, determine the
following:
– Atoms of hydrogen: 3.0e24 atoms H
– Atoms of oxygen: 1.5e24 atoms O
– Neutrons from oxygen: 1.2e25 neutrons
– Total number of protons: 2.6e25 protons
– Molecules of water: 1.5e24 molecules H2O
Congratulations!
• You have just mastered one of the most
powerful (and useful) tools I will be
teaching you this year.
• Don’t forget- you can use this method of
solving problems whenever you have
conversion factors.
• DA is useful for a variety of subjects other
than chemistry.