hello and welcome to chemistry solutions
this tutorial is on dimensional analysis
dimensional analysis is often a tricky
concept for students to get but it's a
very important one and one that you'll
use throughout all of your chemistry
classes dimensional analysis is the
process we use to convert a value from
one unit or set of units to another some
important questions to ask yourself when
doing a dimensional analysis problem are
what are the starting units what are the
units that you wish to obtain and what
are the conversion factors that you can
use to get from the starting units to
the desired units in dimensional
analysis you always use conversion
factors to get from one unit or set of
units to another and conversion factors
are fractions where the numerator and
the denominator are an equal quantity
expressed in different units because the
numerator and the denominator are equal
you can also express the conversion
factor using its reciprocal so if one
thousand milliliters is equal to one
liter you could write that as one liter
over one thousand milliliters or one
thousand milliliters over one liter
because those two values are equal the
value of the fraction or the conversion
factor is one regardless of the
direction that the conversion factor is
written let's try a problem if Tim is
six feet tall how tall is he in
centimeters first let's start with
information given six feet we know that
we need to use a conversion factor and
we know that we need units of feet on
the bottom to cancel those units out and
I think it's helpful to put the unit's
that we want on the bottom right away
before we even choose our conversion
factor that we will write our conversion
factor correctly and not accidentally
flip-flop it so let's look at some
conversion factors we have available to
us you can see that one foot is equal to
12 inches we already have feet on the
bottom so we know that we need to put
inches on the top this allows us to
cancel out feet and if we were to stop
this problem right now
we would have an answer in units of
inches but the question is asking us how
tall Tim is and centimeters that means
we'll need to use another conversion
factor just like I did the first time I
can set up my conversion factor with
inches on the bottom because I know that
I want inches to cancel out then looking
at the conversion factors I have
available I can see that one inch is
equal to two point five four centimeters
because inches is already written on the
bottom I know that 2.54 centimeters must
go on the top this allows me to cancel
out the units of inches and now that
inches are cancelled out I'm left with
units of centimeters so to finish
solving my problem I'm going to start
with six and multiply by every number on
the top and then divide by every number
on the bottom this gives me an answer of
183 centimeters let's try another
problem one gallon of milk is equal to
how many milliliters of milk first we
start by writing our given quantity
which is one gallon and we also need to
note that the unit's we wish to obtain
our milliliters so we know that we need
to use a conversion factor and we know
that that conversion factor needs to
have units of gallons in it somewhere
we also know that we want to put the
units of gallons on the bottom so the
gallons from our starting value and the
units of gallons in our conversion
factor will cancel each other out
let's look at the conversion factors we
have available to us you might see that
one gallon is equal to 4 quarts because
we put gallons already on the bottom we
know that 4 quarts needs to go on the
top this allows us to cancel out the
units of gallons because we haven't yet
obtained units and milliliters we need
to keep going and so we need to use
another conversion factor this
conversion factor we'll need to have
units of quarts and similar to our last
step we'll want the units of quarts to
go on the bottom so that they will
cancel each other out looking at the
conversion factors that we've been given
we can see that one quart is
equal to 0.94 six liters because we
already have quarts on the bottom we
know that 0.94 six liters must go on top
this allows us to cancel out the units
of quarts but we're still not in
milliliters yet so we need to do at
least one more step and so we're looking
for another conversion factor and to
make sure we don't write our conversion
factor the wrong direction it's always
helpful to write the unit's that you
need to cancel out directly on the
bottom so we know that we want to cancel
out units of liters and so we know we
need a conversion factor that has litres
in it and will write the leaders on the
bottom looking at the list of conversion
factors that were given we see that one
liter is equal to one thousand
milliliters and because liters is
already written on the bottom we'll
write 1,000 milliliters on the top this
allows us to cancel out liters now in
order to finish the problem we'll take
our 1 and multiply by all of the
numerical values on the top of our
fractions and then divide by all of the
numerical values on the bottom of our
fractions this would tell us that one
gallon of milk is equal to three point
eight times ten to the third milliliters
now looking at the conversion factors
given you might have noticed that
different conversion factors could have
been used to solve this problem and you
are exactly right dimensional analysis
is a process to use to convert something
from one set of units to another but
there's no one correct path to get from
one set of units to another as long as
you use appropriate conversion factors
there's often many paths you could take
to arrive at the correct answer let's
try this problem again using different
conversion factors again we start with
one gallon we know that we'll need to
use a conversion factor and that we want
gallons to be on the bottom and the only
choice we have from the conversion
factors that we're given here is that
one gallon is equal to four quarts so we
put four quarts on top of our conversion
factor and that allows us to cancel out
the units of gallons in choosing our
next conversion factor we know that
quarts needs to go on the bottom in
order for the unit's to cancel out but
you might have also know
our conversion factor choices that one
quart is equal to four cups so for this
conversion factor we have quarts on the
bottom and we'll put four cups on the
top this still allows us to cancel out
units of quarts if we were to stop our
problem right here we would arrive at an
answer with units of cups and because
the problem is asking us for milliliters
we need to go one more step so again
we'll use another conversion factor cups
is the units that we want to cancel out
so we know that we need units of cups on
the bottom and then looking at our
conversion factor choices we see that
one cup is equal to 236 point 6
milliliters and we'll place the 236
point six milliliters on top this allows
us to cancel out the units of cups and
to arrive at our answer we'll start with
one and multiply by all the numbers on
the top of our conversion factors and
divide by all of the numbers in the
bottom of our conversion factors and
what you'll notice is that you end up
with the same answer of 3.8 times 10 to
the third milliliters which lets us know
that although one path may be more
efficient than another there are often
multiple correct ways to solve a
dimensional analysis problem let's make
things a little more complicated if if
space-shuttle can travel at 17,000 miles
per hour how many meters per second does
it travel starting with our original
value of 17,000 miles per hour we know
that we need to change the set of units
to meters per second so we'll need to
change units of miles on the top two
meters but this time we're also going to
need to change units of hours on the
bottom two seconds and so we'll change
each of those units separately let's
start with what we already know how to
do and convert miles to meters we know
that we'll need to use a conversion
factor and in order for the units of
miles to cancel out we'll need to write
miles on the bottom we can then choose
an appropriate conversion factor one
mile is equal to one point six zero nine
kilometers because miles is already
written on the bottom we know
we have to write one point six zero nine
kilometers on top this allows us to
cancel out the units of miles then we
choose another conversion factor to
convert kilometers we know that in order
for kilometers to cancel it must be
written on the bottom and then choosing
an appropriate conversion factor we can
see that one kilometer is equal to one
thousand meters and because kilometres
is written on the bottom one thousand
meters will be written on the top this
allows us to cancel out the units of
kilometers if we were to stop this
problem now we would have units of
meters per hour but since our question
asks for meters per second we now need
to convert units of hours on the bottom
to seconds so we're going to need to
choose an appropriate conversion factor
but because we want to cancel out units
of hours and it's on the bottom we need
to have hours written on the top because
remember units that are the same on the
top and the bottom can cancel out
looking at our conversion factor choices
we know that one hour is equal to 60
minutes and because we already have one
hour written on top we know that we need
to write 60 minutes on the bottom of our
conversion factor this allows us to
cancel out units of hours and to convert
from units of minutes on the bottom we
need to have a conversion factor where
minutes can be written on the top we
also know that one minute is equal to 60
seconds so we can write 60 seconds on
the bottom and now because we have units
of minutes on the top and bottom of our
dimensional analysis problem the units
of minutes will cancel each other out
and now if you look at the units were
left with you'll see that we have meters
on the top and seconds on the bottom
which is what our question is asking us
for so to finish solving our problem
we'll take 17 thousand and multiply by
all of our numerical values on the top
and then divide by all of our numerical
values on the bottom and we'll come up
with an answer of seven point six times
ten to the third meters per second thank
you for watching chemistry solutions we
hope you enjoyed this tutorial
you
