so let's go ahead and work through some conversions using
Avogadro's number. the first question asks you have a 5.0 kilogram brick of
pure gold, how many gold atoms are in the brick? and the second question asks you
have a 40.0 milligrams sample of saccharin which has the formula C7H5NO3S
how many carbon atoms are in the sample? so we're asking about numbers of
atoms that means we're gonna need Avogadro's number, so if this is
confusing go back and check out my tutorial on the mole and Avogadro's
number and that should clarify a few things and when you're ready go ahead and give these a try.
so let's take a look at the first one remember we have a
5.0 kilogram brick of pure gold and we want to know how many gold atoms we have
so naturally with any conversion we're just gonna start with the value that we
have the only information we have is 5.0 kilograms. so how are we going to convert
this? well first of all you want to get this into grams because whenever we're
using molar masses and things like that we're using grams not kilograms so we
need to go from kilograms to grams so let's multiply our value by 1,000 grams
over one kilogram, that will cancel out the kilograms and now we'll have a value
in grams. so what's next? we have grams of gold we have a number
of grams of gold but we need moles because moles tell us about numbers and
the question is asking us about a number of gold atoms so we need to go from
grams to moles and for that we need the molar
mass of gold, and so we can look that up on the periodic table that is one
hundred ninety six point nine six seven so we're gonna put one mole over that
many grams because we need grams on the bottom so that grams will cancel and now
we have a value in moles. we have the number of moles of gold atoms and now
that we have the number of moles which is a number we're going to use
Avogadro's number to get this into the number of atoms so for however many
moles of gold atoms we have we simply multiply by six point zero two two times
ten to the 23rd atoms over one mole because those are equivalent, Avogadro's
number is the number of items of any object that are in one mole those are
equivalent that is the definition of a mole so now moles will cancel and we
have a value in atoms or a number of atoms so we can just go ahead and plug
these into the calculator we have to multiply everything on the top so that's
five point zero times a thousand times six point zero two two times ten to the
23rd and then we divide by everything on the bottom which in this case is just
one hundred ninety six point nine six seven. now here's a tip, if you think this
feels a little complicated because of the 10 to the 23rd what we can do is
just enter in all of the other values first so in your calculator do five
point zero times a thousand times six point zero two two divided by 196 point
nine six seven just do all of that first and click enter
and that will give us 150 and now we can multiply that by the remaining ten to
the 23rd so that simplifies things a little bit. now remember this is an
appropriate scientific notation because we need one digit followed by a decimal
place so we're going to have to adjust this until we have one point five zero
so we can go by powers of ten but every time we change
value we have to change the exponent above the ten so if we go from 150 to 15
we just made that ten times smaller so we have to make 10 to the 23rd 10 times
bigger that's one more power of 10 so we go to
10 to 24 if we do it again and we go from 15 to 1.5 we've made that 10 times
smaller so we're in order to keep the magnitude of this value the same we have
to also get 10 times bigger somewhere else and so we increase the power of 10
by 1 and we have 10 to the 25 so that is our answer this is 1.5 times 10 to the
25 atoms. now moving on to the second
question we have our 40.0 milligrams of saccharin and there is the
molecular formula and we want to know how many carbon atoms are in there so
this is very similar to the first question let's approach it the same way
we have 40 milligrams and once again we want that in grams because all of our
molar masses that are going to enable us to convert into moles use grams so let's
multiply by 1 gram over 1000 milligrams we need milligrams on the bottom so that
milligrams will cancel and we are left with a value in grams now we can
multiply by 1 mol over 180 3.18 grams that is the molar mass of saccharin that
we can get by adding up 7 carbon atoms 5 hydrogen atoms one nitrogen 3 oxygens
and one sulfur, if we take the atomic masses of those and add them up we get
183 point one eight grams that is the molar mass of saccharin. and so by doing
this and by putting grams on the bottom we get grams to cancel and we are
converting into moles of saccharin. now once we have moles of saccharin that is
a number and so in order to get the number of molecules we just need to
multiply by Avogadro's number, Avogadro's number works for atoms or
molecules or tennis balls or anything because a mole is a number and it
implies that there are Avogadro's number worth of objects in that
sample. so if we have some number of moles of saccharine molecules then
multiplying by Avogadro's number will give us the number of molecules that are
in there so we multiply by 6.022 times 10 to the 23rd molecules over 1 mole
moles will cancel and once again if we do 40 times 6.022 divided by a thousand
divided by 180 3.18 we will get this number and we can leave
the 10 to the 23rd separate once again but now we do have to get the decimal
place just to the right of the 1 so if we move the decimal place and get point
0 1 3 1 we have made that 10 times bigger so we also have to make it 10
times smaller by decreasing the exponent on the 10 from 23 to 22 we can do it
again to get point 1 3 1 so that's 10 times bigger
so to also make it 10 times smaller we go from 10 to the 22 to 10 to the 21 and
then we need to do it one more time so 1.3 1 once again we have made that value
10 times bigger so to simultaneously make it 10 times
smaller and keep the value the same we have to go to 10 to the 20 so we have
1.3 1 times 10 to the 20 saccharine molecules so we're almost there but
remember the question asks how many carbon atoms are there not saccharine
molecules so we take our number of molecules and then we just have to
multiply by 7 carbon atoms per molecule because each molecule of saccharin has 7
carbon atoms in it so however many molecules we have we need to multiply
that by 7 to get the number of carbon atoms that there are so molecules will
cancel out and we end up with 9 point 1 7 times 10 to the 20 carbon atoms and that will be our answer for this question.