In this screencast, we're going to look at calculating the average molecular weight
of a mixture, and answer the questions of why you'd want to do that
and how you would do that, given both molar or mass compositions.
So let's start with an example, where we have some composition of oxygen and nitrogen
entering some kind of mixer. We're interested in determining what
the molar flow rate of the composition would be,
and then if we wanted to actually calculate the mole composition of oxygen and nitrogen out.
So we're just doing a nice conversion from mass
on the incoming side, to moles on the outgoing side.
It's important to label our units if we haven't done so, so I'm going to fill that in on the left side here.
So I've written kilograms of oxygen per kilograms of M,
which I'm using for the mixture. At this point we want to get into a molar composition.
We should be familiar with just converting a mass to a mole. So what we could do here is multiply
0.4 times the 100 kilograms an hour entering, to get our total mass of
oxygen coming in per hour. If we want to convert this to a molar flow rate.
We would then divide this by the molecular weight of oxygen
and we would get 1.25 kilomoles per hour of oxygen.
Now, we do the same thing for nitrogen,
we get 2.14 kilomoles per hour of nitrogen. You would add these two together and we would
have a total flow rate of 3,390 moles per hour.
And we still have to calculate what the composition of O2
and N2 would be, so we could take
each of them and divide by the total to get the molar percentages.
Of course, since we've done this in moles, it's important to write out the units.
And we have our final solution. Where we've calculated the total amount of moles coming out of our mixer
and the molar composition of both oxygen and nitrogen.
Now, another way to go about this, to reduce the amount of steps and work
is to calculate the average molecular weight of the mixture. If we take that
average molecular weight and use it with the incoming mass flow rate, we could
easily calculate the outgoing molar flow rate and then we can look at the composition.
Let's do another example to kind of demonstrate how we would calculate it
depending on whether we're given a mass composition or a molar composition.
So we're going to use a simplified example where we're just looking at air
and just accounting for the nitrogen and oxygen making up to 100 percent of the composition.
If we're given the molar composition of air,
and we want to calculate an average molecular weight.
How would we go about that? Our end goal is to get an average molecular weight
of our mixture, let's say grams of the mixture per moles of the mixture.
So we need to use the information we're given for the molar composition of air, and somehow
come up with an equation that's going to give us units of grams of mixture per mole of mixture.
So let's think about what other information we know about these two components.
Well, we have their molecular weights or we could look them up.
So the molecular weight of nitrogen will be 28 grams per mole of nitrogen.
And again, I'm using the units so you can see how we're going to develop this equation.
Let's just assume that we multiply kind of like we've seen before.
So if I were to write it out just as such, without filling in
values here--in the parentheses that are empty-- the question is what would I multiply
the 0.79 value by to cross out the moles of nitrogen, since in that
end product, we have grams mixture per moles mixture.
Well, we want to keep the moles mixture, so if we multiply by the molecular weight,
we see that we cancel out the moles of nitrogen
and the moles oxygen. So now we have grams nitrogen
per mole of mixture and grams oxygen per mole of mixture,
which isn't quite the same units as the grams mixture per moles of mixture.
So what do we do? Well, the nice thing is we see that grams nitrogen
plus grams oxygen is going to be the grams of our mixture.
Though we have the appropriate units on our left side to get the units on the right side.
So when given the molar composition of something, and we want to calculate the average molecular weight.
We just multiply the mole percents by the molecular weights of each species and add them together.
So we write this as the average molecular weight is equal to the summation of the molar fractions times
their respective molecular weights. Now when we look at a mass composition, we can't use the same formula.
So we have to come up with another one.
So let's approach this by using the same setup we had before,
so I have our mass fractions written out and
I have blank parentheses for the molecular weights.
We want to get an average molecular weight with
grams of our mixture per moles of the mixture, so we want to get rid of the
kilograms nitrogen and kilograms oxygen.
Well, this time we can't multiply by the molecular weight, since it has
a mass in the numerator. So if we divide by it instead,
we see that we can now cancel out our kilograms of nitrogen and our kilograms of oxygen.
So we're left with kilomoles of nitrogen over kilograms of the mixture
and kilomoles of oxygen over kilograms of mixture.
so I made the calculations, you see that the units on this
do not equal the units of a molecular weight.
However, we could add the kilomoles up, just like we did above.
This gives us our kilomoles of our mixture, and now,
when we add these up, we get 0.0347 kilomoles of our mixture over kilograms of our mixture.
If we want the average molecular weight,
we need to take the inverse of this, and when we do that, we get the same answer we did above.
So the rule of thumb, when given a mass composition,
is to use the following: where the reciprocal
of the average molecular weight is going to be the summation of
each mass fraction divided by its respective molecular weight.
So you can see that these are vastly different,
and it's important to use the appropriate one depending on the composition that you're given.
So if we went back up to our mixture of oxygen and nitrogen,
and used what we just worked out for an average molecular weight,
we're given a mass fraction and we want a mole fraction.
So let's calculate the average molecular weight of the incoming stream
knowing that we have mass fractions.
If we do this, we would have 0.40 kilograms of oxygen per kilogram of the mixture,
divided by the molecular weight of oxygen, which is 32 kilograms per kilomole.
And we do this for nitrogen as well, and this is going to equal one over the average molecular weight.
So the average molecular weight of our mixture coming into this mixer
is 29.5 kilograms per kilomole.
So 100 kilograms per hour, divided by our molecular weight, gives us
3.39 kilomoles per hour, which is exactly what we got the first time through.
So when doing this for two components, you can see the work is not
that much quicker, but if you had a stream where you had 7 or 8 components,
say, like a distillation column,
calculating the average molecular weight can save you a lot of time.
Hopefully this gives you an idea on how to calculate the average molecular weight
based on both mass and mole fractions.