 # Average Molecular Weight Derivation

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.

## 4 thoughts on “Average Molecular Weight Derivation”

1. LearnChemE says:

This screencast has been reviewed by faculty from other academic institutions.

2. uBoscuBo says:

I am confused on 5:41, you stated that you got the same value, but the units are different. How is that possible?

3. Anthony Pascual says:

I'm really glad you remade a video for this topic. I want to extend my gratitude and thanks for addressing even the most minor of details such as units. For the most part, the previous lady was terrible at explaining and addressing those kinds of important details.

4. Mrudav Raval says:

What did you do at 2:00 for the streams of N2 and O2?