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In this screencast, we're going to demonstrate the use of the
average molecular weight for a mixture to help simplify a problem.
So here, we're given a distillation column and
a composition of gases is leaving the column. If we were interested
in the total volumetric flow out of this column, for processes downstream,
then we would need to calculate the molar flow from the column.
To do that, we can either calculate
the moles of each species and then
the total amount of moles leaving the column, or we can calculate an average molecular weight.
So let's do that, given information we worked out in previous screencasts.
So let's just reiterate what we're focusing on: if we have 50 kilograms per hour
leaving the column, and we divide this
by our average molecular weight of our mixture--and this is going to be in
kilograms per kilomoles--and this will give us
kilomoles per hour. So we need to figure out what our average molecular weight is.
Now the equation we've worked out previously was the
the reciprocal of the average molecular weight is equal to the summation
each mass fraction, designated "xi"
for each component, divided by their molecular weights.
And we're using this formula, compared to
the other one we've seen, which is the summation of
the molar fractions times the molecular weights
because we're given a mass flow and we're going to assume that this is
a mass composition. So we're going to use this equation rather than the other one.
We start filling in what we know
for each of these species. Now obviously, you see we're going to need the molecular weight
of each component, so I'm going to start looking those up and filling them in our table.
So here are the molecular weights of each of the gas species.
And we basically follow this equation, where we take the
mass fraction and divided by its molecular weight.
And then we some all these up. So here's a summation of all my products
and this is going to be equal to 1 over the average molecular weight.
Just looking at the chart, we look at our lowest molecular weight, which is 16, and our highest, which is 44.
So we know our average has to be between these two values.
So let's do these calculations, and you see that we get
0.4 moles of mixture per grams of mixture.
We know we need to take the reciprocal of this, and when we do so
we get an average molecular weight of 25.4 grams per mole of our mixture.
So what do we do with this?
Well, we're going to go back to our original problem statement:
take our 50 kilograms and hour, and plug this in, making sure that units are correct
and we can calculate what our molar flow from the column is.
And I get 1.97 kilomoles per hour exiting the column.
So now that we've calculated the molar flow rate, we could use
that to determine the volumetric flow rate by, say, using an ideal gas law.
Another way to do this even quicker is if we use a spreadsheet.
So let's open up Excel. I've typed in my species,
the molecular weight of each species, and the mass percent. It's probably good practice to make
sure that you entered everything in correctly.
So the mass percent should add up 100. If you wanted to calculate
the average molecular weight, given these mass percents,
and then have a spreadsheet that recalculates it
if we change the mass percents up or we could also do it for mole percents
then we need to enter in some equations.
So the first thing we need to do
is convert to mass fractions, so I would type in, say, "mass frac",
and then take each of these and divide by 100.
Then we can use our formula, so I'll just type in "formula".
and set this equal to our mass fraction divided by our molecular weight.
Now we do this for each of them and we sum them up.
We can then take the reciprocal of that,
and we have our molecular weight for this composition.
Now, the nice thing is, say I change oxygen to "10" and nitrogen to "28".
This changes our average molecular weight slightly.
Probably a good check is to keep
our summation up to 100 so that we don't necessarily change one value and then
realize that it would be an impossible composition.
So now we have a spreadsheet that's built to calculate
our average molecular weight, given certain compositions and species.
You could do the same thing for molar flow rate and have
a similar looking spreadsheet.