Look at a diagram of enthalpy versus weight percent, this is for sulfuric acid and water.
And the enthalpy is per kg of the solution and out objective is, use this diagram to
determine, What would be the adiabatic temperature of mixing?
So this is an exothermic process.
The enthalpy of the mixture, you can see is significantly lower than the enthalpy of the
pure components.
So it's non-ideal.
If it were ideal the enthalpy would just be the mass fraction of component one times the
enthalpy per kg of component one plus the mass fraction of component two, enthalpy two.
Enthalpy 2 is here, enthalpy 1 is here, and we could rearrange this by writing it just
in terms of the mass fraction of component one.
And you can see this says for an ideal solution we just expect a straight line, so for example
if we were to start at 38 degrees, here's pure component 2, here's pure component 1,
what we would expect then is for any mixture, the enthalpy is somewhere along this line,
so if we made a 50/50 mixture the enthalpy of the mixture would be here.
If it's adiabatic then we also expect that the enthalpy of the mixture is going to be
equal to the enthalpy of the starting components.
And the difference is now that this is not at the same temperature.
So this is going to be at a higher temperature than the enthalpies of our starting materials.
So let's look at this diagram and do this problem where we're going to mix pure sulfuric
acid and pure water, and our objective is to obtain a 30% solution.
And if the water and the sulfuric acid are both at 15 degrees then the question is, What
is the temperature of the final solution?
And so, this means, 15 degrees, this is our starting value for the water, enthalpy.
And 15 degrees here, starting value for sulfuric acid.
So then all we have to do is say the enthalpy of the mixture has to be along this line.
Because it's adiabatic so the enthalpy has to be the sum of our starting enthalpies.
And then if we determine where 30% is which is right around here, So this would be a 30%.
Then we could just read the temperature off of the diagram where its somewhere between
60 and 82.
We had to make an approximation this is about 70 degrees C. So 70 degrees C that's the adiabatic
temperature when we mix pure materials starting at 15 degrees C. So the other question we
could ask is, suppose we wanted to mix them isothermally, how much heat would we need
to remove from the mixture, per kg of mixture.
And so this means our final enthalpy, if we mix them adiabatically, we can read, about
380 kJ per kg of mixture.
If we wanted it to be isothermal so we wanted the final temperature to be 15 degrees, 30%,
15 degrees is about here.
Which is an enthalpy of about 200.
This is what we desire, make it isothermal.
This is what we would end up if with if we did it adiabatically, so this means we need
to remove 180 kJ.
So we'll do one more problem and this is one you might first try on your own.
But we want to mix 10%, that's weight percent sulfuric acid, at 15 degrees, with a mixture
that's 80 degrees sulfuric acid that's at 38 degrees.
So of course there is one more piece of information that we need, and that's what's our final
composition.
What we're going to do is mix 1.5 kg of the 10% sulfuric acid and 1 kg of the 80%.
So we need to determine what's a composition of the mixture.
So the simplest way is to say well we're going to end up with 2.5 kg, the 10% is going to
provide 1.5 multiplied by .1 so .15 kg sulfuric acid.
And the 80% is 1 so this means we have 0.85 kg of sulfuric acid and our mixture is a total
of 2.5 kg.
So this is 0.38 kg of sulfuric acid per kg of mixture.
Or we are at 38% on our diagram.
So to determine a temperature on our diagram now we can draw a straight line between our
starting points, between this 10% and this 80% at different temperatures.
We're going to be at 38% for our mixture, so the point has to be on this line.
So right here is our mixture and we can then read off the diagram that this is somewhere
between 38 and 60 maybe a little more than halfway so maybe 52 degrees C. But this would
be the adiabatic temperature.