Hello.
Today, we're going to talk about
First Order Phase Transitions and finding the vapor dome. So consider the stability of a substance
described by Van Der Waal's equation local criteria requires that the partial
of p with V at constant T be less than 0. And we discussed that in the last video. Now if we were to plot the P-v-T Relation
in terms of constant temperature lines P versus V and plot constant temperature
lines in the liquid vapor region, it would conceptually
look something like this. This series of lines and in some
cases they have quite a humping them. If we look at one of those
isotherms that has the hump in it. It looks like the plot on the left
here that goes ABCDE it Sarah FK f j k l m n o Etc. So the the local stability criteria are clearly violated over
the region of f 2 km. And what's in the red
circled there suppose, we calculate the Gibbs potential that's
and we know that D mu is equal to minus sdt plus VDP along a line
of constant temperature. That means that new is the integral of
V DP plus some function of temperature which would be a constant line
of constant temperature line. So the change in chemical potential along
an isotherm is equal to the integral say from A to B is equal to the integral
from A to B of V of P DP. So if we carried out that integration, then we get a plot that looks
something like that like this. So this is Mew versus P starting at a and go all the way to S along that isotherm
from a high volume to a low volume. Now, what is interesting about
this is that at the point d o the lines the line crosses and
it was we started a BC we get to D. Then then Mew keeps increasing e to F and then it starts coming back
downhill F jklm turns around and N o QR s but addio these two branches
cross now mu of T and P is the fundamental relation per
mole in the Gibbs representation and so stable points have to have
a minimum value of mu and as the pressure is increased the physical
state will be the one with a minimum value of mu now at the points
Dio where the two sides. Up both the liquid and
Vapor branches are at minimum Mew. So recall the PT diagram classic
PT diagram phase diagram where you have vapor at high temperatures solid at lowest the lowest temperatures
and then you have these transitions. You have the vaporization condensation
line where Vapor goes to liquid if the melting line where
liquid goes to solid and you have the sublimation line
where Vapor goes directly. So solid, so
we would expect a process to proceed as Illustrated in the plot on the top
here you start at high volume. He started a you go ABD when you hit D,
you're going to go, you know, your undergraduate thermal
that you're going to that you're going to start into the saturation region
and you're going to go in a straight line at constant pressure
across 2.0 from To all and then the pressure will increase
again as you reduce the volume. So that is the saturation region. Now you d- me you always equal
to the integral of V of P DP and that has they are the same at so
it's zero and what that means is that the area
under the isotherm from DF Decay that area labeled one
has got to be equal the area. You in that region kmo label to so those two areas are the same and
using that criteria. You can very easily find the vapor Dome. If you have the pvt relationship, you can
get other properties as well saturation properties entropy as sub f g
internal energy use of FGM and enthalpy h of f g is given by
these relations on this slide. So here we construct The Vapor
dome by carrying out this process of calculating
the Gibbs free energy Mew at different at
different temperatures. And that way you can completely
construct the vapour Dome. Here's another way of looking at
it which shows the isotherms and then it shows horizontal
lines where the areas. Of the in the two and
the two portions D2K and k2o are equal and if you just draw those lines sure enough,
you'll plot out the size of the vapour Dome now at some point you
just get an inflection point up there. There will be no reversal of value of the
isotherm of the pressure on the isotherm. And when that just happens,
that's the critical point. So that's it for this video. Thanks for listening and have a great day.