In this screencast I'd like to explain some of the advantages of using screencasts to
introduce and explain an equation, as opposed to having an equation presented in a textbook,
and then having text below the equation explaining the various terms in the equation.
So there are at least three significant advantages to doing this, one is that we can use color
to separate out different terms, the other is we don't have to introduce the equation
all at once, so we can sequentially introduce terms.
The idea is we can introduce the term and explain it as we're introducing it rather
than presenting it all at once and having text below.
Another is that we don't necessarily have to be linear, we don't have to go from the
top of the page to the bottom, but we could have an equation, for example, if we have
this equation which we'll use in our example.
We could have things off this way, and over here, so we use this equation as an example
of how we can take advantage of screencasts in introducing or describing properties and
equations.
So d G, which is the change in Gibbs free energy, differential change is equal to minus
the entropy times differential temperature, plus volume, the specific volume and a differential
pressure.
We can look at, because this is an exact differential, we can write this term, what we'll do is write
it here as the change in the Gibbs free energy with respect to temperature, keeping the other
variable constant, and then we're going to do the same thing for the volume, we're going
to write this as the change in the Gibbs free energy, with respect to pressure, keeping
this variable temperature constant.
And then one of the things that we can do is the property of this exact differential,
we can differentiate, so we're going to differentiate this term, with respect to the other variables,
and they all differentiate with respect to pressure while keeping temperature constant
and we're going to differentiate this term with respect to temperature, keeping pressure
constant.
And then what we're going to say is that these two terms are equal, and they're equal to
the second derivative of the Gibbs free energy with respect to temperature, respect to pressure.
The order of the differentiation doesn't matter and so therefore we can write it either way.
This is what makes these two terms equal.
The other thing we can then do is substitute this derivative of pressure, well this term
in brackets in minus entropy and that derivative is at constant temperature, and this term,
the derivative of volume with respect to temperature at constant pressure, and this we'd write
as minus derivative of entropy with respect to pressure at constant temperature and so
these two terms are equal, and this is one of the Maxwell relations that we can derive,
and they have some advantages in thermodynamics, but the point here is that we can relate terms
more easily and we can introduce the terms sequentially to make it easier for students
to see where these terms came from.