In this video I'll discuss just
a few empirical equation of states. There are almost
an endless number of them. They can be found in the literature. I'm sure they're whole PhD thesis, many whole PhD thesis
developing equations of state. Probably less you'll find fewer today than in the past back when I
was an undergraduate. It was still a topic of great interest,
and I had professors that
that's what they did. So, some are well founded on theory,
others are purely empirically based on an essentially
regressions on experimental data. And here we'll look at a few examples. So there's Van Der Waals' equation. We just talked about that. It was first proposed in 1873, and it was one of the first to perform
much better than the ideal gas law. And it's based on a physical understanding
of the role of attractive and repulsive forces as we mentioned. It's not just very accurate as
you approach the liquid state, and nowadays it's really only used for
illustrative purposes. Another well known equation
introduced in 1949 is Redlich-Kwong. You'll find this in
undergraduate thermo books. It was a significant improvement
over other equations of states, doesn't work very well for
liquid-gas equilibria. It's okay for the gas phase when p
over Pc is less than T over 2Tc, where Pc and Tc are the critical
pressures in temperatures. And it doesn't look that
dissimilar from from Van Der Waals equation, just a slight
modification from Van Der Waals equation. So it's a two-parameter equation of state. Another two-parameter equation
of state is Peng-Robinson. It was introduced in 1976,
which is the year after I finished my PhD. It is similar accurate to Redlich-Kwong,
but predicts liquid densities better,
especially for polar molecules. Again, it is also a two-parameter equation of state that the parameter alpha is, as given by the function
at the bottom of the slide. Omega is something called
the eccentric factor. I will not discuss it more here, but
you can find a little bit about it in a textbook, and
in the wonderful book by Wark. Then the final equation I'll
mention is Benedict-Webb-Rubin, and this is an example of
a multi-function fit. It's a more complex relationship and
involves eight constants, eight parameters. And you can see it has 1 over v,
1 over v squared, 1 over v cubed, and 1 over v6 terms. And then in effect,
you get up to essentially 12th order, two times two, times or
I'm sorry, 3 times 2, times 2. So this is a much more complex relation. Nowadays, you would probably do these
kinds of fit using genetic programming methods where you can pull in lots of
different kinds of functions, and so on. And it becomes hard to relate each of the terms to
particular physical meeting. But in the end, it's all about
attractive and repulsive forces, and that's what you're trying
to take into account. So again, there are many, many, many
of these empirical equations of state. If you work in the oil and
gas industry, for example, you'll find lots of them that probably
are proprietary, a lot of them. And just know that what it all comes
down to is it does the functional relationship work well for
the particular substance, or mixture of substances that you're
dealing with over the range of thermodynamics states
that you're interested in. So that's it for this video. Thanks very, very much for
listening, and have a great day.