So, let me just plot those now on a common scale, those are all the numbers I

showed on the last slide in units of joule per kelvin per mole.

Plotted as the log of the mass of the molecule itself.

And so here are the noble gasses. And you see that, sure enough, it's

roughly linear in the log rhythm of the mass.

Here are the dihalogen molecules. And they too are still linear in the

mass. So, Since all were var-, well, we're also

varying a little rotation, but it doesn't show up here.

it, i-, it has the expected increase in entropy associated with an increase in

mass. There will also be some associated with

the change in moment of inertia, but it must be increasing also with log mass.

And then finally we have the HX series. So each of these series is unique -

comparisons being made within itself. And when we were focusing on things

having similar mass, like these three, two molecules and an atom.

They all have a similar mass, and yet the noble gas has the lowest entropy because

all it has is translational entropy. The hydrogen halide has more because it

can rotate, but it's moment of inertia is smaller than the dihalogens, and so, it

gets a little extra rotational entropy. And it's highest here, on the axis of

entropy. So, variation within a series primarily

dictated by mass, relationships between series differentiated by rotational

entropy. Alright, well having made those points,

let me give you a chance to exercise your intuition on a series and then we'll come

back. Well let's wrap up with just a few more

comparisions. I want to look now at some polyatomic

gases. So still at, at 298 k, I've got carbon

dioxide, nitrogen dioxide, methane, acetylene, ethylene and ethane.

And one of the purposes of this slide is to illustrate again, the amazing

agreement between calculated entropies. So using only properties of the

individual molecules, their mass, their rotational temperature, their vibrational

temperature, and the degeneracy of their electronic ground state.

These are the calculated entropies. These are the experimental entropies.

And you observer that to within 0.1 joules per kelvin per mole, spot-on

quantitative agreement. But, our goal here was to look a bit more

at trends. So let's look at CO2 versus NO2.

So CO2 and NO2 weigh very, very nearly the same.

Carbon has a mass of 12 in it's most abundant isotope, nitrogen a mass of 14,

but there's a difference of 27 26.5. I guess, if we want to be careful, in

their entropies. So, why is that?

Well, the issue is that carbon dioxide is a linear molecule.

Nitrogen dioxide is a bent molecule. It's non-linear.

And, so remember, that a linear molecule only has two rotational degrees of

freedom. The non linear nitrogen dioxide has three

rotational degrees of freedom. So that's an extra rotational degree of

freedom where there can be a lot of disorder because of closely spaced

rotational energy states. And that's enough to contribute to

substantially higher entropy. If we now look at the various

hydrocarbons here, methane, acetylene ethylene ethane, there is a steady

increase in the entropy. It's not huge, amongst, say, the, the C2

isomers. And mostly it's just associated with

increasing mass. So I keep adding hydrogen, or, or carbon

atoms. and that adds to translational entropy.

And then there's also a small increase in the rotational moments of inertia for the

heavier hydrocarbons. And so I'll, I'll just emphasize one more

time that you know a real demonstration of the power of statistic thermodynamics

is that it's possible to derive from first principles these third-law

entropies, and derive them with alarming accuracy.