The constant terms that are appearing, the B terms are known as Virial

coefficients. They are indexed by two subscripts, the

first subscript tells us at what term in the expansion do they appear, the second,

the third, the fourth. The second subscript is simply are the

volume term or a pressure term. And so, for instance, we would call B2V a

second virial coefficient. They are functions of temperature, and

that's emphasized by including temperature in parentheses afterwards.

So if and when we determine them, we'll do them at certain temperatures.

And we'll get to that momentarily. Do, take a moment to look at the

equations and think about sort of the sensibility at certain limits, and in

particular, let's consider the limit as pressure is going to 0.

So as the pressure is going to 0, and as my molar volume as a result of there

being essentially no pressure is going to infinity.

And if the molar volume is going to infinity, then, of course, the density is

going to zero because they're inversely related.

Well, what would happen to these equations?

If pressure is going to zero, pressure cubed goes to zero really quickly.

All the terms are going to zero, including even this one.

Although most slowly, this one. And ultimately, I'll get compressibility

equal to one. Which is what I expect.

I should have ideal gas behavior as my pressure goes to zero.

Similarly, if molar volume is going to infinity, I keep dividing things by

infinity or powers of infinity, they all go away, and I'm left with

compressibility equals one. So the virial equation of state has the

correct form to show ideal gas behavior at that limit of infinitely low pressure.

Well, where do these virial coefficients come from?

So, let me show you a plot of compressibility against pressure.

Very near no pressure at all, so these are [COUGH] very low pressures, somewhere

between zero and a tenth of a bar. [COUGH] And remember that a tenth of a

bar is about a tenth of an atmosphere. And what we see for these isotherms.

And this is for ammonia gas. Is zero degrees C, 100 degrees C, 200

degrees C. If I extrapolate, they're all going to

ideal gas behavior at the lowest pressure.

More over they're all linear. And let me think about what happens at

very low pressure. So as the pressure is going to 0, as

numbers go to 0 if you square or cube or take their fourth power they go to 0 much

faster obviously. So the only term that will survive that

contains P within it. In the virial expansion, it will be Z is

equal to 1 plus B2P, function of temperature, divided, sorry, multiplied

times P. So that is a linear equation, in P.

And sure enough, these are lines, as a function of P.

And so the slope, of this experimental measurement, gives you, B2P.

And each of these isotherms, has a different slope.

And that illustrates to you that yes the coefficient does depend on temperature.

But, by doing a series of measurements, you can infact determine B2PT, for a

given gas at given tempuratures. You can also manipulate the two virial

expansions. It takes about maybe 15 lines of

equations to do this. It's not awful, but it's a little much to

put on a slide, so I'll just give you the result.

You can show that B2V(T), so that's the coefficient that appears in the molar

volume, virial expansion, Is equal to something quite simple.

It's just R times T times B2PT. So by doing this experiment, you

simultaneously get both of these two virial coefficients, second virial

coefficients. Let's just look at, how the various terms

in the virial expansion. What sort of orders of magnitude they

take on,at, under different sets of conditions.

So here are some data for argon at room tempature, 298 Kelvin.

I'll show you for one bar, so that's atmospheric pressure, 10 bar, and 100

bar. And these all are the B 2V component of

that stage, as in units of volume per mole.

'Cuz remember compressibility is unitless.

1 is certainly unitless. So, if we're going to divide by Vbar,

B2VT has to have the same units as Vbar, in order to be unitless.

What are the units on Vbar? Volume per mole.

In any case, at one bar, argon, it really behaves pretty ideally, so, the one comes

in. And the second term in the virial

expansion is 0 point 00064. So the deviation from the ideal behavior

is 6 times 10 to the minus 4th. Pretty small.

And the third and fourth terms don't contribute at all.