[Il Divo]

Greetings all,

so just recently I started reviewing my Thermodynamics and Kinetics. The problem is I'm running to is that I feel alot of it becomes math-heavy and difficult to comprehend or grasp the utility of.

Currently I"m trying to get the idea behind virial coefficients, related to the ideal gas law, in the form of:

pVm = RT (1+ B'p + C'p^2...).

Parts of it are fairly intuitive. Everything in the parentheses is related to Z, which is just compression factor explaining how far from ideal the gas is behaving, and the different coefficients have different affects on the relation.

But unlike compression factor, I'm not getting why the virial equation is itself useful? For example, how are we able to use the virial equation to derive other applicable versions of the gas law? (Ex: the van Der Waals equation). Or why do we need other versions of the gas law, if we have the virial equation, which is applicable in all instances?

If there are any simple explanations on this, I'd really appreciate it, thanks!

Edit: And if it helps, I'm getting all this from Atkins.

[Jorriss]

But unlike compression factor, I'm not getting why the virial equation is itself useful? For example, how are we able to use the virial equation to derive other applicable versions of the gas law? (Ex: the van Der Waals equation). Or why do we need other versions of the gas law, if we have the virial equation, which is applicable in all instances?

I do not know much about this but I can make a few comments.

There are several reasons it is important/things to note.

1) It has a firm theoretical basis.

2) It is far more general than the van der waals, ideal gas, redlich kwong, etc equations of state. It is a better working point for new theory on liquid and gas behavior.

2) By keeping more terms, the virial equation of state can be made arbitrarily accurate - even if in principle evaluating coefficients becomes impossible for more complicated potentials/distributions/etc. This is true in general in physics and chemistry - power series are used constantly. To me, the power series for e^x is worth more than e^x itself.

3) While the virial equation is applicable, the actual coefficients one gets from it depend on the starting assumptions. If you assume your system obeys a van der waals equation of state, you will get different coefficients than if you assume ideal gas behavior. You can formulate a new virial expansion for individual potentials. I would suspect the van der waals equation can be derived from assuming hard-sphere behavior + another assumption.

[Enthalpy]

The virial equation is very inaccurate and has no physical meaning nor justification at all.
Any other meaningful equation (not so much Van der Waals, but Redlich-Kwong and many more, see Wiki) is far less bad.

Imagine, the virial equation doesn't even give the density of a highly compressed gas to 30% accuracy, despite tables give a set of coefficients for a very limited condition span. It's cr*p, useless, forget, garbage bin.

Sensible equations of state hold for very varied conditions with only one set of values which, for some equations, may even be computed from known data like the critical point; and the a 5% deviation would be considered bad.

But the virial equation is taught because one can give exercises to the students using it, like deduce expressions for H, S and the like.

[Jorriss]

The virial equation is very inaccurate and has no physical meaning nor justification at all.

What could you possibly mean by this? It is completely theoretically justified and it obviously has physical meaning - it's an equation of state.

[curiouscat]

De gustibus non est disputandum

There's so many different situations and uses for an EOS that it's hard to say any one as "good" or "bad".

Computational complexity, range of applicability, ease of solution, number of fitted parameters, mathematical elegance, fundamental relevance, non-linear behavior, convergence to the relevant solution and a lot of other aspects matter.

[Enthalpy]

[The virial equation] is completely theoretically justified and it obviously has physical meaning.

The virial equation is nothing more than PV=nRT with a polynomial correction.

It has no better physical meaning than PV=nRT. Other equations of state do add physical meaning, for instance a residual volume, for instance an attraction force linked with the critical temperature, or for instance a proportion of untied molecules depending on exp(-Hv/RT).

Because the virial equation adds no modification related physically with the behaviour of gases, it also makes bad previsions, bad interpolations. I tried to use it and it's very inaccurate and inconvenient - imagine a set of coefficients for every small temperature range and still 30% error!

Any other equation of state beyond PV=nRT is less bad than the virial, with some being rather good.

[Jorriss]

[The virial equation] is completely theoretically justified and it obviously has physical meaning.

The virial equation is nothing more than PV=nRT with a polynomial correction.

It has no better physical meaning than PV=nRT. Other equations of state do add physical meaning, for instance a residual volume, for instance an attraction force linked with the critical temperature, or for instance a proportion of untied molecules depending on exp(-Hv/RT).

Because the virial equation adds no modification related physically with the behaviour of gases, it also makes bad previsions, bad interpolations. I tried to use it and it's very inaccurate and inconvenient - imagine a set of coefficients for every small temperature range and still 30% error!

Any other equation of state beyond PV=nRT is less bad than the virial, with some being rather good.

I've never worked with these professionally but from what I remember the virial equation can be derived as an, in principle, exact of equation of state from the canonical and grand canonical partition functions. It is far more than the ideal gas law with a polynomial correction (particularly as an infinite series is not a polynomial). And it does have physical significance in many ways, the virial coefficients can be related to cluster expansions. It's clearly more meaningful than the ideal gas law which assumes no potential interaction.

I don't know if it's a very practical equation of state but it is a pedagogically good model.

Anyhow, outside of some math and derivations, this is about the extent of what I know of it

[fledarmus]

Virial coefficients are very much like Kepler's "circles within circles" descriptions of the orbits of planets around the earth. They may give you very precise descriptions of the behaviors of your materials, but only within the parameters that have already been measured. Their application to other materials or conditions are only coincidental.

The various other alternatives try to include some theoretical underpinning rather than a numerical expansion, and result in wider applicability with perhaps less absolute accuracy in particular applications.