A single value, value of molar volume that satisfies the cubic equation.

And, once we know that critical volume, we can determine the van der Waals

constants, a and b from that position, the critical volume, the critical

temperature, and the critical pressure. The Law of Corresponding States says that

all gases have the same properties if compared at their corresponding states.

Where under corresponding conditions that is, where corresponding conditions means

that the conditions for that gas, relative to its own critical conditions

are the same as the conditions for a different gas relative to its critical

conditions. So that is a corresponding state or

corresponding conditions. We also saw another equation of state,

the virial equation of state. And the virial expansion that appears in

that equation of states has coefficients that are directly related to

intermolecular interactions. The second virial coefficient, B2v,

measures the deviation of the volume of a real gas compared to an ideal one under

the same temperature and pressure conditions.

So, if B2v is a positive number, that describes how much more volume the real

gas occupies than an ideal one would under the same conditions and vice versa.

When B2v is negative, that means that the gas is occupying less volume than an

ideal one at the same conditions. We looked at the sorts of attractive

interactions that could occur between two different molecules in a real gas.

And in particular, we looked at the Lennard-Jones potential where the

intermolecular u is expressed in terms of the strength of a molecular interaction

epsilon and the size of the molecules themselves sigma.

And the Lennard-Jones potential has an attractive term that drops off as r to

the 6th, that is, as it goes as r to the minus 6, and a repulsive terms that goes

as r to the minus 12. So at very short distances, the repulsion

rises very, very steeply. The Lennard-Jones parameters themselves,

epsilon and sigma, can be determined through analysis of experimental second

virial coefficient values. We also talked about the physical

underpinnings of the r to the minus 6 attractive term and the most dominant

contribution to it, namely dispersion. We talked about London's development of a

quantum mechanical explanation for dispersion.

And his equation that produces its magnitude based on atomic or molecular

ionization potentials and the polarizabilities of the individual

molecules. Finally, we looked at some simpler

potentials than the Lennard-Jones potential that allowed us to solve for

the second virial coefficient analytically in order to gain some

physical insight. One potential was the square-well

potential, where there is a region of attractive interaction followed by a hard

wall. And determined that the predicted second

virial coefficients are not bad. And compare reasonably favorably to the

full Lennard-Jones potential over reasonable ranges of temperature and

pressure, and I showed an example for nitrogen gas.