Real gases excluded volume

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

The results of the last section showed that, for any macroscopic container at normal pressures, it is not reasonable to conclude that the molecules proceed from wall to wall without interruption. However, if the interaction potential energy between molecules at their mean separation is small compared to the kinetic energy, the speed distribution and the average concentration of gas molecules is about the same everywhere in the container. In this limit, the only real effect of collisions is the excluded volume occupied by the molecule, which effectively shrinks the size of the container. At 1 atm, only about 1/1000 of the space is occupied (remember the density ratio between gas and liquid), so each additional molecule sees only 99.9% of the container as free space. On the other hand, if the attractive part of the interaction potential cannot be totally neglected, the molecules which are very near the wall will be pulled slightly away from the wall by the other molecules. This tends to decrease the pressure. 

When generalizing this reasoning to include ring formation, we may expect that the ideal tree model with no excluded volume and no ring formation will be realized in the limit C. The classical gel theory corresponds to this limiting case (C—). That is thus comparable to the classic status of the ideal gas law (C—>0) to the real gas. 

The summary of dimensions at the bottom of Fig. 1.33 demonstrates the effects of the various restrictions on the model compound. The random flight model gives for polyethylene already an answer within a factor of about 2.5. Comparison with the experiment is possible by analyzing the dimensions of a macromolecule in solution, as will be discussed in Chap. 7. One can visualize a solution by filling the vacuum of a random flight of the present discussion with the solvent molecules. The 0-temperature listed as condition for the experiment is the temperatme at which the expansion of the molecule due to the excluded volume is compensated by compression due to rejection of the solvent out of the random coil. This compensation of an excluded volume is similar to the Boyle-temperature of a real gas as illustrated in... 

Actually, the ideal volume of a gas is equal to the difference between the real volume occupied by the gas less the excluded volume of the gas entities. The excluded volume corresponds to the excluded molar volume or covolume, denoted historically by the letter b, and expressed in m mo times the amount of substance, n, in mole. 

When the system has N identical spheres in the volume of V, there are N /2 pairs of excluded-volume interaction. Then, the change in the total free energy due to the excluded volume is AA/k T = (N /2)vJV. The change per sphere, (AA/k T)/N, is proportional to the density N/V. At low concentrations, the excluded volume is negligible. As the concentration increases and Nv approaches V, the effect becomes stronger. The same effect appears in the van der Waals equation of state for a real gas The correction to the volume is equal to the excluded volume. 

Figure 1.4 van der Waals considered molecular interaction and molecular size to improve the ideal gas equation, (a) The pressure of a real gas is less than the ideal gas pressure because intermolecular attraction decreases the speed of the molecules approaching the wall. Therefore preai = P ideal — 8p- (b) The volume available to molecules is less than the volume of the container due to the finite size of the molecules. This excluded volume depends on the total number of molecules. Therefore... 

All these difficulties are circumvented if measurements on the polymer solution are conducted under conditions such that the effects of excluded volume are suppressed. The resistance of atoms to superposition cannot, of course, be set aside. But the consequences thereof can be neutralized. We have only to recall that the effects of excluded volume in a gas comprising real molecules of finite size are exactly compensated by intermolecular attractions at the Boyle temperature (up to moderately high gas densities). At this temperature the real gas masquerades as an ideal one. 

For the macTomolecule in solution, realization of the analogous condition requires a relatively poor solvent in which the polymer segments prefer self-contacts over contacts with the solvent. The incidence of self-contacts may then be adjusted by manipulating the temperature and/or the solvent composition until the required balance is established. Carrying the analogy to a real gas a step further, we require the excluded volume integral for the interaction between a pair of segments to vanish that is, we require that 3=0. This is the necessary and sufficient condition.6,13... 

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