The Effect of Intermolecular Forces

The correction subtracts the quantity a(n/from the pressure, where n is the number of moles, V is the volume, and fl is a constant that depends on the gas (see Table 5.5). Notice that the correction factor increases as n/V (the number of moles of particles per unit volume) increases because a greater concentration of particles makes it more likely that they will interact with one another. We can rearrange the corrected equation as  

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We can assess the effect of intermolecular forces quantitatively by comparing the behavior of real gases with that expected of an ideal gas. One of the best ways of exhibiting these deviations is to measure the compression factor, Z, the ratio of the actual molar volume of the gas to the molar volume of an ideal gas under the same conditions ... 

In this investigation, you will examine the differences between molecules that contain different functional groups. As you have learned, the polarity and hydrogen bonding abilities of each functional group affect how these molecules interact among themselves and with other molecules. You will examine the shape of each molecule and the effects of intermolecular forces in detail to make predictions about properties. 

In the case of the flux of mass, the result is the normal component of pua. But for the flux of momentum and energy, in general the flux density is not the normal component of a vector or tensor function of (t, x), since it will depend on the extended shapes of if and Y. But in the case of short-range forces and slowly varying p, ua, E, it can be shown to have this form with sufficient approximation. Thus one is led to the familiar pressure tensor and heat flow vector Qa, both as functions of (t, x). It is to be emphasized that the general expression of these quantities involves not only expected values of products of momenta (or velocities), but the effect of intermolecular forces. 

Most papers dealing with the spectrum of the carbonate ion CO7 neglect to mention the important paper by Decius, Malan and Thompson 42> on the effect of intermolecular forces on molecules in the crystalline state which refer specifically to the out-of-plane bending mode of CO In this paper they derive the dependence of this mode upon the 12C-13C isotopic ratio. Sterzel and Chlorinski 43) also discuss the effect of isotopes upon the CO2" vibrations these two papers should be consulted when assigning C03 -spectra because the modes depend very much upon the t2C-13C ratio. Orville-Thomas 20> has discussed the dependence of the C03 force constants upon the C-O distance, and shows that this leads to a bond intermediate between a single and a double bond. 

A number of other topics are collected together in Section 5. These include systems containing more than two molecules, the hydrogen bond, the effect of intermolecular forces on the properties of molecules, and a brief survey of recent work on empirical potentials and on van der Waals molecules, which are likely in the future to provide more and more detailed information for the theoretician to interpret. 

We have already encountered the effects of intermolecular forces in our discussion of precipitates and solubility. Here the intermolecular attractions between water molecules are instrumental in the ion-cage formation that allows some salts to go into aqueous solution. The glycerin molecule shares some similarities with water, but the individual glycerin molecules are still strongly attracted to each other and admit water to their ranks only when there is sufficient provocation. In this demonstration, the provocation occurs in the form of stirring, but no amount of stirring will force the canola oil into the glycerin solution until soap is added. 

From equation (2.30) it can be seen that p E, t) is dependent on cj, the form of P(EIE ) and k E). o is most often taken to be the Lennard-Jones collision frequency i.e., the hard sphere collision frequency which is rectified for the effects of intermolecular forces by the inclusion of a collision integral factor. 

The models most frequently used to describe the concentration dependence of diffusion and permeability coefficients of gases and vapors, including hydrocarbons, are transport model of dual-mode sorption (which is usually used to describe diffusion and permeation in polymer glasses) as well as its various modifications molecular models analyzing the relation of diffusion coefficients to the movement of penetrant molecules and the effect of intermolecular forces on these processes and free volume models describing the relation of diffusion coefficients and fractional free volume of the system. Molecular models and free volume models are commonly used to describe diffusion in rubbery polymers. However, some versions of these models that fall into both classification groups have been used for both mbbery and glassy polymers. These are the models by Pace-Datyner and Duda-Vrentas [7,29,30]. 

The calculations of TSM have recently been extended to include the effects of intermolecular forces by Tasumi and Shimanouchi 35). Estimates for the magnitude of intermolecular force constants for these calculations were obtained from the small splitting observed for higher-frequency modes. It was shown that intermolecular forces split every mode into two components belonging to different symmetry species. The acoustic modes vj and of TSM were also affected by intermolecular forces. For an isolated chain, these correspond to deformation and torsional vibrations respectively, but in crystals, they are mixed. Further, the zero and n phases of the acoustic modes predicted for an isolated chain correspond to zero frequency. In the crystal, non-zero values corresponding to rotary and translational lattice vibrations are obtained. 

The effects of intermolecular forces upon the thermodynamic properties of a gas can be conveniently summarized in terms of the fugacity of the gas. ... 

Our goal in this work is to explore the effects of intermolecular forces on the phase behavior of supercritical systems. For this purpose, it is preferable to use simple intermolecular potential functions, with as few parameters as possible. The Lennard-Jones (6,12) intermolecular potential function,... 

The much greater separation between molecules in the gas phase than in the condensed phases dramatically influences the effect of intermolecular forces. The repulsive force, although very strong, is very short-ranged and becomes... 

Compression factor (Z) -> A measure of the effect of intermolecular forces... 

Eor real gases, the pV product attains the limiting value RT at zero pressure, where intermo-lecular potential energy vanishes. The extrapolation to zero pressure frees the pVproduct from the effect of intermolecular forces. An ideal gas is defined as one that is free of intermolecular potential energy at finite pressures. For the ideal gas, the pv product equals RT at all pressures, and is called the ideal-gas equation. 

Now the ordinary absorption coefficient a, defined for atmospheric pressure with the path expressed in centimeters, is evidently related to jff by the expression given below, provided the integral absorption coefficient would be unchanged in value if the effect of intermolecular forces could be decreased to zero. 

To understand the effect of intermolecular forces on the properties of liquids... 

Since the denominator in the above equation is smaller than the denominator in the ideal gas equation, the size effect by itself increases the pressure above the ideal value. According to this equation it is the empty space between the molecules, the free volume, that follows the ideal gas law. Second, the effect of intermolecular forces, Eq. (3.5),... 

Table 3.3 shows the effects of intermolecular forces (expressed as cohesive energy density) on the value of Tg for three vinyl polymers. 

The theory of Isotope effects In condensed phase systems, especially vapor pressure Isotope effects (VPIE) Is briefly reviewed. It Is pointed out that the VFIE can be enqtloyed as one measure of the effect of Intermolecular forces on the motions of molecules In condensed phases. This Is Illustrated with a number of examples from the recent literature and from our own laboratory. A more detailed description of our recent work on thermodynamic solvent Isotope effects In aqueous systems Is presented. Experiments on vapor pressures, freezing points, and heats of solution and dilution of solutions of electrolytes In HOH and DOD are described. Implications are discussed with respect to the aqueous solvent structure problem. 

Consideration of these characteristics makes it clear that only very special liquid pairs could conceivably form ideal solutions. It would be necessary that the molecules of the constituents be very similar in every respect, for example in structure, size, and chemical nature. Thus, solutions of optical isomers, adjacent members of an homologous series, and similar mixtures would be expected to be nearly ideal, but actually all solutions can at best only approach ideality as a limit. Solutions which form immiscible liquid phases are of necessity extremely nonideal, and extraction operations depend upon this. The extent to which solutions depart from ideality is manifested by deviations of the properties of the solutions from the characteristics listed above, and a study of these deviations will permit us to some extent to predict their behavior in extraction operations. The most useful characteristics of the ideal solution for these purposes is that of vapor pressure, since considerable information has now been accumulated for many mixtures on this and related properties such as boiling points of solutions, azeotropism, and vapor-liquid equilibria. Classifications of compounds according to the effect of intermolecular forces on properties of mixtures also provide much useful material, but the second and third characteristics in the list above are of limited value owing to lack of experimental data to which we can refer. 

The effect of intermolecular forces is seen in the values of the vapor pressure, the short, horizontal dashed lines intersecting the vertical (pressure) axis at a given tanperature (vertical dashed line at 20°C) the intermolecular forces in diethyl ether (highest vapor pressure) are weaker than those in ethanol, which are weaker than those in water (lowest vapor pressure). 

Nathan Hammer, a chemistry professor at the University of Mississippi, uses vibrational Raman spectroscopy to help us understand the effects of intermolecular forces on molecular structure and behavior. The vibrational spectrum provides a valuable probe of the electron distribution in the molecules as well. In the spectra shown, for example, a vibrational transition in normal pyrimidine shifts from roughly 1570 cm (where it overlaps with another transition in the spectrum at left) to a clearly distinct peak at over 1580 cm (the spectrum at right) when a water molecule attaches to one of the nitrogen atoms. This upward shift occurs because some electron density transfers from a... 

The major practical use of Equation 11.1 lies in testing whether a film of uniform initial thickness ho remains stable or eventually become unstable with time. The solution of linearized equations of motion incorporating the effect of intermolecular forces, can be simplified by the lubrication approximation (non-inertial laminar flow in thin films) and admit space periodic solutions for the film thickness h(x, t) = ho + E sin(kx) exp(cof) with ... 

This first attempt at a prediction of D was followed by a series of more elaborate theories, which took account of the finite size of the gas molecules, as well as the effect of intermolecular forces. Probably the most popular among current prediction methods is that due to Fuller, Schettler, and Gid-dings, who proposed the following expression for the calculation of gas diffusivities ... 

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