Real Gases Intermolecular Forces

The ideal gas law, PV = nRT, is a particularly simple example of an equation of state—an equation relating the pressure, temperature, number of moles, and volume to one another. Equations of state can be obtained from either theory or experiment. They are useful not only for ideal gases but also for real gases, liquids, and solids. 

Real gases follow the ideal gas equation of state only at sufficiently low densities. Deviations appear in a variety of forms. Boyle s law, PV = C, is no longer satisfied at high pressures, and Charles s law, V T, begins to break down at low temperatures. Deviations from the predictions of Avogadro s hypothesis appear for 

FIGURE 9.17 A plot of z = PV/nRT against pressure shows deviations from the ideal gas law quite clearly, for an ideal gas, z is represented by the straight horizontal line, (a) Deviation of several real gases at 25°C. (b) Deviation of nitrogen at several temperatures. 

When 2 differs from 1 (Fig. 9.17), the ideal gas law is inadequate, and a more accurate equation of state is necessary. 

One of the earliest and most important improvements on the ideal gas equation of state was proposed in 1873 by the Dutch physicist Johannes van der Waals. The van der Waals equation of state is  

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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 ... 

The presence of intermolecular forces also accounts for the variation in the compression factor. Thus, for gases under conditions of pressure and temperature such that Z > 1, the repulsions are more important than the attractions. Their molar volumes are greater than expected for an ideal gas because repulsions tend to drive the molecules apart. For example, a hydrogen molecule has so few electrons that the its molecules are only very weakly attracted to one another. For gases under conditions of pressure and temperature such that Z < 1, the attractions are more important than the repulsions, and the molar volume is smaller than for an ideal gas because attractions tend to draw molecules together. To improve our model of a gas, we need to add to it that the molecules of a real gas exert attractive and repulsive forces on one another. 

The attraction of the gas particles for each other tends to lessen the pressure of the gas since the attraction slightly reduces the force of the collisions of the gas particles with the container walls. The amount of attraction depends on the concentration of gas particles and the magnitude of the intermolecular force of the particles. The greater the intermolecular forces of the gas, the higher the attraction is, and the less the real pressure. Van der Waals compensated for the attractive force by the term P + an2/V2, where a is a constant for individual gases. The greater the attractive force between the molecules, the larger the value of a. 

The macroscopic properties of the three states of matter can be modeled as ensembles of molecules, and their interactions are described by intermolecular potentials or force fields. These theories lead to the understanding of properties such as the thermodynamic and transport properties, vapor pressure, and critical constants. The ideal gas is characterized by a group of molecules that are hard spheres far apart, and they exert forces on each other only during brief periods of collisions. The real gases experience intermolecular forces, such as the van der Waals forces, so that molecules exert forces on each other even when they are not in collision. The liquids and solids are characterized by molecules that are constantly in contact and exerting forces on each other. 

Equations of state (EOS) offer many rich enhancements to the simple pV = nRT ideal gas law. Obviously, EOS were developed to better calculate p, V, and T, values for real gases. The point here is such equations are excellent vehicles with which to introduce the fact that gases cannot be really treated as point spheres without mutual interactions. Perhaps the best demonstration of the existence of intermolecular forces that can also be quantified is the Joule-Thomson experiment. Too often this experiment is not discussed in the physical chemistry course. It should be. The effect could not exist if intermolecular forces were not real. The practical realization of the effect is the liquefaction of gases, nitrogen and oxygen, especially. 

Let us now look at the situation in which we deal with real gases, that is, with a situation in which intermolecular forces between the molecules cannot be neglected (as will be even more the case for liquids and solids, see below). These forces influence the (partial) pressure of the gas molecules, but not the amount of the gaseous compound(s). This real pressure is called fugacity. 

Just as the ideal gas law discussed in Section 9.3 applies only to "ideal" gases, Raoult s law applies only to ideal solutions. Raoult s law approximates the behavior of most real solutions, but significant deviations from ideality occur as the solute concentration increases. The law works best when solute concentrations are low and when solute and solvent particles have similar intermolecular forces. 

Because deviations from ideal gas behavior result from intermolecular forces, which go to zero as the average distance between molecules gets very large, we expect Z to approach unity for real gases, as their molar density, n/ V = 1 / Vm, approaches zero. This suggests that it might be useful to expand Z in a series of powers of the molar density ... 

So, it is necessary to apply suitable corrections to the ideal gas equation to make it applicable to real gases, vander Waals introduced two correction terms in the ideal gas equation to account for the errors introduced as a result of neglecting the volume of the molecules and intermolecular forces of attraction. 

Intramolecular interactions were introduced for the first time by van der Waals in 1873 he thus attempted to explain the deviation of the real gas from the ideal gas. In order to apply the ideal gas law equation to the behavior of real gases, allowance should be made for the attractive and repulsive forces occurring between molecules. From that time on, the dipole moment theory of Debye (1912) and the dispersion energy or induced dipole theory by London (1930) were the main driving forces of the research about intermolecular interactions. 

In 1873, J.D. van der Waals proposed his famous equation of state for a non-ideal, i.e., real gas. He modified the ideal gas equation by suggesting that the gas molecules were not mass points but behave like rigid spheres having a certain diameter and that there exist intermolecular forces of attraction between them. The two correction terms introduced by van der Waals are described below. 

One of his corrections assumed that the molecules in a real gas attracted one another. In his honor, the attractive forces between molecules are now called van der Waals forces. And these are the intermolecular forces that hold liquids together. Keep in mind that van der Waals forces exist in gases and solids, too. But their effect is most obvious in liquids, and that is where our discussion will be focused. 

For discussion of intramolecular forces it is essential to remove from consideration effects due to intermolecular forces, that is, to have heats of formation referring to the ideal gas state. In general the correction of heats of formation of real gases at 1 atmosphere pressure to the ideal gas state is very small compared with the accuracy to which heats of formation are known for example, approximately 0-002 kcal mole for methane and 0-02 kcal for methyl chloride. This means that for all purposes connected with bond energies, a knowledge of the heat of formation of the real gas is adequate. Thus for substances whose heat of formation is known directly for the liquid or solid, a knowledge of the heat of vaporization at the appropriate temperature is required. Strictly, however, the quantity concerned is the heat of vaporization to the ideal gas state. 

Real gases deviate from behavior predicted for ideal gases because the particles of a real gas occupy volume and are subject to intermolecular forces. 

Second, molecules in a real gas do exhibit forces on each other, and those forces are attractive when the molecules are far apart. In a gas, repulsive forces are only significant during molecular collisions or near collisions. Since the predominant intermolecular forces in a gas are attractive, gas molecules are pulled inward toward the center of the gas, and slow before colliding with container walls. Having been slightly slowed, they strike the container wall with less force than predicted by the kinetic molecular theory. Thus a real gas exerts less pressure than predicted by the ideal gas law. 

D is correct. An ideal gas has a PV/RT equal to one. Real volume is greater than predicted by the ideal gas law, and real pressure is less than predicted by the ideal gas law. Volume deviations are due to the volume of the molecules, and pressure deviations are due to the intermolecular forces. Thus, a negative deviation in this ratio would indicate that the intermolecular forces are having a greater affect on the nonideal behavior than the volume of the molecules, (see the graph on page 27)... 

Just as the ideal gas forms a convenient point of reference in discussing the properties of real gases, so does the hard-sphere fluid in discussing the properties of liquids. This is especially true at low densities, where the role of intermolecular forces in real systems is not so important. In this limit, the hard-sphere model is useful in developing the theory of solutions, as will be seen in chapter 3. 

The classical insight concerns intermolecular forces, as clearly formulated by van der Waals in 1873 in his equation of state for gases, to explain why real gases do not obey the ideal gas law (PV=nR T),... 

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. 

While the intermolecular forces are absent, the internal structure of the molecule and the attendant energies remain unaltered from the real gas. The internal energy, heat capacity, and related functions retain their specific values for each substance in the ideal-gas state. 

Although we can assume that real gases behave like an ideal gas, we cannot expect them to do so under all conditions. For example, without intermolecular forces, gases could not condense to form liquids. The important question is Under what conditions will gases most likely exhibit nonideal behavior ... 

To study real gases accurately, then, we need to modify the ideal gas equation, taking into account intermolecular forces and finite molecular volumes. Such an analy-... 

This idea is a consequent transfer of the three-dimensional van der Waals equation into the interfacial model developed by Cassel and Huckel (cf. Appendix 2B.1). The advantages of Frumkin s position is a more realistic consideration of the real properties of a two-dimensional surface state of the adsorption layer of soluble surfactants. This equation is comparable to a real gas isotherm. This means that the surface molecular area of the adsorbed molecules are taken into consideration. Frumkin (1925) additionally introduced, on the basis of the van der Waals equation, the intermolecular interacting force of adsorbed molecules represented by a . 

Fig. 16.2. Intermolecular forces within a real gas separated from a vacuum by a solid face. Within the center of the gas, average forces on each molecule are uniform. At the face, forces are all inward, reducing effective gas pressure outward.

At the Boyle temperature the Z versus p curve is tangent to the curve for the ideal gas at p = 0 and rises above the ideal gas curve only very slowly. In Eq. (3.8) the second term drops out at 7, and the remaining terms are small until the pressure becomes very high. Thus at the Boyle temperature the real gas behaves ideally over a wide range of pressures, because the effects of size and of intermolecular forces roughly compensate. This is also shown in Fig. 3.4. The Boyle temperatures for several different gases are given in Table 3.2. 

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