The van der Waals Equation

The van der Waals Equation.—Though rarely used nowadays for fitting to experimental p, V, T data, the van der Waals equation (2) is frequently cited in current discussions of gas + liquid phenomena. Indeed, allusion has been made to a renaissance in the use of the equation  

A fundamental requirement of any equation of state for use around the critical point is that it should be capable of showing a point of inflexion in the / , V isotherm for the critical temperature, Ti, since 

The van der Waals equation can show such an inflexion, and by application of (3) and (4) to (2) a set of three equations containing a and b is obtained. Alternative solutions for a and b are obtainable as follows  

By substitution of these pairs of values of a and b into equation (2), three alternative formulations of the van der Waals equation in terms of the critical parameters are obtained  

Yet further formulations of the van der Waals equation are possible by introduction of reduced pressures p, reduced molar volumes K, and reduced temperatures T, defined by p, = pjp = yjy and Ti = Tin. Thus from equation (5) 

In 1873, J. D. van der Waals recognized deficiencies in the ideal gas equation and developed an equation to eliminate two problems. First, the volume of the container is not the actual volume available to the molecules of the gas because the molecules themselves occupy some volume. The first correction to the ideal gas equation was to subtract the volume of the molecules from V, the volume of the container, to give the net volume accessible to the molecules. When modified to include the number of moles, n, the corrected volume is (V - nb) where b is a constant that depends on the type of molecule. 

From the ideal gas equation, it is found that for 1 mole of gas, PV/KT = 1, which is known as the compressibility factor. For most real gases, there is a large deviation from the ideal value, especially at high pressure where the gas molecules are forced closer together. From the discussions in previous sections, it is apparent that the molecules of the gas do not exist independently from each other because of forces of attraction even between nonpolar molecules. Dipole-dipole, dipole-induced dipole, and London forces are sometimes collectively known as van der Waals forces because all of these types of forces result in deviations from ideal gas behavior. Because forces of attraction between molecules reduce the pressure that the gas exerts on the walls of the container, van der Waals included a correction to the pressure to compensate for the lost pressure. That term is written as w2a/V2, where n is the number of moles, a is a constant that depends on the nature of the gas, and V is the volume of the container. The resulting equation of state for a real gas, known as van der Waals equation, is written as 

In van der Waals equation, it is the term n2a/V2 that is of interest in this discussion, because that term gives information about intermolecular forces. Specifically, it is the parameter a that is related to inter-molecular forces rather than the number of moles, n, or the volume, V. It should be expected that the 

Variation in boiling points of nonpolar molecules with van der Waals a parameter. 

It is apparent that for these nonpolar molecules the correlation is satisfactory. In this case, a characteristic of the liquid state (the boiling point) is correlated with a parameter from an equation that was developed to explain the behavior of gases. The liquid and gaseous states are referred to as fluids, and van der Waals equation can be considered as an equation that applies to fluids as well as to gases through the use of the reduced variables (see references at the end of this chapter). Table 6.4 gives values for the van der Waals a parameter for molecules most of which are nonpolar. 

The general gas law is an approximation that becomes more exact the more diluted a gas is. If a gas becomes denser, deviations from the general gas law become increasingly noticeable. We will use an example to illustrate this Oxygen can be obtained in steel cylinders at pressures up to 20 MPa (200 bar). Under these circumstances, the particles have almost no space to move in, their packing density is similar to a liquid. The characteristics of such a compressed gas are naturally different from those of a dilute gas. 

The Dutchman Johannes Diderik van der Waals came up with an enlightening idea for understanding the behavior of gases at higher densities. He based it upon two very simple assumptions  

Every particle possesses a certain spatial extension and therefore occupies a 

The particles attract each other. The forces of attraction are weak but increase 

The short-range repulsive interaction effect was taken into account by van der Waals by the first of the assumptions above. This assumption implies that the gas particles do not have the entire volume of a container available for motion. The volume needed to be reduced by a contribution determined by the volume from which the particles exclude each other. This volume unavailable for molecular motion is called the co-volume (van der Waals volume) of a gas. The assumed attraction in point 2 leads to the gas particles moving more closely together, just as if there was additional pressure upon them. This pressure or pull (or tensile stress ) caused by the forces of attraction is called the internal pressure or cohesion pressure of a gas. Van der Waals assumed that the general gas law should continue to be valid, except for the following two changes, a lessening of volume by the 

The ideal gas law does not work well for real gases at even moderate pressures. Two of its main problems were recognized by van der Waals in the last century, and appropriate corrections were incorporated into his famous equation of state of 1873  

There have been hundreds of modifications of the van der Waals equation over the intervening century. Most of these follow the same approach, correcting for repulsive and attractive forces. The Redlich-Kwong equation considered below is one of the more successful modifications, and now it too has been modified a great many times. Historically, this is one of the two main directions taken in the search for increasingly better gas equations of state. 

The second direction has been statistical-mechanical. This usually follows the kind of reasoning described above for virial equations of state (equation 15.34). The interactions of all particles are calculated by summing interactions of pairs, triples, quadruples, and so on. In fact, what results is another regression equation with an appropriate shape and adjustable parameters with possible physical significance. Such equations are fit to real data at the present time it is not feasible to calculate the parameters theoretically and then predict the behavior of a specific gas (except for highly idealized systems). With the advent of very fast computers it has been possible 

Because there are conditions under which use of the ideal gas equation would result in large errors (i.e., high pressure and/or low temperature), we must use a slightly different approach when gases do not behave ideally. Analyses of real gases that took into account nonzero molecular volumes and intermolecular forces were first carried out by J. D. van der Waals in 1873. Van der Waals s treatment provides us with an interpretation of the behavior of real gases at the molecular level. 

Johannes Diderik van der Waals (1837-1923). Dutch physicist. Van der Waals received the Nobel Prize in Physics in 1910 for hi.s work on the properties of gases and liquids. 

The other correction concerns the volume occupied by the gas molecules. In the ideal gas equation, V represents the volume of the container. However, each molecule actually occupies a very small but nonzero volume. We can correct for the volume occupied by the gas molecules by subtracting a term, nb, from the volume of the container  

Incorporating both corrections into the ideal gas equation gives us the van der Waak equation, with which we can analyze gases under conditions where ideal behavior is not expected. 

The van der Waals constants a and b for a number of gases are listed in Table 11.6. The magnitude of a indicates how strongly molecules of a particular type of gas attract one another. The magnitude of b is related to molecular (or atomic) size, although the relationship is not a simple one. 

TABLE 11.6 I Van der Waals Constants of Some Common Gases 

Engineers and scientists who work with gases at high pressures often cannot use the ideal-gas equation because departures from ideal behavior are too large. One useful equation developed to predict the behavior of real gases was proposed by the Dutch scientist Johannes van der Waals (1837—1923). 

Van der Waals recognized that the ideal-gas equation could be corrected to account for the effects of intermolecular attractive forces and for molecular volumes. He introduced two constants for these corrections a, a measure of how strongly the gas molecules attract one another, and b, a measure of the finite volume occupied by the molecules. His description of gas behavior is known as the van der Waals equation  

000 atm. Use the van der Waals equation and Table 10.3 to estimate the pressure exerted by 

Analyze We need to determine a pressure. Because we will use the van der Waals equation, we must identify the appropriate values for the constants that appear there. 

Check We expect a pressure not far from 1.000 atm, which would be the value for an ideal gas, so our answer seems very reasonable. 

The term accounts for the attractive forces. The equation adjusts the pressure 

Comment Notice that the term 10.26 atm is the pressure corrected for molecular volume. This value is higher than the ideal value, 10.00 atm, because the volmne in which the molecules are free to move is smaller than the container volume, 22.41 L. Thus, the molecules collide more frequently with the container walls and the pressure is higher than that of a real gas. The term 1.29 atm makes a correction in the opposite direction for intermolecular forces. The correction for intermolecular forces is the larger of the two and thus the pressure 8.97 atm is smaller than would be observed for an ideal gas. 

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Van der Waals Equations of State. A logical step to take next is to consider equations of state that contain both a covolume term and an attractive force term, such as the van der Waals equation. De Boer [4] and Ross and Olivier [55] have given this type of equation much emphasis. 

It must be remembered that, in general, the constants a and b of the van der Waals equation depend on volume and on temperature. Thus a number of variants are possible, and some of these and the corresponding adsorption isotherms are given in Table XVII-2. All of them lead to rather complex adsorption equations, but the general appearance of the family of isotherms from any one of them is as illustrated in Fig. XVII-11. The dotted line in the figure represents the presumed actual course of that particular isotherm and corresponds to a two-dimensional condensation from gas to liquid. Notice the general similarity to the plots of the Langmuir plus the lateral interaction equation shown in Fig. XVII-4. 

Ross and Olivier [55], in their extensive development of the van der Waals equation of state model have, however, provided a needed balance to the Langmuir picture. 

In 1873, van der Waals [2] first used these ideas to account for the deviation of real gases from the ideal gas law P V= RT in which P, Tand T are the pressure, molar volume and temperature of the gas and R is the gas constant. Fie argried that the incompressible molecules occupied a volume b leaving only the volume V- b free for the molecules to move in. Fie further argried that the attractive forces between the molecules reduced the pressure they exerted on the container by a/V thus the pressure appropriate for the gas law isP + a/V rather than P. These ideas led him to the van der Waals equation of state ... 

The importance of the van der Waals equation is that, unlike the ideal gas equation, it predicts a gas-liquid transition and a critical point for a pure substance. Even though this simple equation has been superseded, its... 

It is interesting to note that, when the van der Waals equation for a fluid. 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

Flere b corresponds to the repulsive part of the potential, which is equivalent to the excluded volume due to the finite atomic size, and a/v corresponds to the attractive part of the potential. The van der Waals equation... 

A2.5.3 ANALYTIC TREATMENT OF CRITICAL PHENOMENA IN FLUID SYSTEMS. THE VAN DER WAALS EQUATION... 

Although the previous paragraphs hint at the serious failure of the van der Waals equation to fit the shape of the coexistence curve or the heat capacity, failures to be discussed explicitly in later sections, it is important to recognize that many of tlie other predictions of analytic theories are reasonably accurate. For example, analytic equations of state, even ones as approximate as that of van der Waals, yield reasonable values (or at least ball park estmiates ) of the critical constants p, T, and V. Moreover, in two-component systems... 

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

Flalf a century later Van Konynenburg and Scott (1970, 1980) [3] used the van der Waals equation to derive detailed phase diagrams for two-component systems with various parameters. Unlike van Laar they did not restrict their treatment to the geometric mean for a g, and for the special case of b = hgg = h g (equalsized molecules), they defined two reduced variables. 

Figure A2.5.11. Typical pressure-temperature phase diagrams for a two-component fluid system. The fiill curves are vapour pressure lines for the pure fluids, ending at critical points. The dotted curves are critical lines, while the dashed curves are tliree-phase lines. The dashed horizontal lines are not part of the phase diagram, but indicate constant-pressure paths for the T, x) diagrams in figure A2.5.12. All but the type VI diagrams are predicted by the van der Waals equation for binary mixtures. Adapted from figures in [3].

The previous seetion showed how the van der Waals equation was extended to binary mixtures. However, imieh of the early theoretieal treatment of binary mixtures ignored equation-of-state eflfeets (i.e. the eontributions of the expansion beyond the volume of a elose-paeked liquid) and implieitly avoided the distinetion between eonstant pressure and eonstant volume by putting the moleeules, assumed to be equal in size, into a kind of pseudo-lattiee. Figure A2.5.14 shows sohematieally an equimolar mixture of A and B, at a high temperature where the distribution is essentially random, and at a low temperature where the mixture has separated mto two virtually one-eomponent phases. 

The leading tenn in equation (A2.5.17) is the same kind of parabolic coexistence curve found in section A2.5.3.1 from the van der Waals equation. The similarity between equation (A2.5,5t and equation (A2.5.17) should be obvious the fomi is the same even though the coefficients are different. 

For T shaped curves, reminiscent of the p, isothemis that the van der Waals equation yields at temperatures below the critical (figure A2.5.6). As in the van der Waals case, the dashed and dotted portions represent metastable and unstable regions. For zero external field, there are two solutions, corresponding to two spontaneous magnetizations. In effect, these represent two phases and the horizontal line is a tie-line . Note, however, that unlike the fluid case, even as shown in q., form (figure A2.5.8). the symmetry causes all the tie-lines to lie on top of one another at 6 = 0 B = 0). 

For simple fluids Nq is estimated to be about 0.01, and Kostrowicka Wyczalkowska et aJ [29] have vised this to apply crossover theory to the van der Waals equation with interesting resnlts. The critical temperature is reduced by 11% and the coexistence curve is of course flattened to a cvibic. The critical density is almost unchanged (by 2%), bnt the critical pressure p is reduced greatly by 38%. These changes redvice the critical... 

While the phase rule requires tliree components for an unsymmetrical tricritical point, theory can reduce this requirement to two components with a continuous variation of the interaction parameters. Lindli et al (1984) calculated a phase diagram from the van der Waals equation for binary mixtures and found (in accord with figure A2.5.13 that a tricritical point occurred at sufficiently large values of the parameter (a measure of the difference between the two components). 

Pegg I L, Knobler C M and Scott R L 1990 Tricritical phenomena in quasibinary mixtures. VIII. Calculations from the van der Waals equation for binary mixtures J. Chem. Phys. 92 5442-53... 

TABLE 5.29 Van der Waals Constants for Gases The van der Waals equation of state for a real gas is ... 

Fig. 3.24 Test of the tensile strength hysteresis of hysteresis (Everett and Burgess ). TjT, is plotted against — Tq/Po where is the critical temperature and p.. the critical pressure, of the bulk adsorptive Tq is the tensile strength calculated from the lower closure point of the hysteresis loop. C), benzene O. xenon , 2-2 dimethyl benzene . nitrogen , 2,2,4-trimethylpentane , carbon dioxide 4 n-hexane. The lowest line was calculated from the van der Waals equation, the middle line from the van der Waals equation as modified by Guggenheim, and the upper line from the Berthelot equation. (Courtesy Everett.)...

The procedure outlined in this example needs only one modification to be applicable to the critical point for solution miscibility. In Fig. 8.2b we observe that there are two inflection points in the two-phase region between P and Q. There is only one such inflection point in the two-phase region of the van der Waals equation. The presence of the extra inflection point means that still another criterion must be added to describe the critical point The two inflection points must also merge with each other as well as with the maximum and the minima. 

The virial equations are unsuitable forhquids and dense gases. The simplest expressions appropriate (in principle) for such fluids are equations cubic in molar volume. These equations, inspired by the van der Waals equation of state, may be represented by the following general formula, where parameters b, 9 5, S, and Tj each can depend on temperature and composition ... 

Reduced Properties. One of the first attempts at achieving an accurate analytical model to describe fluid behavior was the van der Waals equation, in which corrections to the ideal gas law take the form of constants a and b to account for molecular interactions and the finite volume of gas molecules, respectively. 

Whereas this two-parameter equation states the same conclusion as the van der Waals equation, this derivation extends the theory beyond just PVT behavior. Because the partition function, can also be used to derive aH the thermodynamic functions, the functional form, E, can be changed to describe this data as weH. Corresponding states equations are typicaHy written with respect to temperature and pressure because of the ambiguities of measuring volume at the critical point. 

Some formulas, such as equation 98 or the van der Waals equation, are not readily linearized. In these cases a nonlinear regression technique, usually computational in nature, must be appHed. For such nonlinear equations it is necessary to use an iterative or trial-and-error computational procedure to obtain roots to the set of resultant equations (96). Most of these techniques are well developed and include methods such as successive substitution (97,98), variations of Newton s rule (99—101), and continuation methods (96,102). 

The effects of the constants in the van der Waals equation become more marked as the pressure is increased above atmospheric. Early measurements by Regnault showed tlrat the PV product for CO2, for example, is considerably less tlran that predicted by Boyle s law... 

Opschoor (1974) applied the Van der Waals equation of state to estimate the maximum superheat temperature for atmospheric pressure from the critical temperature (TJ (i.e., that temperature above which a gas cannot be liquefied by pressure alone) as follows ... 

Reid (1976) and many other authors give pure propane a superheat temperature limit of 53 C at atmospheric pressure. The superheat temperature limit calculated from the Van der Waals equation is 38°C, whereas the value calculated from the Redlich-Kwong equation is S8°C. These values indicate that, though an exact equation among P, V, and 7 in the superheat liquid region is not known, the Redlich-Kwong equation of state is a reasonable alternative. 

It is difficult to combine these two equations. The van der Waals equation is cubic in volume and equation (3.87) requires explicit values for the volume. Approximations must be made to the van der Waals equation before substitutions for Vm and (0Vm/dT)p are made in equation (3.87). 

Figure 3.7(a) compares the experimental inversion curve for nitrogen gas with the van der Waals prediction. Considering the approximations involved, it is not surprising that the quantitative prediction of the van der Waals equation is not very good. Equation (3.91) is quadratic in T and hence, predicts two values for the inversion temperature, which is in qualitative agreement with the experimental observation.1... 

The fugacity coefficient can be calculated from other equations of state such as the van der Waals, Redlick-Kwong, Peng-Robinson, and Soave,d but the calculation is complicated, since these equations are cubic in volume, and therefore they cannot be solved explicitly for Vm, as is needed to apply equation (6.12). Klotz and Rosenburg4 have shown a way to get around this problem by eliminating p from equation (6.12) and integrating over volume, but the process is not easy. For the van der Waals equation, they end up with the relationship... 

In both equations, k and k are proportionality constants and 0 is a constant known as the critical exponent. Experimental measurements have shown that 0 has the same value for both equations and for all gases. Analytic8 equations of state, such as the Van der Waals equation, predict that 0 should have a value of i. Careful experimental measurement, however, gives a value of 0 = 0.32 0.01.h Thus, near the critical point, p or Vm varies more nearly as the cube root of temperature than as the square root predicted from classical equations of state. 

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