Waals Isotherm

Example 1.8 van der Waals isotherms Use the critical parameters for ammonia and estimate the van der Waals constants a and b. Plot the van der Waals isotherms for ammonia at Tl = 200K, T2 = 406K, and 73 = 550K, when the volume changes from 0.04 to 0.25 L. 

Plot[ p[Y200],p[Y406], p[Y550], Y0.04,0.2, Frame - True, 

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Phase equilibrium requires that A2 = Al and hence that the integral vanish. All conditions are satisfied if the points 1 and 2 are located such that the areas A = B. This geometry defines the Maxwell construction. It shows that stable liquid and vapour states correspond to minima in free energy and that AL = Ay when the external pressure line cuts off equal areas in the loops of the Van der Waals isotherm. At this pressure that corresponds to the saturated vapour pressure, a first-order phase transition occurs. 

Fig 1. Van der Waals Isothermals Solid Curve-Derived from Eq 14 Broken Curve-Derived from Eq 11... 

Figure 2.8 Representative supercritical (T = 31 OK) and subcritical (T = 280K) Van der Waals isotherms for C02, showing the liquid-gas (L + G) condensation plateau (P = 52 atm) for T = 280K, and outlining the 2-phase liquid-gas coexistence dome (dotted line) topped by the critical point (x) at Tc = 304K, Pc = 73 atm.

It was shown by J. C. Maxwell that a horizontal line can be drawn through the Van der Waals loop region in such a way that the area enclosed above the line in the upward loop exactly matches that enclosed below the line in the downward loop ( Maxwell s equal-area construction ). As shown in Fig. 2.10b, this horizontal line (say, at pressure P0) can be taken as the Van der Waals approximation to the actual condensation plateau, bounded on the left by the steeply sloping liquid branch, and on the right by the more gently sloping gaseous branch of the isotherm. The three points where this horizontal line P = P0 crosses the Van der Waals isotherm may be obtained as the roots of the cubic polynomial P = P(V) for P = P0, i.e., as solutions of the equation... 

If we were to utilize Maxwell s equal-area construction, since the H-h plot in Figure 1.15(d) resembles a van der Waals isotherm in liquid-vapor equilibria [93], then Figures 1.15(c) and 1.15(d) would be qualitatively similar. Related issues can be further investigated via a stability analysis as described in Section II.D. 

Figure 90. Typical van der Waals Isotherms for Finely Ground Activated Carbons.

FIGURE 2.3.5 Detail of van der Waals isotherm taken from Fig. 2.3.4. 

Figure 1.20a. Adsorption isotherms according to Hill and De Boer (two-dimensional Van der Waals Isotherms).

Verschaffelt" showed that if the surface tension is proportional to J e—T) , the value of n on the assumption that the van der Waals isotherm is continuous at the critical point is 1-5, but if it is not continuous the value is found to be 1-17, in agreement with the observed value 1-2. 

The parameter 5i characterizes the thickness of the adsorption layer 8, can be set (approximately) equal to the length of the amphiphilic molecule. represents the maximum possible value of the adsorption. In the case of localized adsorption (Langmuir and Frumkin isotherms) 1/F.. is the area per adsorption site. In the case of nonlocalized adsorption (Vohner and van der Waals isotherms) 1/F is the excluded area per molecule. 

Now, let us apply the same general scheme, bnt this time to the derivation of the van der Waals isotherm, which corresponds to nonlocalized adsorption of interacting molecules. (Expressions corresponding to the Volmer isotherm can be obtained by setting p = 0 in the respective expressions for the van der Waals isotherm.) Now the adsorbed molecules are considered a two-dimensional gas. The corresponding expression for the canonical ensemble partition function is... 

The theoretical model contains four parameters, (3, F K, and K2, whose values are to be obtained from the best fit of the experimental data. Note that all 11 curves in Figure 5.2 are fitted simultaneously." In other words, the parameters (3, F K, and K2 are the same for all curves. The value of F, obtained from the best fit of the data in Figure 5.2, corresponds to 1/ F = 31 A. The respective value of is 82.2 mVmol, which in view of Equation 5.49 gives a standard free energy of surfactant adsorption = 12.3 kT per DS- ion, that is, 30.0 kJ/mol. The determined value of K2 is 8.8 X 10 mVmol, which after substitution in Equation 5.49 yields a standard free energy of counterion binding = 1.9 kT per Na" " ion, that is, 4.7 kJ/mol. The value of the parameter P is positive, 2p TJkT = h-2.89, which indicates attraction between the hydrocarbon tails of the adsorbed surfactant molecules. However, this attraction is too weak to cause two-dimensional phase transition. The van der Waals isotherm predicts such transition for l TJkT> 6.75. 

This inequality, called the condition of mechanical stability, implies that if at a constant temperature the pressure in a system increases, then its volume decreases. It follows from Fig. 37 that along the section BC the van der Waals isotherm does not satisfy the condition (3.15a). This signifies that the isotherms obtaind from equation (3.14) are, at least partially, non-physical, and the corresponding states are physically unattainable. Thus, equation (3.14) is only an approximate state equation. 

The non-physical van der Waals isotherm may be improved using the so-called Maxwell construction. It involves drawing the horizontal section AD, for which (8p/8V)T = 0, joining the two branches of the isotherm, EA and DF, corresponding to the liquid and gaseous phase of a system, respectively. It follows from the condition of equality of chemical potentials at a critical point that the section AD should be thus selected that the areas and S2 be equal. Between the points A and D the system is nonhomo-geneous, i.e. separated into two phases coexisting in equilibrium. The... 

Comparison of the van der Waals isotherms with those of a real gas shows similarity in certain respects. The curve at 7 in Fig. 3.7 resembles the curve at the critical temperature in Fig. 3.5. The curve at T2 in Fig. 3.7 predicts three values of the volume, V, V", and F ", at the pressure p. The corresponding plateau in Fig. 3.5 predicts infinitely many volumes of the system at the pressure p. It is worthwhile to realize that even if a very complicated function had been written down, it would not exhibit a plateau such as that in Fig. 3.5. The oscillation of the van der Waals equation in this region is as much as can be expected of a simple continuous function. 

The section BC of the van der Waals isotherm cannot be realized experimentally. In this region the slope of the p-V curve is positive increasing the volume of such a system would increase the pressure, and decreasing the volume would decrease the pressure States in the region BC are unstable slight disturbances of a system in such states as B to C would produce either explosion or collapse of the system. 

We return once more to the van der Waals isotherms (Fig. 11.3) The van der Waals curves describe fairly exactly compression of gases up to the dew point. If the volume is decreased beyond this point, pressure does not increase, but condensation sets in. This occurs at constant pressure, meaning that the corresponding piece of the curve must be horizontal until the gas phase completely disappears at the boiling point. These lines have been constructed so that the areas enclosed by the van der Waals curves above and below the straight line are equal (Maxwell construction or equal area rule) (compare Fig. 11.3 where the two areas in the case of the T2 isotherm are dark gray). Both gas and liquid exist simultaneously along these lines. The subsequent steep rise in pressure with further decrease of volume is characteristic of the low compressibility of liquids. 

As to evaporization waves an analogous discussion can be given except there is no possibility of shock splitting. This is due to the asymmetry of the van der Waals isotherms and hence the lack of existence of a state playing the role of V4 above. 

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