Waals Equation of State

As compared with, the equation of state for perfect gases, van der Waals equation of state for actual gases, given in the text (p. 20), contains two correction terms, a volume correction and a pressure correction. Here we shall seek to show, at least qualitatively, how these terms arise. 

The probability that a molecule will be found in a definite portion of volume v is of course proportional to v, as above. If we introduce a second molecule into this region, the space available for it is only V — the space left for a third molecule is — 2 X 8 , and so on. The probability of finding n molecules in v, then, is proportional, not to but to the product 

V is accordingly replaced by tbe root of this product. We can easily calculate this as is very small and 8v n still small in comparison with V, we can replace the product approximately by 

Taking the -th root of this, we see that if the finite volume of the molecules is to be taken into account v must be replaced by 

All phenomena involving deviation from a mean value depend on the formula 

Let us consider a non-ideal gas in the AF T ensemble. Assume that the gas particles interact with the Sutherland potential (Fig. 8.3), expressed 

A reasonable assumption for high-temperature systems is that z/ksT 1. Then, 

Assuming that A4ttct /3 C T, a reasonable assumption for low-density systems, yields 

Defining a molecular volume u = l/p = V/Awe arrive at the celebrated van der Waals equation of state (Fig. 8.4) 

Therefore, below the critical temperature, the van der Waals equation of state predicts three distinct systems with different molar volumes at the same pressure and temperature. It turns out that only the two molar volumes at the ends are thermodynamically stable (an elegant proof of the instability of the middle molar volume is given in Sandler s handbook, see Further reading). The lower molar volume represents a hquid state, whereas the higher molar volume represents a vapor gas state. These two states are in equihbrium. 

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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. 

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 ... 

Figure A2.3.3 P-Visothemis for van der Waals equation of state. Maxwell s equal areas mle (area ABE = area ECD) detemiines the volumes of the coexisting phases at subcritical temperatures.

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

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 ... 

In 1893, it was shown that corresponding states are not unique to van der Waals equation of state (73). Rather, for any equation of state having not more than three constants, corresponding states are only a mathematical consequence. 

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 ... 

Since most equations of state are pressure-explicit, Eqs. (6) and (99) are often more convenient than Eqs. (5) and (98). With these equations, basing his calculations on van der Waals equation of state, Temkin (Tl) showed that gas-gas immiscibility may occur if the van der Waals constants a and b... 

The Van der Waals equation of state is perhaps the best-known example of a mean-field theory. It was first proposed in the form... 

Figure 5 P — V diagram of the Van der Waals equation of state. The solutions to these simultaneous equations are...

Even though the van der Waals equation is not as accurate for describing the properties of real gases as empirical models such as the virial equation, it has been and still is a fundamental and important model in statistical mechanics and chemical thermodynamics. In this book, the van der Waals equation of state will be used further to discuss the stability of fluid phases in Chapter 5. 

In Section 2.2 we introduced the van der Waals equation of state for a gas. This model, which provides one of the earliest explanations of critical phenomena, is also very suited for a qualitative explanation of the limits of mechanical stability of a homogeneous liquid. Following Stanley [17], we will apply the van der Waals equation of state to illustrate the limits of the stability of a liquid and a gas below the critical point. 

For sub-critical isotherms (T < Tc), the parts of the isotherm where (dp/dV)T < 0 become unphysical, since this implies that the thermodynamic system has negative compressibility. At the particular reduced volumes where (dp/dV)T =0, (spinodal points that correspond to those discussed for solutions in the previous section. This breakdown of the van der Waals equation of state can be bypassed by allowing the system to become heterogeneous at equilibrium. The two phases formed at T[Pg.141]

Figure 5.11 The p-T(a) and the T-p (b) phase diagrams of H2O calculated using the van der Waals equation of state.

The Dutch scientist van der Waals was well aware that the ideal-gas equation was simplistic, and suggested an adaptation, which we now call the van der Waals equation of state ... 

Take, for example (12), the problem of solving for the P-V-T properties of a real gas obeying the van der Waals equation of state. 

Figure 1 shows the Rule Sheet for a TKISolver model REALGAS.TK (12. The first rule is the van der Waals equation of state. The second defines the gas constant, and the third rule defines Ae number density. The fourth defines the compressibility factor z, a dimensionless variable which measures the amount of... 

According to the van der Waals equation of state, the value of compressibility at the critical point should be 3/8 = 0.375. When does a real gas depart significantly from an ideal gas We can write equation (4.9) as the reduced equation of state, with the reduced temperatures, pressures, and volumes = TITc, Pi = P/Pc, Vr = V/Vc- Then, all gases would have the same equation of state in the form of reduced parameters ... 

When the functional form of the correlation is suggested by theory, there is a great deal more confidence that the correlation can be extrapolated into regions of P that have no experimental data, and can be used for other families of compounds other than the training set S. Examples of theory-suggested functional forms include the van der Waals equation of state for gases, the Langmuir isotherm for adsorption and catalysis, and the Clausius-Clapeyron equation for the vapor pressure of liquids. 

For a gas that follows the van der Waals equation of state, the extraction of the values of the parameters a and b from a set of experimental data is facilitated by reformulating the equation to obtain... 

One has to design the experiment to take a set of data designed to facilitate the task of parameter extraction. If a set of data is taken under constant volume conditions, and the pressure is plotted against the temperature, then there will be an intercept of —alV and a slope of R/ V — b). The van der Waals equation of state is the simplest of the equations of state beyond the perfect gas law, and the task of extracting parameter values from experimental data for the more complicated equations of state would require more ingenuity. The Redlich-Kwong equation has two parameters, A and B ... 

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