The van der Waals equation of state

Comparing this with (11.14) we see that the second virial coefficient has the form 

The free energy F of a van der Waals gas is obtained by substituting (11.33) into (11.7) to give 

The entropy of the ideal gas at the same temperature and volume is the observed entropy less the term nR In (F - nb)jV, It may be noted 

The internal energy may be obtained using the Gibbs-Helmholtz equation 

Thus a van der Waals gas has the same specific heat at constant volume as the perfect gas at the same temperature. This result is common to all equations of state in which the pressure is linearly related to the temperature. This can be shown quite generally since (c/. 4.2) 

Consider a pure fluid of volume V containing N molecules, each with a hard-core diameter a. To develop an expression for the free volume, we subtract from the total volume, the volume not accessible to the molecules, which is referred to as the excluded volume. 

To calculate the latter, consider a molecule moving around in a fluid of low density so that it has only one closest neighbor. Since the closest distance between the centers of these two molecules is r, the space that belongs to their centers is a sphere of radius a with a volume equal to (4ir/3)ff. The free volume is, thus  

To calculate the potential energy ( f /2) associated with each molecule we use Eq. 17.4.4 and replace the lower limit by a, since g(r) is zero for r a. van der Waals, in effect, assumed that g r) is independent of temperature and volume for all values of r as a result, the integral is a constant, which van der Waals expressed as (-2alN ), where a is a positive constant (Vera and Prausnitz). Thus, on a molar basis  

Similarly, the assumption that g(r) is independent of volume and temperature is also restrictive, and Vera and Prausnitz demonstrate how relaxing this assumption can lead, for example, to the Redlich-Kwong EoS. 

We discuss, next, how use of molecular simulation results leads to a new generation of equations of state. 

We will now use our knowledge of intermolecular interactions to modify the ideal gas model for situations when potential interactions between the species are important. In this section, we will use the Sutherland potential function to describe the intermolecular interactions, that is, use the hard sphere model to account for repulsive forces and van der Waals interactions to describe attractive forces. This development leads to the van der Waals equation of state. This equation is particularly well suited for illustrating how the molecular concepts we learned about in Section 4.2 can be related to macroscopic property data. However, it should be emphasized that more accurate equations of state have been developed and will be covered next. 

let s consider the size of the molecules based on the hard sphere model. The entire volume of the system will no longer be available to the molecules. We can account for this effect by replacing the volume term in the ideal gas model with one for available volume. Recall that in the hard sphere model, the molecules have a diameter (J. Thus, the center of one molecule cannot approach another molecule closer than a distance (j. The excluded volume of the two molecules is then (4/3) jrcr. Dividing by 2 and multiplying by Avogadro s number, we get one mole of molecules occupying a volume b = (2/3) 7Tcr lVA- To correct for size, we modify the ideal gas model to include only the unoccupied molar volume, v — h). Hence, we get  

We still need to take into account attractive intermolecular forces. In the absence of net electric charge, the attractive forces in the gas phase can include dispersion, dipole-dipole, and induction, all of which have an dependence. However, we do not have distance as a parameter in our equations, but rather volume, which is proportional to the cube of the distance (o r ). We can say, therefore, that all of these terms are proportional to v.  

But how do we incorporate this into our equation of state As we saw in Section 4.2, the variable most related to potential energy is pressure. So we correct the pressure by including a term that accounts for attractive forces. Attractive forces should decrease the pressure, since the molecules will not bang into the container as readily hence, we subtract a correction term as follows  

This equation was first proposed by the Dutch physicist van der Waals in 1873. Since it assumes a 1/r dependence for all attractive forces, any force with this functionality (be it dispersion, dipole-dipole, or induction) has been termed a van der Waals force. The parameter a in Equation (4.15) can be related to molecular constants by integrating the Sutherland potential function. This calculation gives a = 2ttN Cq)/. In practice, a and b are treated as empirical constants that account for the magnitude of the attractive and repulsive forces. Can you think of how we might find values for the constants a andfc  

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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 ... 

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 ... 

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 ... 

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 ... 

This is certainly true of the simple properties, such as the molecular weight of a mixture. For the van der Waals equation of state, the parameter b stands for the excluded volume due to the molecule, which suggests that the linear additive relation, or the arithmetic mean, may be appropriate for an ideal mixture ... 

In this lab, the students determine the compression factor, (9) Z = PV/nRT, for Argon using the hard sphere model, the soft sphere model, and the Lennard-Jones model and compare those results to the compression factor calculated using the van der Waals equation of state and experimental data obtained from the NIST (70) web site. Figure 3 shows representative results from these experiments. The numerical accuracy of the Virtual Substance program is reflected by the mapping of the Lennard-Jones simulation data exactly onto the NIST data as seen in Figure 3. 

One particularly powerful insight students gain from this assignment is the limitations of the van der Waals equation of state. Often in undergraduate chemistry courses, the van der Waals equation is presented as the universal correction to the ideal gas law, perhaps owing to its straightforwardness and the ease with which it can be understood. Recognizing its limitations leads students to consider other equations of state, where each expression has its own set of assumptions. While students are initially uneasy with the notion that the van der Waals equation has drawbacks and that decisions about which EOS to use depends on the system or context, this unease is not uncommon in the execution of real science. 

P (r) is the attractive contribution important at large separations. (This same notion is used in the van der Waals equation of state, in which the constant a accounts for molecular attraction and b for molecular repulsion.)... 

We have already seen from Example 10.1 that van der Waals forces play a major role in the heat of vaporization of liquids, and it is not surprising, in view of our discussion in Section 10.2 about colloid stability, that they also play a significant part in (or at least influence) a number of macroscopic phenomena such as adhesion, cohesion, self-assembly of surfactants, conformation of biological macromolecules, and formation of biological cells. We see below in this chapter (Section 10.7) some additional examples of the relation between van der Waals forces and macroscopic properties of materials and investigate how, as a consequence, measurements of macroscopic properties could be used to determine the Hamaker constant, a material property that represents the strength of van der Waals attraction (or repulsion see Section 10.8b) between macroscopic bodies. In this section, we present one illustration of the macroscopic implications of van der Waals forces in thermodynamics, namely, the relation between the interaction forces discussed in the previous section and the van der Waals equation of state. In particular, our objective is to relate the molecular van der Waals parameter (e.g., 0n in Equation (33)) to the parameter a that appears in the van der Waals equation of state ... 

EXAMPLE 10.2 The Dispersion Force and Nonideality of Gases. The nonideality of gases arises from the repulsive and attractive forces between atoms. As a consequence, the deviation of the properties of a gas from ideal gas behavior can be traced to the interatomic or intermolec-ular forces. Assume that methane follows the van der Waals equation of state at sufficiently low densities. It is known from experiments that (see Israelachvili 1991)... 

Pressure is a manifestation of the kinetic energy of gas molecules. According to the van der Waals equation of state (see Eq. (36)), the pressure of 1 mole of gas must be increased by an amount a/v2 due to intermolecular attractions that decrease the pressure from what it would be if ideal. Use the term a/v2 as a general expression for the attraction between a pair of molecules and, based on this, reconstruct the argument leading to Equation (52). 

In the new version, Chapter 10 focuses exclusively on van der Waals forces and their implications for macroscopic phenomena and properties (e.g., structure of materials and surface tension). It also includes new tables and examples and some additional methods for estimating Hamaker constants from macroscopic properties or concepts such as surface tension, the parameters of the van der Waals equation of state, and the corresponding state principle. 

On the other hand, Kilian 50) having analysed the strain-induced volume dilation 24 91 using the van der Waals equation of state (Fig. 6) emphasized that only pressure dependence of the interchain parameter, a, is required for a full explanation of the relative volume changes. He arrived at a conclusion that non-crystalline rubbers are anisotropic equilibrium liquids and a higher compressibility of NR was only necessary for fitting the extension data. Hence, on using the van der Waals approach, there is no need of postulating volume dependence of the front factor as proposed by Tobolsky and Shen. 

Kilian 50 describes work and heat effects at very high extensions of SBS linear and star thermoelastoplastics241 using the van der Waals equation of state with the following set of parameters in Eqs. (81)—(85) = 8.5 (star SBS) and Xm — 12... 

In 1873, Dutch physicist J. D. Van der Waals (Sidebar 2.9) presented (in his doctoral thesis) the celebrated equation of state that now bears his name. The Van der Waals equation of state may be written in a form... 

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