Van der Waals Equation

10 Key excerpts on "Van der Waals Equation"

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  Engineering and Chemical Thermodynamics

  • Milo D. Koretsky(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  238 ► Chapter 4. Equations of State and Intermolecular Forces As we have just seen, the Van der Waals Equation is an example of a cubic equation of state because its highest term in volume is raised to the third power. The Van der Waals Equation is presented because of the clear way in which it incorporates the attractive and repulsive interactions we have discussed. We will see that many modern cubic equations have the same basic form as the Van der Waals Equation, but are considerably more accurate. In other words, if you need an accurate answer, there are better equations to use than the Van der Waals Equation. In fact, hundreds of different cubic equations of state exist. All these equations are approximate. They merely fit experimental data. Yet, in general, many can provide rea- sonable values for both the vapor and liquid regions of hydrocarbons and the vapor region for many other species. In this section, we will not attempt to go through a critical review of all the available cubic equations of state; rather, we will illustrate the scientific concepts and engineering application through a few commonly used cubic equation. The majority of other equations that have been proposed are variations of the forms we will study. The general form of a cubic equation is: v 3 1 f 1 1 T, P 2 v 2 1 f 2 1 T, P 2 v 1 f 3 1 T, P 2 5 0 where f i 1 T, P 2 represent a function that can contain fitting parameters as well as the properties T and P. The three characteristic roots in volume follow the same trends as those we discussed with the Van der Waals Equation in relation to Figure 4.12. Table 4.3 illustrates some examples of the form: P 5 RT v 2 b 2 Attr All these equations use the same “repulsive” term as the Van der Waals Equation. The term indicated by “Attr” quantifies attractive interactions. In general, these terms are empirically established to best fit experimental data.

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  Classical and Geometrical Theory of Chemical and Phase Thermodynamics

  • Frank Weinhold(Author)
  • 2009(Publication Date)
  • Wiley-Interscience

    (Publisher)

  from experiment are still evident, and one may conclude that the Van der Waals Equation provides only the first tier of improvements to the ideal gas picture. The success of the Van der Waals Equation inspired many attempts to develop improved equations of state based on more complex functional forms and additional empirical fitting parameters. Sidebar 2.10 lists some modified equations of state (and associated empirical parameters) that have been proposed and found useful in engineering applications. Although increasingly parametrized empirical equations of state can “fit” selected real gas properties with ever-increasing accuracy, the fundamental significance of these par- ameters is problematic. The parameters usually lack theoretical significance that would permit their numerical evaluation from deeper principles (i.e., quantum mechanics or other physical models and measurements lying outside the fitting procedure itself). For any given parametrization, the accuracy tends to vary from one PVT region to another, but if the accuracy is inadequate, there is no systematic way to improve the description. Thus, we seek an alternative formulation of the equation of state that is more directly connected to the underlying theory of intermolecular forces, is systematically improvable to any desired accuracy, and (at least in principle) requires no empirical “fitting” parameters to fix its form. SIDEBAR 2.10: EMPIRICAL EQUATIONS OF STATE A few of the simpler and more commonly known empirical equations of state are shown below for comparison with the Van der Waals Equation [(P þ a/V 2 )(V 2 b) ¼ RT] of 1873. Many other such equations (some with 20 or more parameters!) have appeared in the literature.

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  Principles of Colloid and Surface Chemistry, Revised and Expanded

  • Paul C. Hiemenz, Raj Rajagopalan, Paul C. Hiemenz, Raj Rajagopalan(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  ■ * * *

  In this section we have examined the three major contributions to what is generally called the van der Waals attraction between molecules. All three originate in dipole-dipole interactions of one sort or another. There are two consequences of this: (a) all show the same functional dependence on the intermolecular separation, and (b) all depend on the same family of molecular parameters, especially dipole moment and polarizability, which are fairly readily available for many simple substances. Many of the materials we encounter in colloid science are not simple, however. Hence we must be on the lookout for other measurable quantities that depend on van der Waals interactions. Example 10.2 introduces one such possibility. We see in Section 10.7 that some other difficulties arise with condensed systems that do not apply to gases.

  In the next section we take a preliminary look at the way van der Waals attractions scale up for macroscopic (i.e., colloidal) bodies. This will leave us in a better position to look for other measurements from which to estimate the van der Waals parameters.

  10.5  VAN DER WAALS FORCES BETWEEN LARGE PARTICLES AND OVER LARGE DISTANCES

  The interaction between individual molecules obviously plays an important role in determining, for example, the nonideality of gas, as illustrated in Example 10.2 . It is less clear how to apply this insight to dispersed particles in the colloidal size range. If atomic interactions are assumed to be additive, however, then the extension to macroscopic particles is not particularly difficult. Moreover, when dealing with objects larger than atomic dimensions, we also have to consider interactions over appropriately large distances. In the case of the London attraction, forces over large distances show a more rapid decay than indicated by the inverse sixth-power equations derived in Section 10.4 . This is known as (electromagnetic) retardation. We discuss these two important issues in this section before developing the equations for interactions between macroscopic bodies in Section 10.6

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  Physical Chemistry

  Understanding our Chemical World

  • Paul M. S. Monk(Author)
  • 2005(Publication Date)
  • Wiley

    (Publisher)

  Firstly, interparticle interactions are usually attractive, encouraging the particles to get closer, with the result that the gas has a smaller molar volume than expected. Secondly, since the particles have their own intrinsic volume, the molar volume of a gas is described not only by the separations between particles but also by the particles themselves. We need to account for these two factors when we describe the physical properties of a real gas. The Dutch scientist van der Waals was well aware that the ideal- The a term reflects the strength of the interaction between gas particles, and the b term reflects the particle’s size. gas equation was simplistic, and suggested an adaptation, which we now call the Van der Waals Equation of state: p + n 2 a V 2 (V − nb ) = nRT (2.2) where the constants a and b are called the ‘van der Waals constants’, the values Equation (2.2) simpli- fies to become the ideal-gas equation (Equation (1.13)) whenever the volume V is large. of which depend on the gas and which are best obtained experimentally. Table 2.4 contains a few sample values. The constant a reflects the strength of the interaction between gas molecules; so, a value of 18.9 for benzene suggests a strong interaction whereas 0.03 for helium represents a negligible interaction. Incidentally, this latter value reinforces the idea that inert gases are truly inert. The magnitude of the constant b reflects the physical size of the gas particles, and are again seen to follow a predictable trend. The magnitudes of a and b dictate the extent to which the gases deviate from ideality. Note how Equation (2.2) simplifies to become the ideal-gas equa- tion (Equation (1.13)) if the volume V is large. We expect this result, because a large volume not only implies a low pressure, but also yields the best conditions for minimizing all instances of interparticle collisions.

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  Physics of Matter

  • George C. King(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  Over a period of time, these forces of attraction cancel out. However, molecules that strike the walls of the container experience only the attractive forces from the molecules behind them. This means that the resultant attractive force is not balanced. Consequently, the kinetic energy of the molecules striking the wall is reduced and hence the pressure exerted on the container walls is lower than the pressure in the interior of the gas. Real gases 161 again. Thus, the attraction of the walls has no net effect on the momentum change due to collisions with the wall, i.e. on the pressure exerted on the walls. Combining the above two corrections to the ideal gas law, we obtain the Van der Waals Equation: P + a V 2 V b RT for one mole of gas. At high pressure, the number density of molecules is high, and then the volume factor b and the pressure defect a/V 2 become important. Conversely, at low pressure, implying large volume, the molecules are far apart on average and the gas behaves like an ideal gas obeying PV = RT. For n moles of gas, the Van der Waals Equation becomes P + n 2 a V 2 V nb nRT (5.4) We see that the units of a are (volume) 2 (pressure)/(mol) 2 ≡ (m 6 Pa)/mol 2 , and the units of b are (volume)/ (mol) ≡ m 3 /mol. The Van der Waals Equation does not represent an exact law; it is only an approximate equation of state. The constants a and b are best regarded as empirical parameters for a particular molecule rather than as precisely defined molecular parameters. The Van der Waals Equation does, however, have the advantages that it is a relatively simple equation and it only introduces two extra parameters to the ideal gas equation. In addition, it provides physical insights into how intermolecular interactions contribute to the deviations of a real gas from the perfect gas law. We can judge the reliability of the Van der Waals Equation by compar- ing its predictions with experimental data, as we will do in Section 5.2.

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  Thermodynamics with Chemical Engineering Applications

  • Elias I. Franses(Author)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  p ideal ð Þ V ideal ð Þ ¼ p þ a V 2 V − b ¼ RT : ð 5 : 34 Þ This is the famous Van der Waals Equation of state. In this equation, one presumes that the ideal gas equation of state is still valid, but the effective or ideal values of p and V are different from those in the ideal case. The deviations are described by two parameters: b (m 3 /mol), which depends on the excluded volume of the molecules, and is four times the actual total volume of molecules for spherical molecules; and a , which has units Pa m 6 /mol 3 and is a measure of how the average attractive interaction between a pair of molecules affects the actual pressure compared with the ideal gas pressure of non-interacting molecules. 5.3.1 Additional notes on the Van der Waals Equation of state (for the advanced student) 1. The parameters b and a refer to how two molecules interact, either in excluding the available space, or in attracting each other. Only “ binary interactions ” (i.e. interactions of two molecules) are considered. Thus, this equation would be expected to be accurate for dilute, nearly ideal gases, and become inaccurate, or even fail, for dense gases, condensed liquid phases, supercritical fl uids, or fl uids close to the critical point. For these systems, many-molecule interactions need to be considered. 2. The parameter a provides a gross average representation of intermolecular attractive forces, which depend on the intermolecular distances, molecular structure, and relative molecular orientations. The dependence of a on the distance is accounted for, in part and indirectly, by the dependence of A on ρ 2 , since as p increases the correction term increases. 3. Equation (5.34) is applied to all kinds of molecules, with the parameters a and b determined “ empirically ” by fi tting p − V − T data to the equation, and then using the equation for other conditions of p and T .

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  An Introduction to Applied Statistical Thermodynamics

  • Stanley I. Sandler(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  In this way, a direct relationship is obtained between the parameters in the thermody- namic models and molecular parameters. An alternative is to use the Generalized van der Waals model in reverse and work backward from commonly used macroscopic thermodynamic models to understand the assumptions on which they are based. This latter procedure has been used to understand the assumptions that underlie some of the commonly used engineering equations of state that are shown in Table 16.2-1. Notice that some use the simple van der Waals free-volume term, while others use the free-volume term that comes from the Carnahan-Starling expression derived from the Percus-Yevick equation, which is a better representation of hard-core free volume. Table 16.2-1 Examples of Square-Well Based Equations of State van der Waals V f = V − b; N c = 4π 3 σ 3 (R 3 sw − 1)ρ = Cρ →  = − 2π 3 σ 3 ε(R 3 sw − 1)ρ = − εCρ s ; which leads to P = RT V − b − a V 2 with a = N 2 Av Cε 2 = bN Av ε(R 3 sw − 1) Temperature-dependent van der Waals V f = V − b; N c = 4π 3 σ 3 (R 3 sw − 1)e ε/kT →  = − 4πσ 3 3 kT (R 3 sw − 1)ρ(e ε/kT − 1) which leads to P = RT V − b − a(T ) V 2 with a(T ) = C 2 kT (e ε/kT − 1) = bRT (e ε/kT − 1) Redich-Kwong V f = V − b; N c = C  T o T ln (1 + βρ) →  = − Cε 3  T o T ln  1 + b V  which leads to P = RT V − b − a/ √ T V (V + b) with a = N Av Cε 3 b  T o Soave-Redlich-Kwong V f = V − b; N c = C 2 (T ) ln(1 + βρ) which leads to P = RT V − b − a(T ) V (V + b) with a given above Peng-Robinson V f = V − b; N c = C 3 (T )ρ atan  N  2N bρ − b 2 ρ 2   2N bρ − b 2 ρ 2 = C 3 (T ) atan  V √ 2NbV − b 2  √ 2NbV − b 2 which leads to P = RT V − b − a(T ) V (V + b) + b(V − b) with a(T ) (or C 3 (T )) empirically fit to vapor pressure data Note: C = 4π 3 σ 3 (R 3 sw − 1) and b = 2πN Av 3 σ 3

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  Classical and Quantum Thermal Physics

  • R. Prasad(Author)
  • 2016(Publication Date)
  • Cambridge University Press

    (Publisher)

  It is only under the extreme conditions of low pressure and high temperature that the above-mentioned ideal gas equation may be applied to a real gas, and that too, over a limited range of pressure and temperature. An important experimental observation in this regard is that the ideal gas equation may be applied to a real gas with about 1% accuracy to the pressure and temperature range where the specific molar volume v is >5 liter/mole for diatomic gases and > 20 liter/mole for other gases. Several attempts have been made to develop equation of state for real gases, mostly by modifying either semi-empirically or empirically the ideal gas equation. Such attempts resulted in the development of: 1. Cubic equations of state, 2. Virial equations of state and 3. Compressibility factor equation. Ideal to a Real Gas, Viscosity, Conductivity and Diffusion 51 Van der Waals Equation of state for real gases is a typical example of cubic equations. A cubic equation gives three roots for specific volume for each set of pressure and temperature, at least one of which is real, and can thus explain the co-existence of different phases. 2.2 Modification of Ideal Gas Equation: Van der Waals Equation of State The equation of state for an ideal gas is written as PV = RT 2.1 Here P is the gas pressure, V the volume, the number of moles of the gas contained in volume V, R the gas constant and T the Kelvin temperature of the gas. The gas equation per mole may be obtained by dividing both sides of Eq. 2.1 by the number of moles and putting specific molar volume v = V to get, Pv = RT 2.2 2.2.1 Reduction in pressure In the case of ideal gas, the pressure exerted by the gas molecules was calculated assuming that the molecules are free point particles and have no attraction between them. In a real gas, however, molecules attract each other and, hence, the force with which the molecules hit the walls of the container is less than that in the case of the ideal gas.

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  Molecular Theory of Capillarity

  • J. S. Rowlinson, B. Widom, B. Widom(Authors)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  3 THE THEORY OF VAN DER WAALS

  3.1 Introduction

  ‘I am more than ever an admirer of van der Waals’; so wrote Lord Rayleigh1 in a private letter in 1891. It was a view shared by many of the best nineteenth-century physicists and chemists. Twenty years earlier Maxwell had taught himself Dutch in order to be able to read van der Waals’s thesis, while Ostwald translated his work for republication in German. What commanded the admiration of such men was his development of powerful and far-sighted methods of approximation in the newly developing subject of statistical thermodynamics. We have already touched in Chapter 1 on his contribution to the development of the mean-field approximation; here we describe his application of these ideas to the state of matter within the gas-liquid surface.

  In the quasi-thermodynamics of the last chapter we introduced a local free-energy density ψ(z), although its identification in (2.90 ) proved unsatisfactory; it led to a vanishing interfacial tension σ and to a density profile ρ(z) that was a step-function. What is missing in (2.90 ) is a term that imposes a characteristic length on the fluid, and thus leads, after minimization of the integrated excess free-energy density, to a density profile consistent with an interface of non-zero thickness. The theory was given such a form by van der Waals;2 Landau and Lifshitz’s theory of magnetic domain structure3 is equivalent, and the theory has been reformulated and extended by Cahn and Hilliard.4 In this chapter we present van der Waals’s theory and some of its elaborations.

  Let ψ(z) continue to represent the local free-energy density of (initially) a one-component system, though now not necessarily given by (2.90 ). Whatever the correct ψ(z), let Ψ(z

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  Phase Behavior of Petroleum Reservoir Fluids

  • Karen Schou Pedersen, Peter Lindskou Christensen, Jawad Azeem Shaikh(Authors)
  • 2024(Publication Date)
  • CRC Press

    (Publisher)

  4 Equations of State

  DOI: 10.1201/9780429457418-4

  The majority of calculations of the pressure-volume-temperature (PVT) relation carried out for oil and gas mixtures are based on a cubic equation of state. Cubic equations date back more than 100 years to the famous Van der Waals Equation (van der Waals 1873 ). The most commonly used equations in the petroleum industry today are similar to the Van der Waals Equation, but it took almost a century for the industry to accept this type of equation as a valuable engineering tool. The first cubic equation of state to obtain widespread use was the one presented by Redlich and Kwong (1949) . Soave (1972) and Peng and Robinson (1976 and 1978 ) further developed this equation in the 1970s. In 1982, Peneloux et al. (1982) presented a volume-shift concept for improving liquid density predictions of the two former equations. Computer technology has made it possible, within seconds, to perform millions of multicomponent phase equilibrium and physical property calculations with an equation of state as the thermodynamic basis. This chapter presents some of the most popular cubic equations of state as well as the noncubic PC-SAFT and GERG-2008 equations of state. Chapter 6 describes the application of cubic equations of state in phase equilibrium (flash) calculations, Chapter 8 the derivation of physical properties from cubic equations of state, and Chapter 16 the application of cubic equations of state to mixtures with water and other aqueous components as well as an extension of cubic equations with an association term.

  4.1 Van der Waals Equation

  When deriving the first cubic equation of state, van der Waals used the phase behavior of a pure component as the starting point. Figure 4.1 shows schematically pressure (P) versus molar volume (V) curves for a pure component at various temperatures. At temperatures far above the critical (T1 in Figure 4.1

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