The van der Waals constants

An approximate value for dc in the equation for tire Lennard-Jones potential, quoted above, may be obtained from the van der Waals constant, b, since... 

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 adds two correction terms to the ideal gas equation. Each correction term includes a constant that has a specific value for every gas. The first correction term, a fV, adjusts for attractive intermolecular forces. The van der Waals constant a measures the strength of intermolecular forces for the gas the stronger the forces, the larger the value of a. The second correction term, n b, adjusts for molecular sizes. The van der Waals constant b measures the size of molecules of the gas the larger the molecules, the larger the value of b. 

The van der Waals constants for a number of gases appear in Table 11-1. and the magnitude of van der Waals corrections is explored in Example. ... 

This potential was developed to ensure that the molecules inside the sphere never escape and maintain a fully solvated system during molecular dynamics. Here, es, Rs, ew and Rw are the van der Waals constants for the solvent and the wall and rj is the distance between the molecule i and the center of the water sphere, Ro is the radius of the sphere. The quantities A, B and Rb are determined by imposing the condition that W and dW/dr, vanish at r, = Ro. The restraining potential W is set to zero for r, < R0. The van der Waals parameters Es, ew, Rs and Rw can also be specifically defined for different solvents. The constants Awaii and Cwan are computed using a well depth of es = ew = 0.1 kcal and the radius of Rs = Rw = 1.25 A. For the other set of simulations, especially for the hydride ion transfer, we applied periodic boundary conditions by using a spherical boundary shell of 10.0 A of TIP3P40 water to cover the edges of the protein. 

The van der Waals constant a corrects for the attractive forces between gas molecules. The constant b corrects for particle volume. The attractive forces between gas molecules become pronounced when the molecules are closer together. Conditions which favor this are low temperatures (-100°C) and high pressures (5.0 atm). 

Hence, if the van der Waals constants are known, 7) can be calculated. For all gases except hydrogen and hehum, this inversion temperature is above common room temperarnres. 

The van der Waals constants for many molecules are tabulated in units of (L atm)/mol for a, which is associated with intermolecular attraction the unit for b is L/mol, and is associated with molecular volume. Some of the values of a, b, Tc, Pc, Vc are given in table 4.27. 

Van der Waals equation is (p + a/F2) (F — 6) = RT, where p is the pressure, F the molar volume, T the absolute temperature, a the Van der Waals constant to account for attractive forces between molecules, and b the Van der Waals constant to account for the finite volume of molecules. 

Equations of this type are known as virial equations, and the constants they contain are called the virial coefficients. It is the second virial coefficient B that describes the earliest deviations from ideality. It should be noted that B would have different but related values in Equations (26) and (27), even though the same symbol is used in both cases. One must be especially attentive to the form of the equation involved, particularly with respect to units, when using literature values of quantities such as B. The virial coefficients are temperature dependent and vary from gas to gas. Clearly, Equations (26) and (27) reduce to the ideal gas law as p - 0 or as n/V - 0. Finally, it might be recalled that the second virial coefficient in Equation (27) is related to the van der Waals constants a and b as follows ... 

Before considering how the excluded volume affects the second virial coefficient, let us first review what we mean by excluded volume. We alluded to this concept in our model for size-exclusion chromatography in Section 1.6b.2b. The development of Equation (1.27) is based on the idea that the center of a spherical particle cannot approach the walls of a pore any closer than a distance equal to its radius. A zone of this thickness adjacent to the pore walls is a volume from which the particles —described in terms of their centers —are denied entry because of their own spatial extension. The volume of this zone is what we call the excluded volume for such a model. The van der Waals constant b in Equation (28) measures the excluded volume of gas molecules for spherical molecules it equals four times the actual volume of the sphere, as discussed in Section 10.4b, Equation (10.38). 

From the second of these derivatives, evaluate 0C as predicted by this model. Use this value of 6C and the first of these derivatives to evaluate the relationship between Tc and the two-dimensional a and b constants. How does this result compare with the three-dimensional case The van der Waals constant b is four times the volume of a hard -sphere molecule. What is the relationship between the two dimensional b value and the area of a haid-disk molecule ... 

EXAMPLE 4-1 Calculate the van der Waals constants for 3-methyl-hexane. 

Here a and b are the standard Van der Waals constants. The Van der Waals constant a is associated with the isotropic inter-particle attraction. Each intermediate phase, i, is characterized by the anisotropic inter-particle interaction y. In this approach, for each phase i there are Q, l configurations all having y- = 0 and only a single configuration in which the formation of the anisotropic bonds with energy y is allowed. 

The van der Waals constants were introduced to account for mutual attractions of the molecules and the space occupied by each molecule. 

From what we have said, the values of the Van der Waals constants b for the gases of our table look very reasonable. Next we can consider their a s. In Eq. (3.6) of Chap. XXII, we have seen that the Van der Waals interaction energy between molecules of polarizability a, mean square dipole moment /, at a distance r, is... 

The Van der Waals constants Cn(a)A, (oB, R) depend on the Euler angles (oA and (oB specifying the orientation of the monomers in an arbitrary space-fixed frame, and on the polar angles R = (fi, a) determining the orientation of the intermolecular axis (R is assumed to join the monomer centers of mass) with respect to the same space-fixed frame. 

Therefore, knowing the Van der Waals constants is very useful to estimate the interaction energy at large distances, and is necessary to guarantee the correct large R asymptotic behavior of the potential energy surface EM(R, a>A, wB, R). 

Explicit expressions for the Van der Waals constants may be obtained by invoking the well-known147 multipole expansion of the operator V. In an arbitrary space-fixed coordinate system, this expansion can be written as... 

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