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161 Molecular Theory of Fluids
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The hard-core diameter d is a measure of the center-to-center separation for which U, and hence F, becomes infinite It is not subject to precise determination, but plays the role of a modeling parameter in some expressions for U The collision diameter o is defined as the separation for which U = 0 The equilibrium separation ro is the separation for which U attains its minimum value of - E At r = ro, the net intermolecular force is zero Quantity E is called the well depth For a particular class of chemical species (eg, noble gases, or cyclic alkanes), each of these special quantities increases with increasing molecular size Typical J Commonly, ranges of values for o and E are 0 x 3 to 8 x 10-lo m and E x 01 to 10 x ro is about 10 to 15% greater than o Scores of expressions have been proposed for U' All are essentially empirical, although their functional forms often have some basis in theory The most widely used is the LennardJones (LJ) 12/6pair-potential function:
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Equation (161) provides semiquantitative representations of thermodynamic and transport properties for nonpolar substances of relatively simple molecular structure In Eq (161), the r-l2 term is supposed to represent bimolecular repulsions, and the r p 6term bimolecular attractions Although the attraction term has significant theoretical justification, the r-l2 dependence for repulsions is chosen primarily for mathematical convenience Table 161 shows some values of the dimensionless potential energy U / r as a function of the dimensionless separation r l o , as implied by Eq (161) Clearly, the effects of intermolecular forces are significant only over a modest range of separations Although they operate over an infinite range (U and F together approach zero only in the limit as r + oo), both U and F for neutral molecular pairs are numerically negligible for separations greater than about 10 molecular diameters
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Contributions to the Intermolecular Potential Energy
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The origins of intermolecular forces stem from the essential concept that a molecule is a distribution of charges: a collection of positively charged nuclei, surrounded by a cloud of negatively charged electrons Intermolecular repulsions at sufficiently small separations therefore result from the overlap of electron clouds of interacting molecules, giving rise to a Coulombic repulsion At still smaller separations the positively charged nuclei "see" each other, again promoting repulsion The origins of intermolecular attractions are less obvious, and several mechanisms can contribute First, consider the electrostatic interaction of two rigid charge distributions A and B By Coulomb's law, the electrostatic potential energy of interaction U(e1) is:
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a compilation of expressions for U see G C Maitland, M Rigby, E B Smith, and W A Wakeham, Intermolecular Forces: Their Origin and Determination, app 1, Clarendon Press, Oxford, 1981
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'A comprehensive discussion of these contributions to U ( r ) is given by J 0 Hirschfelder, C F Curtiss, and R B Bird, Molecular Theory of Gases and Liquids, pp 25-30, 209, 983-1020, John Wiley and Sons, New York, 1954
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CHAPTER 16 Introduction to Molecular Thermodynamics Table 161 Bimolecular Potential Energy as a Function of Separation for the LennardJones 1216 Potential
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Here, qi is a charge in distribution A, qj is a charge in distribution B, and rij is the separation between charges qi and qj Quantity 0 is the electric permittivity of vacuum, a physical constant3 equal to 8854 19 x lo-'' C V-' m-' (The unit of electric charge is the coulomb C, and the unit of electric potential difference is the volt V) The sums are taken over all charges in the distributions Equation (162) is exact, but awkward to use as it stands Application is facilitated when the charge separation rij is replaced by the center-of-mass separation r of the two distributions Statistical averaging of contributions from all orientations of charge distribution yields the following approximate expression for U(e1) for two neutral rigid charge distribution^:^
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J K-l ; p~ and p~ are the permanent Quantity k is Boltzmann's constant, equal to 1381 x dipole moments for charge distributions A and B associated with the molecules This contribution to the pair-potential function vanishes only when one of the permanent dipole moments is zero The averaging procedure which leads to Eq (163) produces several remarkable results:
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Even though the distributions are electrically neutral, there is a net attraction between them The dependence of U(e1) on charge-distributionorientation becomes on averaging an rP6 dependence As given by Eq (163) U(e1) varies with T-' Hence the magnitude of the permanentdipole interaction decreases with increasing temperature Equation (163) is the simplest example of a direct electrostatic potential for two neutral molecules; here, the dipole moment emerges as an important physical property Dipole moments are measures of the net separation of charge within a molecule For a spherically symmetric neutral charge distribution (eg, an atom of argon), p is zero For a molecule in which charge Iq 1 is separated from charge - Iq 1 by distance 1, the dipole moment is p = Iq 1 1 Hence p
3~nrelated the well depth E of Fig 161 to 4 ~ h iresult is only valid for modest dipole moments For a discussion, see T M Reed and K E Gubbins, Applied s Statistical Mechanics, sec 5-7, McGraw-Hill, New York, 1973
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