Geometric mean rule origin

Problem 2.4 The origin of the geometric mean rule Starting from the van der Waals (London) forces, prove first Equation 2.9 and then prove Equation 2.13. What are the most important assumptions in the calculations and what is the significance of the final result for colloid and surface science ... 

S.Eq.l 1.7.2 expresses what is referred to as the geometric mean combining rule, whose origin and limitations are discussed in Section 7.6.4. 4.The performance of these combining rules becomes poorer, as we should expect, with increasing dissimilarity between species i andy. It would indeed be naive to expect that pure component properties alone would suffice for the prediction of mixture properties, considering the difference between intra- and inter- Oike and unlike species) molecular forces. And this difference becomes, of course, more pronounced as the dissimilarity between the species increases. 

In the original Fowkes model [14], only dispersion component of the surface tension was considered, which is caused by London dispersion force. The London dispersion forces arise from the interaction of fluctuating electronic dipoles with induced dipoles in neighboring atoms or molecules [15], It exists in all type of materials and always presents as an attractive force at the liquid-solid interface. The work of adhesion from dispersion interaction has been proved thermodynamically to take the form of the geometric mean according to the Berthelot mixing rule [17, 18]. 

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