Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 4 / d

For the same geometry, separation and intervening medium, the attractive Van der Waals force scales with the effective Hamaker constant. With the constants treated as applying to those same conditions,The silver particles attract more strongly. This comparison is conditional on a common geometry and on using the appropriate constants across the actual medium. Constants for particles in vacuum cannot automatically be applied unchanged across a solvent.

For two equal spheres of radius $R$, the effective radius in the Derjaguin approximation is $R_{\mathrm{eff}}=R/2$. The nonretarded attractive van der Waals potential isDifferentiating with respect to the surface gap givesThus the requested approximation gives an attraction of about $8.3$ pN. Here $h/R=2$, whereas the Derjaguin approximation requires $h\ll R$. This is the formal estimate requested by the hint, not a quantitatively reliable near-contact approximation for the printed dimensions.

The Stern layer is the compact near-surface layer of specifically adsorbed or closely associated ions. The diffuse layer is the more extended, thermally distributed excess of counterions that screens the surface charge.

For water near $25^{\circ }\mathrm{C}$, the Debye length can be writtenFor $0.01$ M KCl, $I=0.01$ M and ${\lambda }_D\simeq 3.0$ nm. For a fully dissociated 2:2 electrolyte at the same formula-unit concentration, $I=4c=0.04$ M, giving ${\lambda }_D\simeq 1.5$ nm, half as large. These are linear-screening estimates, not sharp boundaries of the diffuse layer.