 Solution

ID: 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/ii/solution

For two equal spheres of radius

R

R_{eff} = R /2

. The nonretarded attractive van der Waals potential is

U (h) = - \frac{A R _{eff}}{6 h} = - \frac{A R}{12 h} .

Differentiating with respect to the surface gap gives

F (h) = - \frac{A R}{12 h ^{2}} = - \frac{( 4.0 \times 1 0 ^{- 19} ) ( 1 0 ^{- 9} )}{12 ( 2 \times 1 0 ^{- 9} ) ^{2}} = - 8.3 \times 1 0^{- 12} N .

Thus the requested approximation gives an attraction of about $8.3$ pN. Here

h / R = 2

, whereas the Derjaguin approximation requires

h ≪ R

. This is the formal estimate requested by the hint, not a quantitatively reliable near-contact approximation for the printed dimensions.

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