Perrin's measurement can be rerun on numbers of the kind he had

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Condensed Matter, Materials & Soft Matter · Solid continuum mechanics and complex fluids

Condensed Matter Physics, Ch Soft matter: when entropy, not bonding, sets the structure

Problem

Perrin's measurement can be rerun on numbers of the kind he had. Gamboge grains of radius 0.212 μm0.212\,\mu\mathrm{m} and density excess Δρ=194 kg m−3\Delta\rho=194\,\mathrm{kg\,m^{-3}} over water are suspended at 293 K293\,\mathrm{K}, and counting them at different heights gives a gravitational length ℓg=53 μm\ell_g=53\,\mu\mathrm{m}. ((a)) Find Boltzmann's constant kBk_B from these measurements. ((b)) Find the Avogadro constant from NA=R/kBN_A=R/k_B, with R=8.31 J mol−1 K−1R=8.31\,\mathrm{J\,mol^{-1}\,K^{-1}}. ((c)) The radius is known to ±3\pm3 per cent, Δρ\Delta\rho to ±1\pm1 per cent and ℓg\ell_g to ±2\pm2 per cent. Which measurement limits the accuracy of NAN_A, and roughly how large is the resulting uncertainty?

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