In a Perrin-style count

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Statistical foundations and the equilibrium ensembles

Soft Matter at the Thermal Scale, Ch Brownian particles, sedimentation and the hard-sphere fluid

Problem

In a Perrin-style count, spherical grains of radius a=0.25 μma = 0.25\,\mu\mathrm{m} and density excess Δρ=ρp−ρs=200 kg m−3\Delta\rho = \rho_p - \rho_s = 200\,\mathrm{kg\,m^{-3}} over the surrounding fluid are left to reach sedimentation equilibrium, dilutely and without interacting, in a column held at T=300 KT = 300\,\mathrm{K} (measured with a thermometer). Counting grains in a fixed small volume at the bottom of the column (z=0z=0) and again at a height z=90 μmz = 90\,\mu\mathrm{m} above it gives 800800 grains and 4949 grains respectively. Take g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}} and the gas constant R=8.314 J mol−1 K−1R = 8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}. ((a)) Starting from the equilibrium condition that the total chemical potential of a grain is independent of height, write the gravitational length ℓg\ell_g in terms of the height difference and the count ratio, and find the value implied by the two counts. ((b)) Use the two counts to determine Boltzmann's constant kBk_B and hence Avogadro's constant NAv=R/kBN_{\rm Av} = R/k_B from this measurement.

Browse the problem library