Perrin-style measurements on gum-resin granules of radius

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Physical foundations of the living cell · Physical foundations of the living cell

Forces, Machines and Signals, Ch Diffusion: friction, capture and the crowded cytoplasm

Problem

Perrin-style measurements on gum-resin granules of radius a=0.30 μma=0.30\,\mathrm{\mu m}, suspended in water of viscosity η=0.85×10−3 Pa s\eta = 0.85\times10^{-3}\,\mathrm{Pa\,s} at T=300 KT=300\,\mathrm{K}, recorded the displacement of a single granule along one axis every Δt=20 s\Delta t = 20\,\mathrm{s}. Eight consecutive displacements (in μm\mathrm{\mu m}) were: 6.1, −4.8, 7.3, −3.0, 8.5, −2.1, 4.9, −6.9.6.1,\ -4.8,\ 7.3,\ -3.0,\ 8.5,\ -2.1,\ 4.9,\ -6.9. ((a)) Find the mean square one-axis displacement ⟨x2⟩\langle x^2\rangle and hence the diffusion coefficient DD of the granule. ((b)) Taking the drag coefficient as ζ=6πηa\zeta = 6\pi\eta a (quoted, for a sphere), use Dζ=kBTD\zeta = k_BT to find Boltzmann's constant kBk_B. ((c)) Hence estimate Avogadro's number NA=R/kBN_A = R/k_B with the molar gas constant R=8.314 J mol−1 K−1R = 8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}, and compare it with the accepted value 6.022×1023 mol−16.022\times10^{23}\,\mathrm{mol^{-1}}.

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