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Kinetic theory of gases. Boltzmanns Law and Maxwell distribution

Problem 2.103

When examining the suspended gamboge droplets under a microscope, their average numbers in the layers separated by the distance h=40μmh=40 \mu \mathrm{m} were found to differ by η=2.0\eta=2.0 times. The environmental temperature is equal to T=290 KT=290 \mathrm{~K}. The diameter of the droplets is d=0.40μm,d=0.40 \mu \mathrm{m}, and their density exceeds that of the surrounding fluid by Δρ=0.20 g/cm3\Delta \rho=0.20 \mathrm{~g} / \mathrm{cm}^{3}. Find Avogadro's number from these data.

Reveal Answer
NA=(6RT/πd3Δρgh)ln⁡η≈6.4⋅1023 mol−1N_{A}=\left(6 R T / \pi d^{3} \Delta \rho g h\right) \ln \eta \approx 6.4 \cdot 10^{23} \mathrm{~mol}^{-1}
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