Figure shows the results of a series of observations of a migrating Brownian particle. The observations were made at intervals of 30 s, the temperature of water was `25^@C`, the radius of a Brownian particle is `4.4 xx 10^(-7)m`. Measuring the "steps" of the particle in the scale specified, find the square of the r.m.s. displacement for a given time, and calculate the Boltzmann constant and the Avogadro number. The scale is as follows: 1 mm on the graph corresponds to a i displacement of `1.25 mu m`.
 

Explanation

Consider 10 successive steps starting with the lower left. Measure their lengths in millimetres as accurately as possible and transform them to scale into actual dimensions (see Table).

Hence the square of the r.m.s displacement is `Delta^2 = 1.04 xx 10^(-9) m^2`. Substituting `Delta^2 and t = 300 s`. into Einstein.s formula we obtain
`k (pi eta r Delta^2)/(Tt) = (pi xx 8.9 xx 10^(-4) xx 4.4 xx 10^(-7) xx 1.04 xx 10^(-9))/(300 xx 300) = 5.9 xx 10^(26) "k mole"^(-1)`.
We would advise the reader to carry out similar calculations using other parts of the graph and to assess the inherent error in the method.