(Data Analysis) Einstein’s theory of Brownian motion connects microscopic properties (eg, radius, diffusivity) of the beads to macroscopic properties (eg, temperature, viscosity) of the fluid in which the beads are immersed. This amazing theory enables us to estimate Avogadro’s constant with an ordinary microscope by observing the collective effect of millions of water molecules on the beads. 1. Estimating the self-diffusion constant. The self-diffusion constant D characterizes the stochastic movement of a molecule (bead) through a homogeneous medium (the water molecules) as a result of random thermal energy. The Einstein- Smoluchowski equation states that the random displacement of a bead in one dimension has a Gaussian distribution with mean zero and variance o2 = 2DAT, where At is the time interval between position measurements. That is, a molecule's mean displacement is zero and its mean square displacement is proportional to the elapsed time between measurements, with the constant of proportionality 2D. We estimate o2 by computing the variance of all observed bead displacements in the x and y directions. Let (delta x1, delta y1),...,(deltat x1, delta y1) be the n bead displacements, and let 11,...,, denote the radial displacements. Then
2 (delta x1. +...+delta x2n) + (delta y21+ ... + delta y2n)/2n
= r2n+ + r2n/2n
For our data, delta t = 0.5 so our estimate for o2 is an estimate for D as well. Note that the radial displacements in the formula above are measured in meters. The radial displacements output by your bead_tracker. py program are measured in pixels. To convert from pixels to meters, multiply by 0.175x10^6 meters per pixel). The value of n is the count of the total number of displacements read.
2. Estimating the Boltzmann constant. The Stokes-Einstein relation asserts that the self-diffusion constant D of a spherical particle immersed in a fluid is given by D = T, where, for our data T (absolute temperature) is 297 degree Kelvin (room temperature), n (viscosity of water) is 9.135x10 Nsm-2 (at room temperature), p (radius of bead) is 0.5x10-6 and k is the Boltzmann constant. All parameters are given in SI units. The Boltzmann constant is a fundamental physical constant that relates the average kinetic energy of a molecule to its temperature. Use k = bare as an estimate of Boltzmann's constant.
3. Estimating Avogadro's constant. Avogadro's constant NA is defined to be the number of particles in a mole. By definition, k = R. where the universal gas constant R is approximately 8.31457 JK 'mol. Use NA = as an estimate of Avogadro's constant. Implement a client program data analysis. py that accepts the displacements (output of bead_tracker. py) from standard input; computes an estimate of Boltzmann's constant and Avogadro's constant using the formulae described above; and writes the values separated by a space..