Measuring Water Temperature & Diffusion Coefficient with Pollen

[alfab]

Problem:
We wish to find the temperature and the diffusion coefficient of water by measuring the velocity of pollen grains in the medium, due to brownian motion and other forces.

Attempt:
We have a video clip and are using the program Tracker to measure the position of the pollen grains over time. We are using Einsteins equation for the diffusion coefficient

D = kT/γ

where k is Boltzmann's constant, T is the absolute temperature, and γ is the friction coefficient.

Using the equipartition theorem:

(1/2)m<v²>=(1/2)kT
We found:

D = m<v²>/6πηr
where η is the viscosity of the water, r is the radius of the pollen grain, m is the mass, and v is the velocity of each grain.

The issue:

After doing initial calculations we realized that using the velocities of the pollen grains give a very low temperature as the equipartition function is for the molecules of the water. Is there a way to rewrite this to get the temperature with respect to the grains.

 

[gtoguy97]

Solution:Yes, there is a way to rewrite the equation to get the temperature with respect to the pollen grains. We can use the Stokes-Einstein equation to calculate the temperature based on the diffusion coefficient and the grain radius: T = 6πηDr/k This equation can be used to calculate the temperature of the water directly from the measured velocities of the pollen grains.