(a) Consider a sphere of radius R in water. Due to random...

(a) Consider a sphere of radius $R$ in water. Due to random collisions with the water molecules, the sphere will rotationally diffuse. The diffusion law in this case is analogous to the one obtained for translational motion, \[\left\langle\Delta \theta^{2}\right\rangle=2 D_{\mathrm{r}} t\] What are the units of the rotational diffusion coefficient $D_{r} ?$ Write down the formula for $D_{\mathrm{r}}$ using the Einstein relation and the rotational friction coefficient obtained in Problem $12.5,$ and convince yourself that the units are correct. (b) Estimate how long it takes for an $E$. coli to diffuse over an angle equal to 1 radian. What is the distance traveled by the bacterium during that time?

(a) Consider a sphere of radius $R$ in water. Due to random collisions with the water molecules, the sphere will rotationally diffuse. The diffusion law in this case is analogous to the one obtained for translational motion,
\[\left\langle\Delta \theta^{2}\right\rangle=2 D_{\mathrm{r}} t\]
What are the units of the rotational diffusion coefficient $D_{r} ?$ Write down the formula for $D_{\mathrm{r}}$ using the Einstein relation and the rotational friction coefficient obtained in Problem $12.5,$ and convince yourself that the units are correct.
(b) Estimate how long it takes for an $E$. coli to diffuse over an angle equal to 1 radian. What is the distance traveled by the bacterium during that time?

Key Concepts

Estimation Techniques in Diffusion

Estimation techniques in diffusion problems involve using the diffusion law to approximate timescales and displacements over which significant diffusion occurs. By setting the mean squared angular displacement to a specific value (such as 1 radian squared) and solving for time, one can estimate how long rotational diffusion takes. Further, by relating the angular displacement to linear motion, approximate distances traveled by the particle can be determined, demonstrating the interconnectedness of rotational and translational dynamics.

Einstein Relation in Rotational Dynamics

The Einstein relation connects the diffusion coefficient of a particle to its mobility, providing a link between thermal energy and frictional forces. In the context of rotational diffusion, this relation typically takes the form Dr = kBT/?r, where kB is Boltzmann's constant, T is the absolute temperature, and ?r is the rotational friction coefficient. This relation not only provides insight into the underlying physics of thermal fluctuations affecting rotations but also ensures that the units balance correctly.

Rotational Friction Coefficient

The rotational friction coefficient represents the resistance a rotating body experiences due to its interaction with the surrounding medium. It is an analogue to the translational friction coefficient but for rotational motion. In problems involving spheres or other geometries, this coefficient is derived based on hydrodynamic considerations and depends on the particle’s size and the viscosity of the medium, linking into the Einstein relation for rotational motion.

Rotational Diffusion

Rotational diffusion is the process by which the orientation of a macroscopic or microscopic object changes over time due to random collisions with surrounding molecules. It is analogous to translational diffusion but concerns angular displacement rather than linear displacement. Its dynamics are usually described by the relation ???²? = 2Dr t, where Dr is the rotational diffusion coefficient and has units that capture how quickly angular variance accumulates over time.

Units and Dimensional Analysis

Dimensional analysis is a key concept that involves checking the consistency of units in physical equations. For the rotational diffusion coefficient Dr, given that the mean square angular change ???²? is measured in radians squared and time in seconds, the units of Dr must be in radians squared per second (rad²/s). This ensures that the product Dr t has the correct units to match the observed angular displacement.

Transcript

00:01 What is the diffusion coefficient given the following equation and values? so the diffusion coefficient is equal to the boltzman constant times the temperature over 6 pi aeta times the radius.

00:21 And so we need to find the radius, but the density of a sphere is given by this equation.

00:28 Density equals mass over volume, and the volume is 4 thirds, pi, are cubed.

00:42 So we can solve for the volume first because we know the mass and the density, but let's convert the mass into grams because the one value, the density isn't grams too, so we need to cancel those.

00:57 So we have m equals 100 kilo -daltons.

01:04 So let's convert from kill -daltons to dalton's.

01:10 So again, kilo -dil -daltons is 1 ,000.

01:13 So we have this.

01:14 And then 1 -dalton.

01:18 Is equal to 1 .66 times 10 of the negative, 27 kilograms.

01:27 And then let's convert from kilograms to grams, because again, the densities in grams.

01:33 So we have one kilogram is 1 ,000 grams.

01:42 And doing the math here, we have that this answer is 1 .66.

01:55 Actually, let me move that.

02:00 So we have 1 .66.

02:03 Times 10 and the negative 19 grams.