Question

So, how do we describe diffusive motion? If a particular object moves a distance Ar from its original location (displacement) in a time Δ t, we cannot write down an equation that precisely relates dr and Δ, since the direction and distance moved by the object between collisions with the fluid particles is random and the time between collisions is also random. Because of these random effects, a particular object has very little chance of ever returning to the position where it started, so at any point in time, it is very unlikely for Ar to be zero-this is qualitatively responsible for the "spreading out" effect of diffusion. If we can't predict the trajectory of any particular particle as it moves randomly, what can we predict? As you will investigate for yourselves, we can characterize the average displacement of all the objects in a large collection, (Ar), where the brackets indicate an average over the entire collection of objects. Observations (like the ones you'l make in Lab 3) and statistical theories indicate that for sufficiently large numbers of particles, where now we average the squared displacement (mean square displacement -MSD) of the collection of objects. 1 In this equation, α = 2 for one dimension (motion constrained to a line), a-4 for two dimensions (motion constrained to a plane), and α = 6 for three dimensions (motion unconstrained in three dimensions). D is the diffusion constant which depends on several factors, as you will investigate in lab.

Answer 1

1) Units of D are m²/s.

Because we can rewrite the formula as follows:

$D= \frac{<\Delta r^2>}{\alpha\Delta t}$

$\alpha$ is dimentionless. r and t have dimentions of length and time respectively. Therefore units are m²/s.

2) We can take the data by capturing a picture of the microscope slide under microscope at some known time intervals. We can measure the displacement of each particle using the number of pixels moved by the particles. We can use some smaller scale to calibrate the pixels.

$(\Delta r_i)^2=(\beta_x\Delta x_i)^2+(\beta_y\Delta y_i)^2$

where $\Delta x_i$ and $\Delta y_i$ are number of pixels moved by i^th particle in x and y directions respectively and $\beta_x$ and $\beta_y$ are calibrateed distances per pixel.

3) $<\Delta r^2>=\frac{1}{N}\sum_{i=1}^{N}(\Delta r_i)^2$ It is just the average over all particles.

We get diffusion constant from equation in first part.

4) The spheres will move away from their respective starting positions but their average displacement will be zero. $\Delta r=0$ but $<\Delta r^2>\neq0$ .

Answer 2

1) Units of D are m²/s.

Because we can rewrite the formula as follows:

$D= \frac{<\Delta r^2>}{\alpha\Delta t}$

$\alpha$ is dimentionless. r and t have dimentions of length and time respectively. Therefore units are m²/s.

2) We can take the data by capturing a picture of the microscope slide under microscope at some known time intervals. We can measure the displacement of each particle using the number of pixels moved by the particles. We can use some smaller scale to calibrate the pixels.

$(\Delta r_i)^2=(\beta_x\Delta x_i)^2+(\beta_y\Delta y_i)^2$

where $\Delta x_i$ and $\Delta y_i$ are number of pixels moved by i^th particle in x and y directions respectively and $\beta_x$ and $\beta_y$ are calibrateed distances per pixel.

3) $<\Delta r^2>=\frac{1}{N}\sum_{i=1}^{N}(\Delta r_i)^2$ It is just the average over all particles.

We get diffusion constant from equation in first part.

4) The spheres will move away from their respective starting positions but their average displacement will be zero. $\Delta r=0$ but $<\Delta r^2>\neq0$ .