Q: A scientist measures the root mean square distance of a diffusing particle in one dimension to be $$x_{\mathrm{rms}} = 5 cm $$after $$t = 10 s. If $$the diffusion coefficient is $$D$$, what is its value?

$$D = \frac{5^2$}{2 \times 10}$$

Q: A pollen grain is observed to diffuse in water. If the diffusion coefficient is $$D = 2 \times 10$$^{-9}$$ \ \mathrm{m}^2/\mathrm{s}$$, what is the expected root mean square displacement after $$t = 100 s in $$three dimensions?

$$r_{\mathrm{rms}} = \sqrt{6 \times 2 \times 10$$^{-9}$$ \times 100}$$

Question 1

A particle undergoes a random walk in one dimension, taking $$n$$ steps of length $$d$$ each. What is the root mean square (rms) distance from the origin after $$n$$ steps?

Accepted Answer

$$x_{\mathrm{rms}} = \sqrt{n} d$$

Question 2

Given the total time $$t = n \Delta t$$ for $$n$$ steps of duration $$\Delta t$$ each, and step length $$d$$, what is the expression for the root mean square velocity $$v_{\mathrm{rms}}$$?

Accepted Answer

$$v_{\mathrm{rms}} = \frac{d}{\Delta t}$$

Question 3

The diffusion coefficient $$D is $$defined as $$D = \frac{n d^2$}{2 t}. $$What does the diffusion coefficient physically represent?

Accepted Answer

The rate at which particles spread out from their initial position over time

Question 4

If the mean square distance is $$\langle x^2$ \rangle = 2 D t$$, what is the root mean square distance $$x_{\mathrm{rms}}$$ after time $$t$$?

Accepted Answer

$$x_{\mathrm{rms}} = \sqrt{2 D t}$$

Question 5

A scientist measures the root mean square distance of a diffusing particle in one dimension to be $$x_{\mathrm{rms}} = 5 cm $$after $$t = 10 s. If $$the diffusion coefficient is $$D$$, what is its value?

Accepted Answer

$$D = \frac{5^2$}{2 	imes 10}$$

Question 6

In three dimensions, the mean square displacement after time $$t is $$given by $$\langle r^2$ \rangle = 6 D t. $$What is the root mean square displacement $$r_{\mathrm{rms}}$$?

Accepted Answer

$$r_{\mathrm{rms}} = \sqrt{6 D t}$$

Question 7

A pollen grain is observed to diffuse in water. If the diffusion coefficient is $$D = 2 	imes 10$$^{-9}$$ \ \mathrm{m}^2/\mathrm{s}$$, what is the expected root mean square displacement after $$t = 100 s in $$three dimensions?

Accepted Answer

$$r_{\mathrm{rms}} = \sqrt{6 	imes 2 	imes 10$$^{-9}$$ 	imes 100}$$

Question 8

Which of the following best describes the relationship between the mean square displacement and time for a diffusing particle?

Accepted Answer

It is proportional to time

Question 9

If a molecule diffuses a root mean square distance of $$x_{\mathrm{rms}} in $$time $$t$$, how long will it take to diffuse a distance of $$2 x_{\mathrm{rms}}$$?

Accepted Answer

$$4 t$$

Question 10

For a random walk in three dimensions, the mean square displacement is given by $$\langle r^2$ \rangle = \langle x^2$ \rangle + \langle y^2$ \rangle + \langle z^2$ \rangle. If$$ $$\langle x^2$ \rangle = \langle y^2$ \rangle = \langle z^2$ \rangle = 2 D t$$, what is $$\langle r^2$ \rangle$$?

Accepted Answer

$$\langle r^2$ \rangle = 6 D t$$

Question 11

A student claims that the diffusion coefficient $$D$$ increases if the step length $$d is $$decreased, keeping all other parameters constant. Is this correct?

Accepted Answer

No, because $$D is $$directly proportional to $$d^2$$

Question 12

Which of the following equations correctly relates the mean square displacement to the diffusion coefficient in one dimension?

Accepted Answer

$$\langle x^2$ \rangle = 2 D t$$

Question 13

If the time interval per step $$\Delta t is $$doubled while keeping the step length $$d$$ constant, what happens to the diffusion coefficient $$D$$?

Accepted Answer

It is halved

Question 14

A biologist observes that a certain protein diffuses with $$D = 1 	imes 10$$^{-12}$$ \ \mathrm{m}^2/\mathrm{s}. $$What is the expected mean square displacement after $$t = 1$$ hour in one dimension?

Accepted Answer

$$\langle x^2$ angle = 2 	imes 1 	imes 10$$^{-12}$$ 	imes 3600$$

Question 15

Which of the following statements about diffusion is correct?

Accepted Answer

The mean square displacement increases linearly with time

Question 16

A molecule diffuses in a cell, and its root mean square displacement after $$t$$ seconds is $$x_{\mathrm{rms}} = \sqrt{2 D t}. If $$the diffusion coefficient is doubled, what happens to $$x_{\mathrm{rms}}$$?

Accepted Answer

It increases by a factor of $$\sqrt{2}$$

Random Walk and Diffusion: Root Mean Square Distance and Diffusion Coefficient

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