Q: A molecule undergoes diffusion in water at 25°C with a diffusion coefficient $$D = 2.3 \times 10$$^{-9}$$ \ \mathrm{m}^2/\mathrm{s}. $$What is the root mean square (rms) distance $$r_{\mathrm{rms}} it $$will typically diffuse in $$t = 1$$ second?

$$r_{\mathrm{rms}} = \sqrt{6Dt} = \sqrt{6 \times 2.3 \times 10$$^{-9}$$ \times 1} \approx 3.7 \times 10$$^{-5}$$ \ \mathrm{m}$$

Q: A particle diffuses a typical distance of $$10\ \mu\mathrm{m} in $$one second. Roughly how far will it diffuse in ten seconds?

$$r_{\mathrm{rms}} = 10\ \mu\mathrm{m} \times \sqrt{10} \approx 32\ \mu\mathrm{m}$$

Q: A glucose molecule diffuses in water at $$25^\circC $$with $$D = 7.1 \times 10$$^{-10}$$ \ \mathrm{m}^2/\mathrm{s}. $$How long would it take to diffuse a typical distance of $$10\ \mu\mathrm{m}$$?

$$t = \frac{L^2$}{6D} = \frac{(10 \times 10$$^{-6}$$)^2$$}{6 \times 7.1 \times $$10^{-10}$$} \approx 2.35\ \mathrm{s}$

Q: A cell is $$20\ \mu\mathrm{m} in $$diameter. Using $$D = 2.3 \times 10$$^{-9}$$ \ \mathrm{m}^2/\mathrm{s}$$ for oxygen in water, estimate the time required for oxygen to diffuse across the cell.

$$t = \frac{L^2$}{6D} = \frac{(20 \times 10$$^{-6}$$)^2$$}{6 \times 2.3 \times $$10^{-9}$$} \approx 0.029\ \mathrm{s}$

Question 1

A molecule undergoes diffusion in water at 25°C with a diffusion coefficient $$D = 2.3 	imes 10$$^{-9}$$ \ \mathrm{m}^2/\mathrm{s}. $$What is the root mean square (rms) distance $$r_{\mathrm{rms}} it $$will typically diffuse in $$t = 1$$ second?

Accepted Answer

$$r_{\mathrm{rms}} = \sqrt{6Dt} = \sqrt{6 	imes 2.3 	imes 10$$^{-9}$$ 	imes 1} \approx 3.7 	imes 10$$^{-5}$$ \ \mathrm{m}$$

Question 2

According to Fick’s First Law, which of the following correctly expresses the rate of molecular diffusion $$N/\Delta t$$ through a channel of length $$L$$ and cross-sectional area $$A$$?

Accepted Answer

$$\frac{N}{\Delta t} = \frac{DA}{L} \Delta c$$

Question 3

A typical odor molecule in air has a diffusion coefficient $$D = 0.1 \ \mathrm{cm}^2/\mathrm{s}. If $$your friend peels an orange 1 meter away in still air, why do you smell the orange?

Accepted Answer

Both diffusion and air currents contribute with roughly equal probability.

Question 4

A particle diffuses a typical distance of $$10\ \mu\mathrm{m} in $$one second. Roughly how far will it diffuse in ten seconds?

Accepted Answer

$$r_{\mathrm{rms}} = 10\ \mu\mathrm{m} 	imes \sqrt{10} \approx 32\ \mu\mathrm{m}$$

Question 5

A glucose molecule diffuses in water at $$25^\circC $$with $$D = 7.1 	imes 10$$^{-10}$$ \ \mathrm{m}^2/\mathrm{s}. $$How long would it take to diffuse a typical distance of $$10\ \mu\mathrm{m}$$?

Accepted Answer

$$t = \frac{L^2$}{6D} = \frac{(10 	imes 10$$^{-6}$$)^2$$}{6 	imes 7.1 	imes $$10^{-10}$$} \approx 2.35\ \mathrm{s}$

Question 6

Which of the following is a reason why cells cannot be arbitrarily large, based on diffusion?

Accepted Answer

Diffusion time increases with the square of distance, so large cells would not receive nutrients quickly enough.

Question 7

Which of the following equations correctly relates the average velocity of fluid flow through a pipe to the pressure difference $$\Delta p$$, viscosity $$\eta$$, and pipe dimensions?

Accepted Answer

$$v_{\mathrm{avg}} = \frac{A}{8\pi \eta L} \Delta p$$

Question 8

If the cross-sectional area $$A of a $$channel is doubled while all other parameters remain constant, how does the rate of diffusion $$N/\Delta t$$ change according to Fick’s Law?

Accepted Answer

It doubles.

Question 9

Which of the following substances has the highest diffusion coefficient in water at $$25^\circC$$, based on the provided data?

Accepted Answer

Urea

Question 10

A cell is $$20\ \mu\mathrm{m} in $$diameter. Using $$D = 2.3 	imes 10$$^{-9}$$ \ \mathrm{m}^2/\mathrm{s}$$ for oxygen in water, estimate the time required for oxygen to diffuse across the cell.

Accepted Answer

$$t = \frac{L^2$}{6D} = \frac{(20 	imes 10$$^{-6}$$)^2$$}{6 	imes 2.3 	imes $$10^{-9}$$} \approx 0.029\ \mathrm{s}$

Question 11

Which of the following best describes the relationship between Brownian motion and entropy?

Accepted Answer

Brownian motion is a manifestation of the random thermal motion of particles, which increases the system's entropy over time.

Question 12

If the concentration gradient $$\Delta c$$ across a membrane is doubled, what happens to the rate of diffusion $$N/\Delta t$$ according to Fick’s Law?

Accepted Answer

It doubles.

Question 13

Which of the following is NOT a similarity between Fick’s Law, Fourier’s Law for heat conduction, and the Hagen-Poiseuille equation for fluid flow?

Accepted Answer

All require the transported quantity to be electrically charged.

Question 14

A red blood cell has an average volume of $$100\ \mu\mathrm{m}^3. If $$the cell is roughly spherical, what is its approximate diameter?

Accepted Answer

$$d = 2 \left( \frac{3V}{4\pi} \$$right)^{1/3}$$ = 2 \left( \frac{3 	imes 100}{4\pi} \$$right)^{1/3}$$ \approx 5.8\ \mu\mathrm{m}$$

Question 15

Which of the following best explains why oxygen diffuses more rapidly in air than in water?

Accepted Answer

The diffusion coefficient of oxygen in air is much larger than in water.

Question 16

If the length $$L of a $$diffusion channel is halved, what happens to the rate of diffusion $$N/\Delta t$$ according to Fick’s Law?

Accepted Answer

It doubles.

Question 17

A cell relies on diffusion to obtain nutrients. If the cell’s diameter increases by a factor of 3, by what factor does the time required for diffusion across the cell change?

Accepted Answer

It increases by a factor of 9.

Question 18

Which of the following best describes the trend in dissolved oxygen concentration in a lake during the summer, based on the provided graph?

Accepted Answer

Oxygen concentration tends to decrease as temperature increases.

Question 19

Which of the following is the best estimate for the size of a typical mammalian cell based on the provided scale?

Accepted Answer

$$10^{-5}$$\ \mathrm{m}$$

Diffusion and Fick’s Law: Physics with Calculus Study Notes

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