Unit 39: Diffusion & the Random Walk
Practice Problems
Note: problem difficulty is ranked using a star system.
(*) One-star problems are fundamental to the unit, and can be done relatively quickly. Use these problems to introduce yourself to the material.
(**) Two-star problems are more difficult, and require an understanding of one or two key concepts. Use these problems to test your understanding of the material.
(***) Three-star problems are the most difficult, and require some creative thinking in addition to a deep familiarity with multiple key concepts. Use these problems to challenge yourself; if you can complete one of these, you’re on your way to mastering the material.
*Q39.1) Consider a sugar molecule of radius 0.5nm diffusing freely through 0°C water. You may treat the sugar molecule as spherical.
a) What’s the molecule’s diffusion coefficient?
b) What will be the molecule’s root-mean-squared displacement after 10 minutes of diffusion?
**Q39.2) Consider a population of phospholipids embedded in a water-oil interface with their hydrophilic heads in the water and their hydrophobic tails in the oil. They diffuse freely around the interface with a diffusion coefficient of 1mm²/s. How long will it take for one third of the population to have diffused at least 1cm from their initial positions?
**Q39.3) Cells use diffusion as a form of passive transport for a variety of nutrients including sugar. How long would it take one third of a population of sugar to diffuse across the entire length of a cell of length 10μm at 37°C? You may treat the sugar molecules as spherical with a radius of 0.1nm. Assume cytoplasm to have five times the viscosity of water.
***Q39.4) Your friend lights a scented candle 4m away from you. It takes some time, but the scent eventually diffuses across the room to where you’re standing. As soon as you notice the smell, you back away an additional 2m from the candle and find that it takes 2 minutes for the smell to catch up to you. What’s the diffusion coefficient in air for the molecules you smell? Ignore the effects of air currents.