Questions: Diffusion Coefficients and Kinetic Molecular Theory

Question 1

A gas has a diffusion coefficient D at pressure P and temperature T. If the pressure is doubled while temperature is held constant, what happens to D?

• A. D doubles, because more collisions carry molecules farther
• B. D is halved, because the mean free path is halved by higher pressure
• C. D is unchanged, because pressure doesn't affect molecular speed
• D. D increases by T^(3/2), because temperature still dominates

Answer: (B)

From kinetic theory, D ≈ ⅓λū, where λ is the mean free path. Doubling pressure doubles the number density of molecules, which cuts the mean free path in half (molecules collide more frequently before traveling far). Average molecular speed ū is set by temperature and doesn't change. So D is halved. The common misconception is that more pressure means more 'push,' but diffusion is driven by concentration gradients and random walks, not bulk pressure.

Question 2

Which gas would you expect to diffuse most rapidly through air at the same temperature and pressure?

• A. Sulfur hexafluoride (SF₆, M = 146 g/mol), because it is a large molecule that sweeps more volume
• B. Carbon dioxide (CO₂, M = 44 g/mol), because it is a common atmospheric gas
• C. Hydrogen (H₂, M = 2 g/mol), because its high average speed and small cross-section give it the longest mean free path
• D. Nitrogen (N₂, M = 28 g/mol), because it dominates air and equilibrates quickly

Answer: (C)

D ∝ T^(3/2)/(Pσ²√m): lighter molecules move faster and, if small, also have smaller collision cross-sections σ². H₂ is both the lightest molecule and physically tiny, giving it the highest average speed and longest mean free path. SF₆ is both heavy and geometrically large — a disastrous combination for diffusion. This is the physical basis of Graham's law of effusion.

Question 3

In a liquid, the Stokes-Einstein equation D = k_BT/(6πηr) implies that a smaller solute molecule diffuses faster than a larger one, all else equal.

• T. True
• F. False

Answer: True

The Stokes-Einstein equation shows D is inversely proportional to the hydrodynamic radius r: smaller r means larger D. This makes physical sense — a smaller molecule needs to push less solvent out of the way as it moves. The same logic applies to viscosity η: more viscous solvents slow diffusion. Both r and η appear in the denominator, so decreasing either increases D.

Question 4

Diffusion is driven by molecules actively moving from high to low concentration in response to a chemical potential gradient, analogous to how a ball rolls downhill.

• T. True
• F. False

Answer: False

Diffusion is driven by random thermal motion, not directed movement toward low concentration. Individual molecules have no 'awareness' of concentration gradients. The net flux toward lower concentration emerges statistically: in a region with many more molecules on one side than the other, random walks produce more crossings from the dense side to the sparse side simply because there are more molecules to cross. Fick's law J = −D(∂c/∂x) describes the macroscopic result of this statistical asymmetry, not a molecular driving force.

Question 5

Why does increasing temperature increase the diffusion coefficient of a gas, while increasing pressure decreases it? What molecular-level mechanisms explain each effect?

Model answer: Higher temperature increases the average molecular speed (ū ∝ √T) and also increases the mean free path slightly through reduced density at fixed volume, resulting in D ∝ T^(3/2). Higher pressure, at fixed temperature, increases the number density of molecules without changing their speed — this shortens the mean free path because molecules collide more frequently, reducing D ∝ 1/P. In short: temperature controls how fast molecules move between collisions; pressure controls how far they travel before the next collision.

The key is that D depends on both molecular speed (set by temperature) and mean free path (set by collision frequency, which depends on density/pressure). These two factors are controlled by different variables. Understanding this distinction — temperature affects kinetic energy, pressure affects collision frequency — is essential for predicting how D changes under different conditions.