A container is divided by a semipermeable membrane that allows water molecules to pass freely. Compartment A has μ_water = −5.2 kJ/mol and compartment B has μ_water = −4.8 kJ/mol. In which direction do water molecules spontaneously flow?
AFrom B to A, because A has the lower chemical potential and particles flow toward lower μ
BFrom A to B, because B has the lower chemical potential — particles flow toward lower μ
CNo net flow occurs because both chemical potentials are negative, indicating equilibrium
DFrom A to B, because compartment A has a more negative value, indicating higher molecular density
Particles flow spontaneously from high chemical potential to low chemical potential, exactly as heat flows from high to low temperature. Here μ_B = −4.8 kJ/mol > μ_A = −5.2 kJ/mol, so B has the higher chemical potential and water flows from B to A. The sign of μ is irrelevant to the direction; only the difference matters.
Question 2 Multiple Choice
Water boils at 100°C and 1 atm, meaning liquid and vapor coexist with equal chemical potentials. If pressure is suddenly increased slightly, what happens and why?
AMore water evaporates, because the increased pressure raises molecular kinetic energy, pushing molecules into the gas phase
BVapor condenses into liquid, because the pressure increase lowers μ of the liquid phase below μ of the vapor
CNothing changes because the boiling point is a fixed property at 100°C regardless of pressure
DBoth phases compress equally, maintaining the same liquid-vapor ratio
At coexistence, μ_liquid = μ_vapor. Increasing pressure lowers the chemical potential of the condensed (liquid) phase relative to the vapor — the liquid becomes the energetically cheaper state for molecules. Particles flow from higher μ (vapor) to lower μ (liquid), and vapor condenses. This is why the Clausius-Clapeyron equation predicts higher boiling points at higher pressures: coexistence requires re-equalizing the two chemical potentials.
Question 3 True / False
Chemical potential plays the same role for particle exchange as temperature plays for heat flow: systems equalize their chemical potentials when particles can move between them, just as they equalize temperature when heat can flow.
TTrue
FFalse
Answer: True
This is the precise thermodynamic analogy. Temperature difference drives heat flow; pressure difference drives volume change; chemical potential difference drives particle flow. At full thermodynamic equilibrium, all three are equalized simultaneously: T₁ = T₂ (thermal equilibrium), P₁ = P₂ (mechanical equilibrium), μ₁ = μ₂ (chemical equilibrium).
Question 4 True / False
Adding a particle to a denser ideal gas costs less free energy (a lower chemical potential) because the gas has more space per molecule at higher density, making insertion easier.
TTrue
FFalse
Answer: False
The opposite is true. For an ideal gas, μ = μ₀(T) + k_BT ln(n/n₀), which increases with number density n. Inserting a particle into a denser gas is more costly because the new particle must compete with more existing particles for accessible microstates. This is why particles flow from dense regions to sparse ones in diffusion — from high μ to low μ.
Question 5 Short Answer
Explain why the condition for two phases to coexist in equilibrium is that their chemical potentials must be equal, rather than their temperatures or pressures being different.
Think about your answer, then reveal below.
Model answer: Temperature equality is required for thermal equilibrium (no heat flow) and pressure equality for mechanical equilibrium (no net volume change). But these two conditions together do not determine whether particles redistribute between phases. Chemical potential equality is specifically the condition for particle-exchange equilibrium: if μ_liquid ≠ μ_vapor, particles will spontaneously migrate to the lower-μ phase until the potentials equalize. At a phase boundary, all three equalities hold simultaneously — but it is the chemical potential equality that governs the coexistence of phases and defines the phase boundary in pressure-temperature space.
The insight is that thermodynamic equilibrium has multiple components, each associated with a different 'generalized force' and its conjugate variable. Chemical potential is the force associated with particle number. Phase coexistence requires particle-exchange equilibrium, which is precisely μ₁ = μ₂.