Questions: Chemical Potential and Thermodynamic Equilibrium

Question 1

A sealed container holds liquid water and water vapor in equilibrium at 25°C. The pressure is then slightly increased, compressing the vapor. In what direction will molecules spontaneously move, and why?

• A. Vapor condenses into liquid, because compression raises the chemical potential of the vapor above that of the liquid
• B. Liquid evaporates into vapor, because compression increases the energy of the liquid phase
• C. No net transfer occurs, because the system was already at equilibrium and pressure does not affect chemical potential
• D. Molecules transfer from liquid to vapor to restore the original pressure

Answer: (A)

Matter spontaneously flows from high chemical potential to low chemical potential — exactly as heat flows from hot to cold. Compressing the vapor raises its chemical potential above that of the liquid, so molecules condense until the potentials re-equalize. This is why the equilibrium condition is μ_liquid = μ_vapor, not equal numbers of molecules or equal pressures on each side. Option C is wrong because the perturbation broke the equality; the system must respond.

Question 2

A student dissolves table salt in water and observes that the boiling point rises. Which explanation correctly uses chemical potential?

• A. Salt raises the chemical potential of water vapor, making it harder for vapor to escape the liquid
• B. Salt lowers the chemical potential of liquid water (via the RT ln x term), so a higher temperature is needed to make μ_liquid equal μ_vapor
• C. Salt increases the kinetic energy of water molecules, so more heat is needed before they can escape
• D. Salt forms hydrogen bonds with water that must be broken before boiling can occur

Answer: (B)

Dissolving a solute lowers the chemical potential of the solvent: μ_water(solution) = μ_water° + RT ln x_water, and since x_water < 1, the RT ln x term is negative. The liquid's chemical potential is now lower than the pure liquid's. For boiling to occur, the liquid and vapor chemical potentials must be equal — but the vapor hasn't changed, so you must heat the solution to raise the liquid's chemical potential until it matches the vapor's. That is why the boiling point rises. Options C and D describe incorrect physical mechanisms.

Question 3

At thermodynamic equilibrium, the chemical potential of a substance is equal in all phases it occupies.

• T. True
• F. False

Answer: True

This is the fundamental equilibrium condition. If chemical potential were higher in one phase, molecules would spontaneously transfer to the lower-potential phase — by definition, not equilibrium. The equality μ_liquid = μ_vapor = μ_solid (for coexisting phases) is the thermodynamic expression of phase equilibrium. It unifies ice melting, evaporation, osmotic equilibrium, and chemical reaction equilibrium under one principle: the system reaches its minimum total free energy when all chemical potentials are equalized.

Question 4

Adding a non-volatile solute to a solvent raises the chemical potential of the solvent, which is why the solvent needs a higher temperature to boil.

• T. True
• F. False

Answer: False

Dissolving a solute LOWERS the chemical potential of the solvent. In an ideal solution, μ_solvent = μ_solvent° + RT ln x_solvent. Since the mole fraction of solvent x_solvent < 1 in a solution, the logarithm is negative, reducing the chemical potential. This lowering is what causes all colligative properties: reduced vapor pressure, elevated boiling point, depressed freezing point, and osmotic pressure. The boiling point rises not because the solvent needs more energy, but because the lower chemical potential of the liquid means a higher temperature is needed for the liquid and vapor potentials to equalize.

Question 5

Why does dissolving any solute in a solvent lower the solvent's vapor pressure, raise its boiling point, and depress its freezing point — all from the same underlying cause?

Model answer: All three colligative properties arise because the solute lowers the chemical potential of the solvent. The mole fraction of the solvent is less than 1 in any solution, so μ_solvent = μ_solvent° + RT ln x_solvent is reduced below the pure solvent's value. Vapor pressure drops because equilibrium between liquid and vapor requires equal chemical potentials — the lower liquid μ is matched at a lower vapor pressure (Raoult's law). Boiling point rises because you must heat the solution to bring the liquid μ back up to meet the vapor's μ. Freezing point drops because the liquid's lower μ now falls below the solid's μ at the normal freezing point, so cooling is needed to bring the solid's μ down to match. The solute's specific identity doesn't matter — only how much it lowers x_solvent.

This is the power of the chemical potential framework: one equation (μᵢ = μᵢ° + RT ln xᵢ for ideal solutions) explains an entire class of phenomena without invoking different physical mechanisms for each. The chemical potential is the thermodynamic 'pressure' driving all spontaneous transfer, and its modification by concentration is the single upstream cause of all colligative effects.