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# Note 6 - Note 6 Equilibrium II Applications of free energy...

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1 Note 6. Equilibrium II : Applications of free energy concepts 6.1 Phase Diagram: Clapeyron equation We’ve talked about phase equilibria in a couple of different ways (i.e. in terms of free energy (pg 5 in Note 5) and in terms of chemical potentials (in pg 14 in Note 5, if you consider A and B are two phases of a substance). The Clapeyron equation connects P , V , T and H in a new way, which is often useful for thinking about phase transitions. Consider two phases, a liquid and its vapor in equilibrium at temperature T and pressure P . If we slightly change T and P , we get the new values T + dT and P + dP . Since dG = V dP S dT we can write the change in free energies for both the liquid and gas phases: dG l = V l dP - S l dT dG g = V g dP - S g dT Under the new conditions (i.e. temperature T + dT and pressure P + dP ), we can calculate equilibrium by dG l = dG g . Equating these two we get V l dP - S l dT = V g dP - S g dT and by rearranging, we can put this in the form vap vap l g l g V S V V S S dT dP Δ Δ = Since at Δ G vap = G g - G l = 0 the two phases are in equilibrium, we can write Δ G vap = Δ H vap T trans Δ S vap = 0 trans vap vap T H S Δ = Δ Note that this is equivalent to say that the chemical potentials for gas and liquid are the same i.e. dG l = dG g μ (liquid) = μ (gas). Taking this and putting it into our previous formula, we get vap trans vap vap vap V T H V S dT dP Δ Δ = Δ Δ =

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