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

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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 equilibrium in a couple of different ways. 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 (gas) in equilibrium at temperature T and pressure P . At equilibrium, the Gibbs free energies for the liquid and the vapor are identical (i.e. G l = G g ), so both phases can co-exist. 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 G l + dG l = G g + dG g . I.e. 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) = (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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2 It is important that T trans is the transition temperature (boiling point in this case). More generally, we can write V T H dT dP eq eq Δ Δ = We write this equation in this form to stress that this doesn’t just have to refer to a liquid boiling and becoming a gas, but rather the equilibrium between two phases in general. It is important to stress that this equation, called the Clapeyron equation , is very general and its derivation has not used any approximations, but simply applied the concepts of equilibrium and the Gibbs free energy. This equation can define the phase boundary in a phase diagram. 6.2 The vaporization of liquids: Phase boundary between liquid and gas 6.2.1 Clausius-Clapeyron equation We can now take the Clapeyron equation and extend it to describe the vapor pressure of gases. To do so, we will make three approximations. We start with the Clapeyron equation: vap vap V T H dT dP Δ Δ = Recall that l g vap V V V = Δ . The volume of gases is much larger than that of liquids. For example, at 300K and 1 atm pressure, V g 24,000 cm 3 while V l 100 cm 3 for one mole of typical substances. Thus, since V g >> V l , we make our first approximation and say that g l g vap V V V V = Δ Also, if we assume that the gas is ideal, then P nRT V g = Thus, putting these together, we get () P nRT H P nRT T H V T H dT dP vap vap vap vap 2 / Δ = Δ Δ Δ = we can rearrange this to get
3 dT P d PdT dP nRT H vap ln 2 = = Δ Finally, if we make the approximation that Δ H is independent of temperature , then we get const nRT H P vap + Δ = ln The equation above is called the Clausius-Clapeyron equation. As you see, it relates the vapor pressure of a liquid to the enthalpy change per mole on vaporization. It is important

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## This note was uploaded on 05/04/2010 for the course CH 43445 taught by Professor Lim during the Spring '10 term at University of Texas.

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Note 6 - Note 6. Equilibrium II: Applications of free...

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