MIT 3.00 Fall 2002 175 c W.C Carter ° Lecture 26 The Gibbs Phase Rule and its Application Last Time Symmetry and Thermodynamics Cf + 2 Variables C(f − 1) Equations for Continuity of Chemical Potential f Gibbs-Duhem Relations (one for each phase) D = C − f − 2 Degrees of Freedom Left Over The Gibbs Phase Rule D+f =C +2 (26-1) The Gibbs phase rule is a very useful equation because it put precise limits on the number of phases f that can be simultaneously in equilibrium for a given number of components. What does Equation 26-1 mean? Consider the following example of a single component (pure) phase diagram C = 1. 176 c W.C Carter ° Liquid Solid P (Pressure) P (Pressure) MIT 3.00 Fall 2002 Gas (Vapor) T (Temperature) T (Temperature) Figure 26-1: A single component phase diagram. On the right figure, the color represents a molar extensive quantities (i.e., blue is a low value of V and red is a large value of V ) that apply to each phase at that particular P and T . Consider a single-phase region: D =2−f +C =2−1+1=2 This implies that two variables (P and T ) can be changed independently (i.e., pick any dP and dT ) and a single phase remains in equilibrium. Consider where two phases are in equilibrium: D = 2 − f + C = 2 − 2 + 1 = 1, There is only one degree of freedom–for the two phases to remain in equilibrium, one variable can be changed freely (for instance, dP ) but then the change in the other variable (i.e., dT ) must depend on the change of the free variables: dP = f (P, T ) dT MIT 3.00 Fall 2002 c W.C Carter ° 177 Finally, consider where three phases are in equilibrium then: D = 2 − 3 + 1 = 0. There can be no change any variable that maintains three phase equilibrium. Various Confusing Issues on Applications of D + f = C + 2 Consider a pure liquid A in contact with the air. The degrees of freedom can be determined in several equilvalent ways. A Consider the system composed of two components, the pure liquid A and air and restrict that the total pressure is 1 atm. (D + f = C + 2) → (D + f = C + 1). Therefore, D = 2 − 2 + 1 = 1. B Considering that the system consists of three components: A, O2 , N2 and has two additional restrictions: 1) ΣP = 1 PO2 /PN2 = constant, then (D + f = C + 2) → (D + f = C + 0). D = 3 − 2 + 0 = 1 as before. MIT 3.00 Fall 2002 178 c W.C Carter ° C Disregard the air: C = 1. f = 2 and therefore D = 1. The liquid has an equilibrium vapor pressure which is a function of temperature. One can pick either the vapor pressure or the temperature independently, but not both. Single Component Phase Equilibria When there is only one degree of freedom in a single component phase diagram, it was shown above that there must be a relation between dP and dT for the system to remain in two phase equilibrium. Such a relation can be derived as follows: 0 = S liquid dT − V liquid dP 0 = S solid dT − V solid dP dP =⇒ dT ¯ ¯ ∆S ∆H ¯ = = ¯ Teq. ∆V equilibrium ∆V (26-2) Equation 26-2 is the famous Clausius-Clapeyron equation. Consider the behavior of the molar free energy (or µ) on slices of Figure 26-1 at constant P and T : 179 c W.C Carter ° P (Pressure) MIT 3.00 Fall 2002 Pls(TA) PB Pvl(TA) TA Tsl(PB) Tlv(PB) T (Temperature) Figure 26-2: Considerations of the molar Gibbs free energy on slices of the single component phase diagram along lines of constant T and constant P . — G (molar Gibbs free energy) P =PB — solid G — liquid G Tsl(PB) Tlv(PB) — vapor G T (Temperature) Figure 26-3: Behavior of G = µ at constant P as a function of T . Where the curvature of G changes sign, the system is unstable. The liquid and vapor curves must be connected to each other and this is illustrated with the ”spiny-looking” curve with opposite curvature. The curve for solid is not connected to the others. 180 c W.C Carter ° MIT 3.00 Fall 2002 T =TA — G (molar Gibbs free energy) — solid G — liquid G — vapor G Pls(TA) Pvl(TA) P (Pressure) Figure 26-4: Behavior of G = µ at constant T as a function of P . Liquid Vsys P (Pressure) Gas (Vapor) Solid Liquid Gas (Vapor) Solid T (Temperature) T (Temperature) Figure 26-5: Example of single component phase diagram plotted with one derived intensive variable. What would the plot look like with two extensive variables plotted?