Phase Equilibria in a One-Component System I

Size: px

Start display at page:

Download "Phase Equilibria in a One-Component System I"

Flora Payne
5 years ago
Views:

1 5.60 spring 2005 Lecture #17 page 1 Phase Equilibria in a One-Component System I Goal: Understand the general phenomenology of phase transitions and phase coexistence conditions for a single component system. The Chemical Potential µ controls phase transitions and phase equilibria, as well as equilibrium in chemical reactions. For example, consider two phases (liquid and solid) of water at a fixed (T, p). If µ s (T, p) = µ l (T, p) then liquid water and ice coexist If µ s (T, p) > µ l (T, p) then the water is in the liquid phase If µ s (T, p) < µ l (T, p) then the water is in the solid phase Phase diagrams Describe the phase properties as a function of state variables, for example in terms of (T, p).

2 5.60 spring 2005 Lecture #17 page 2 For the melting line, for example, solid and liquid coexist, and µ s (T, p) = µ l (T, p) One equation [µ s (T, p) = µ l (T, p)], two variables (T, p). This means that coexistence of two phases is described by T(p) or p(t). e.g. a line in the (T, p) phase diagram. At the triple point, the chemical potential of all three phases are the same solid, liquid and gas coexist. µ s (T, p) = µ l (T, p) = µ g (T, p) Two equations, two variables. This defines a unique point (T t, p t ) in the (T, p) phase diagram. For H 2 O, T t = K and p t = bar (Recall the definition of the temperature scale in Lecture # 2) In the single phase (planar) regions of the diagram, one of the chemical potentials is lower than the other two. T and p can be changed independently without changing phases. The above cases are summarized by a phase rule for a one component system: F = 3 p Where F is the number of independent variables (also called the number of degrees of freedom) and p is the number of phases that coexist. This is a specific case of the "Gibbs' Phase Rule" which will be presented in more detail next week.

3 5.60 spring 2005 Lecture #17 page 3 Can we understand the shape (i.e. slope) of the coexistence lines? For example, can we get an equation for (dp/dt) coexistence? Goal: to be able to predict, using state functions, phase transitions and equilibria. Let α and β be two phases (e.g. α,β are l, s, or g). On a coexistence curve, µ α (T, p) = µ β (T, p) Now take T T + dt and p p + dp, staying on the coexistence line. So then µ α µ α + dµ α and µ β µ β + dµ β AND dµ α = dµ β But since implies that d µ = dg = SdV + Vdp, having dµ α = dµ β S dt + V dp = S dt V dp on the coexistence line. α α β + β That means dp dt coexist S = V β β S α V α ΔS = ΔV α β Another way to write this is using µ = G = H TS so that µ α = µ β on the coexistence line implies H α TS α = H β TS β or ΔH = TΔS. Using this then we obtain the two forms of α β α β the Clapeyron Equation (These are always valid) dp dt coexist = ΔS ΔV α β or dp dt coexist = ΔH TΔV α β

4 5.60 spring 2005 Lecture #17 page 4 Let s now use the Clapeyron equation to obtain a qualitative understanding of the phase diagram. l g dp ΔS ΔS > 0, ΔV >> 0 = > 0, but small dt coexist (not steep) P _ ΔV g g s g T Δ S >> 0, ΔV >> 0 dp ΔS = dt coexist ΔV s g > 0, and steeper than for l g P s g T

5 5.60 spring 2005 Lecture #17 page 5 s l For most substances V l V s (almost equal) and S > S s dp ΔS = > 0, and very steep dt coexist ΔV s P s L _ g T For most substances, raising the pressure above a liquid near the liq to s coexistence line can cause it to freeze Except for one of the most important substances on earth: H 2 O. In dp this case V < V s, so that <0. dt coexist Thus increasing the pressure above ice can cause it to melt, and the bottom of the ocean is not ice, but liquid water at 4 C. Interestingly enough, silicon shows similar behavior (at much higher temperatures). Critical Point and Supercritical Fluids Does l g coexistence curve extend to indefinitely high T and P? NO it stops at critical point (T c, P c )

6 5.60 spring 2005 Lecture #17 page 6 Above (T c, P c ), l and g become indistinguishable: single fluid phase for all T > T c, P > P c. Supercritical fluids are finding remarkably practical applications. Supercritical water (T c = 375 C, P c = 221 bar): organic molecules readily soluble inorganic salts nearly insoluble organic compounds oxidized to CO 2, N 2, mineral salts Supercritical carbon dioxide (T c = 31 C, P c = 75 bar): reaction solvent, replaces chlorinated and volatile organic compounds dry cleaning solvent, replaces perchloroethylene

The Second Law of Thermodynamics (Chapter 4)

The Second Law of Thermodynamics (Chapter 4) First Law: Energy of universe is constant: ΔE system = - ΔE surroundings Second Law: New variable, S, entropy. Changes in S, ΔS, tell us which processes made