CHAPTER 4 Physical Transformations of Pure Substances.

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1 I. Generalities. CHAPTER 4 Physical Transformations of Pure Substances. A. Definitions: 1. A phase of a substance is a form of matter that is uniform throughout in chemical composition and physical state. e.g., gas phase, liquid phase, and various solid phases of different atomic structure. 2. Phase transition is the spontaneous conversion of one phase into another, occurring at a characteristic temperature for given pressure. e.g. At 1 atm, ice is the most stable phase below 0 C, but liquid is most stable above 0 C. That means ice has a lower Gibbs energy per mole than the liquid at temperatures below 0 C, etc. 3. Transition temperature T tr is temperature at which the two phases are in equilibrium and each have the same molar Gibbs energy. 4. This chapter will concern itself with thermodynamics of phase behavior, not kinetics. E.g., one can lower a liquid below its freezing point and yet not see the onset of the phase transition because of the slowness of the molecules in getting reorganized into an ordered phase. This would be called supercooling in this case. 5. A persistent phase which is thermodynamically not the most stable phase at the prevailing conditions is called a metastable state. e.g. diamond is an allotrope of carbon that is thermally less stable than graphite at normal pressures, but kinetically the phase transformation is too slow to perceive. 6. The central themodynamic quantity for treating phase behavior is the chemical potential µ: µ = ( G/ n) p,t Useful for open systems, where amount n of substance is changing. Tells how G changes with n (moles) in each phase. 7. µ of perfect gas: Now since G(p 2 ) = G(p 1 ) + nrt ln p 2 /p 1 if p 1 = 1 bar (standard pressure) = p θ and p 2 = our working pressure p then G(p) = G + nrt ln p/p θ 1

2 Therefore: µ = µ + RT ln p/p θ Shows variation of chemical potential with pressure. 8. Chemical potential of a real gas (the fugacity): ideal gas: µ = µ + RT ln p/p θ real gas: replace pressure p with the fugacity f. f = γp (γ is the fugacity coefficient, γ 1 in most cases) f can be viewed as the "effective pressure". µ = µ + RT ln f/p θ = µ + RT ln p/p θ + RT ln γ where RT ln γ term is the effect of intermolecular forces and, hence, deviations from ideality γ! 1 as p! 0 γ is a measure of non-ideality and can be directly related to: - Z (compression factor) - VdW coefficients - virial coefficients B. Use of µ: pure substance Phase equilibrium exists when µ V = µ l When µ V > µ l, will be a net flow of vapor molecules into liquid. µ V < µ l, vice versa. Transition T is temp at which µ V = µ l. 2

3 II. Phase diagrams for pure substances. A. General features: Figure Phase boundaries = lines on phase diagram where two phases coexist in equilibrium. 2. Triple Point = unique point where solid, liquid and vapor can coexist simultaneously in equilibrium. µ s = µ l = µ V e.g., H 2 O T 3 = p = 4.58 Torr Also T 3 is the lowest p at which liquid phase can exist. 3. Critical Point: critical temperature T c + p c point. T c is temperature above which cannot form a liquid at any pressure. 3

4 4. Vapor pressure = the pressure of the vapor phase in equilibrium with its corresponding liquid. e.g., the liquid-vapor phase boundary from triple point to critical point thus shows how vapor pressure varies with T (governed by Clausius-Clapeyron equation) e.g., the solid-vapor phase boundary thus shows how sublimation vapor pressure varies with T. I 2 (s) "! I 2 (g) 5. Boiling Temperature = T at which vapor pressure of liquid equals the surrounding pressure. T b = normal boiling point = boiling temperature when p = 1 atm. Line from T 3 to T c shows the boiling point of the pure substance as function of pressure above the liquid. 6. Melting (or freezing) temperature = temp at which solid and liquid coexist in equilibrium. T f = normal melting pt. = melting temp when pressure = 1 atm. III. Examples of phase diagrams of pure substances. A. CO 2 - a typical substance CO 2 cannot exist as liquid at room pressure, since 1 atm is well below the triple point pressure of 5.11 atm Thus solid CO 2 sublimes (so-called dry ice ). 4

5 B. Pure H 2 O phase diagram Figure 4A.9 Special feature of pure H 2 O phase diagram: Negative slope is unorthodox. Shows that: Melting temp as p. Density of solid less than liquid phase. (H 2 O expands as it freezes) H-bonding network (see right) Figure 4A.10 5

C. Helium: Note: solid never in equilibrium with vapor TWO liquid phases: normal and superfluid Superfluid He flows without viscosity. IV. Thermodynamics of Phase Transitions of Pure Substances. A.

6 C. Helium: Note: solid never in equilibrium with vapor TWO liquid phases: normal and superfluid Superfluid He flows without viscosity. IV. Thermodynamics of Phase Transitions of Pure Substances. A. Two phases α and β in equilibrium, then: µ α = µ β B. But µ depends on T and p: 1. T-dependence: ( µ/ T) p = -S m (molar entropy) Now since always S>0, µ always as T Slopes on right diagram are = -S m Heating a region always lowers its Chemical potential 6

7 2. P-dependence ( µ/ p) T = V m (molar volume) shows chemical potential always goes up when pressure goes up. Figure 4B.2 Typical subst.: Density of solid > liquid So V m of solid is smaller than liquid. Increase in p causes freezing T to go up. Water: Density of solid < liquid So V m of solid > liquid Increase in p causes freezing T to go down. C. Derive equation for phase boundary lines in phase diagram (p vs. T): First: see µ as function of p and T dµ = ( µ/ p) T dp + ( µ/ T) p dt so dµ = V m dp S m dt Now along a phase boundary line µ α and µ β remain equal, so: dµ α = dµ β (as p and T change) 7

8 Therefore: V m,α dp - S m,α dt = V m,β dp - S m,β dt (S m,β - S m,α )dt = (V m,β -V m,α )dp Finally - Clapeyron Equation: dp dt = ΔS m ΔV m = change in molar entropy/change in molar volume as α β and since entropy change is enthalpy change/transition temp: dp dt = ΔH tr T tr ΔV m True for any phase transition, pure substance. dp/dt is slope of coexistence line on p vs. T diagram: 8

9 D. The solid-liquid line: Replace ΔS m with ΔH fus /T. dp dt = ΔH fus T f ΔV m typical behavior on right ΔH fus > 0 melting requires energy (endothermic) ΔV fus > 0 for most substances (liquid less dense and occupies larger volume than solid) Now we understand H 2 O slope: Figure 4-16 For H 2 O, ΔV fus < 0 solid occupies larger volume than liquid dp/dt = +/- = (-) negative slope exaggerated here 9

10 E. The liquid-vapor line: Substitute ΔS vap with ΔH vap /T dp dt = ΔH vap T ΔV m ΔV m = V m,vap - V m,liq 0 V m,vap RT/p dp dt = ΔH vap T (RT / p) dp dt = p ΔH vap RT 2 (dp / dt) p = ΔH vap RT 2 and so finally we obtain: dlnp dt = ΔH vap RT 2 Clausius-Clapeyron Equation Integrated form: $ p = p exp& ΔH vap & % R $ 1 T 1 '' ) & % T ) ( ) ( Gives vapor pressure at two temperatures T and T* F. The solid-vapor line: Follow all the same arguments. dlnp dt = ΔH sub RT 2 sublimation 10

11 V. Types of Phase Transitions (Ehrenfest s classification) A. First order - most melting, vaporization are examples. 1st derivative of µ vs T are discontinuous at transition. Slope changes! µ T e.g. Volume: Similarly H: 11

12 B. Second-order - order-disorder transitions, ferromagnetism. 1st derivative of µ vs T is continuous. 2nd derivative is discontinuous. µ T ΔV = 0 at transition: V T Figure 4B.10 C p grows before trans T Τ λ gel liquid crystal in lipid bilayers. 12

13 Summary: 1 st order 2 nd order Figure 4B.9 13

14 Notes: 14

Physical transformations of pure substances Boiling, freezing, and the conversion of graphite to diamond examples of phase transitions changes of

Physical transformations of pure substances Boiling, freezing, and the conversion of graphite to diamond examples of phase transitions changes of phase without change of chemical composition. In this chapter