All Excuses must be taken to 233 Loomis before 4:15, Monday, April 30.

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1 Miscellaneous Notes he end is near don t get behind. All Excuses ust be taken to 233 Loois before 4:15, Monday, April 30. he PHYS 213 final exa ties are * 8-10 AM, Monday, May 7 * 8-10 AM, uesday, May 8 and * 1:30-3:30 PM, Friday, May 11. he deadline for changing your final exa tie is 10p, Monday, April 30. Hoework 6 is due uesday, May 1 at 8 a. (NO late turn-in). Course Survey = 2 bonus points (now on SartPhysics due Wed. May 2) Lecture 18, p 1

2 Lecture 20 Phase ransitions Phase diagras Latent heats Phase-transition fun Reading for this Lecture: Eleents Ch 13 Lecture 20, p 2

3 Solids, Liquids and Gases Solid Liquid Gas Solids have fixed relationships aong the atos (or olecules) Liquids have looser relationships aong atos. he liquid has ore entropy, because there are ore ways to arrange atos in the liquid. In liquids there are still soe correlations between atos, but in gases there are essentially none. In ost situations the entropy of the gas is vastly larger than that of the liquid or solid. Lecture 20, p 3

4 Phase ransitions and Entropy If S gas > S liguid > S solid, why are different phases stable at different teperatures? he answer is that we ust also consider the entropy of the environent. hat s what free energy and cheical potential do for us. For exaple: At low enough teperatures a substance like water is a solid. Its entropy is lower than that of the liquid so it ust give up enough energy to its environent to ake the total entropy increase when ice fors: ice Liquid H 2 O Solid: DS tot = DS L +DS S + DS env 0 < 0 > 0 In order for this to work, enough heat ust be given to the environent to ake DS tot 0. OK. So, why is liquid H 2 O favored at higher teperatures? he relative sizes of the DS ters ust be different. water Let s look at the proble ore quantitatively. Lecture 20, p 4

5 Solid-gas equilibriu: vapor pressure Consider solid-gas equilibriu at constant volue and teperature. Soe substances (e.g., CO 2 ) don t exist as liquids at atospheric pressure. he solid has negligible entropy (copared to the gas), so we ll ignore it. In that case: F s = U s -S s = -ND s = -D he gas: g = k ln(n/n Q ) Binding energy of an ato in the solid Set the equal and solve for the equilibriu gas pressure: n p k ln = k ln = D p pvapor pqe n Q p Q he equilibriu pressure is called the vapor pressure of the solid at teperature. If p < p vapor, atos will leave the solid until p gas = p vapor. his is called subliation. For liquids, it s called evaporation. - D/k V, Exaples: Si (28 g/ol): p vapor = ( at)(28) 3/2 e -3eV/.026eV = at CO 2 (44 g/ol): p vapor = ( at)(44) 3/2 e -0.26eV/.026eV = 535 at Soe solids don t subliate. Soe do. Question: Does water ice subliate? Note the different D s Lecture 20, p 5

6 Cheical Potential of an Ideal Gas Reeber: ln n p g k k ln, where pq nqk n p Q It s negative, because (unless highly copressed) n Q >> n and p Q >> p. 4 For heliu at = 300K and p = 1 at: pq p 6 So, He k ln k ln( ) (.026 ev)( 12.69) 0.33 ev p Q All ideal gases will behave siilarly, with a logarithic pressure dependence and an approxiately linear teperature dependence. (he curvature is due to the dependence of n Q.) Q at at Here s a graph: Increase the teperature, holding p constant:: decreases K Increase the pressure, holding constant:: increases ev p > 1 at p < 1 at p = 1 at Lecture 20, p 6

7 Vapor Pressure of a Solid and the p- Phase Diagra Last lecture we saw that solid is constant (-D). We can use this to deterine graphically the equilibriu pressure (vapor pressure) of the solid as a function of. For a given teperature, at what pressure does the curve cross = -D? p > 1 at -D Equilibriu p = 1 at p < 1 at Fro this, we can plot the equilibriu curve, p(), where the two phases can coexist. his graph is called a p- phase diagra. It tells us what regions in the (p,) plane ake a gas and which ake a solid. he equilibriu curve separates the. It s just a way to visualize the equilibriu equation: p g s k ln D p pqe p Q p Solid Gas D/ k At low or high p, equilibriu is all solid. he phases coexist on the equilibriu curve. At high or low p, equilibriu is all gas. Lecture 20, p 7

8 Solids and Liquids, More Accurately We ignored the entropy of the solid, because the entropy of the gas was so uch larger. When coparing solids and liquids, we can t ignore entropy. he entropy is sall, but not copletely negligible. Copare solids and liquids (for a given substance): D S > D L. Atos in the solid are ore strongly bound. S S < S L. Atos in the liquid have ore available icrostates (due to otion). Copare the cheical potentials: Reeber that F = U-S, and is the free energy per particle. At very low, the entropy is not iportant, so S < L. As increases, both cheical potentials decrease, but the liquid decreases faster. herefore, the solid phase is stable at low teperature, while the liquid phase is stable at high teperatures. here is an equilibriu teperature, the elting/freezing teperature. As the substance passes through freeze, there is a phase transition. D L D S freeze Liquid-solid equilibriu L S Lecture 20, p 8

9 Act 1: Solid-Liquid 1) Which point corresponds to a liquid? A B C A B C S L 2) he substance is in state C. What will happen? A) he substance will elt B) Free energy will iniize itself C) Entropy will axiize Lecture 20, p 9

10 Solution 1) Which point corresponds to a liquid? A B C Both points lie on the liquid s () curve. Point A does not lie on the solid s curve, so it ust be liquid A B C L S (but not one in equilibriu [lowest possible ]. At point B the liquid and solid are in equilibriu. We don t actually know how uch of each there is. FYI: Global equilibriu condition: Miniu G = N g g + N s s + N l l G = Gibbs Free Energy Except at a coexistence teperature, c, two of the N s will be zero in equilibriu. 2) he substance is in state C. What will happen? A) he substance will elt B) Free energy will iniize itself C) Entropy will axiize Lecture 20, p 10

11 Solution 1) Which point corresponds to a liquid? A B C Both points lie on the liquid s () curve. Point A does not lie on the solid s curve, so it ust be liquid A B C L S (but not one in equilibriu [lowest possible ]. At point B the liquid and solid are in equilibriu. We don t actually know how uch of each there is. FYI: Global equilibriu condition: Miniu G = N g g + N s s + N l l G = Gibbs Free Energy Except at a coexistence teperature, c, two of the N s will be zero in equilibriu. 2) he substance is in state C. What will happen? A) he substance will elt B) Free energy will iniize itself C) Entropy will axiize B and C are equivalent. hat s what systes do unless we interfere. A is also correct, because L < S at that teperature. Lecture 20, p 11

12 he p- Diagra with hree Phases p 1 p 2 p 3 Gas Solid Liquid p p 3 p 2 p 1 Phase Diagra: Solid Liquid Gas At a given p,, the syste will reside in the lowest. Phase transitions occur where the () curves cross. If we raise at constant p, the sequence of phase transitions depends on the value of p. he solid and liquid curves don t change with pressure, because solids and liquids are nearly incopressible. Lecture 20, p 12

13 he p- Diagra with hree Phases p 1 p 2 1 p 3 Gas Solid Liquid p p 3 p 2 p 1 Phase Diagra: Solid Liquid Gas At low pressure, the liquid phase is not stable for any. he substance sublies at teperature 1. If we increase the pressure, the gas curve oves to the right. here is a critical pressure for which all three curves pass through a point. he three phases can coexist only at ( 2,p 2 ). At high pressure, there are two phase transitions: solid-liquid at 3, and liquid-gas at 4. Lecture 20, p 13

14 Act 2: Out of Equilibriu 1) A syste starts in the state shown. What happens? A) Substance freezes B) Substance elts C) Substance evaporates Liquid Gas Solid 2) How could we ake this happen ore quickly? Lecture 20, p 14

15 Solution 1) A syste starts in the state shown. What happens? A) Substance freezes B) Substance elts C) Substance evaporates Liquid Gas It s a liquid out of equilibriu. he cheical potential of the gas phase is lower, so the liquid evaporates Solid 2) How could we ake this happen ore quickly? Lecture 20, p 15

16 Solution 1) A syste starts in the state shown. What happens? A) Substance freezes B) Substance elts C) Substance evaporates Liquid Gas It s a liquid out of equilibriu. he cheical potential of the gas phase is lower, so the liquid evaporates Solid 2) How could we ake this happen ore quickly? he rate depends on the cheical potential difference. So: We could increase, because the gas curve falls faster than the liquid curve. hat s why war water evaporates ore quickly. We could decrease p (aking the gas curve ove to the left). Note: If several gases are present (as in air) we only need to lower the pressure of the one we are interested in. I didn t justify this, so you ll have to take it on faith. Lecture 20, p 16

17 Phase ransitions: OOPS! he photos below show the results of stea cleaning a railroad tank car, then sealing it before the stea had cooled Lecture 20, p 17

18 Phase Diagras p solid liquid On special transition lines, two phases are stable. gas At very special triple points, those line cross and three phases are stable. Notice that in alost all the (p.) plane, only ONE phase is stable. Why is that? Lecture 20, p 18

19 AC 3: Phase ransitions Let s think a bit about energy flow during phase transitions. Suppose heat flows slowly into 0 0 C water containing soe ice cubes (aintaining equilibriu). What happens to the energy added to the syste? A) It wars the ice. B) It wars the water. C) It breaks ice bonds. ice Water at 0 0 C Q Lecture 20, p 19

20 Solution Let s think a bit about energy flow during phase transitions. Suppose heat flows slowly into 0 0 C water containing soe ice cubes (aintaining equilibriu). What happens to the energy added to the syste? A) It wars the ice. B) It wars the water. C) It breaks ice bonds. ice Water at 0 0 C he teperature will reain constant until all the ice is elted. Q Lecture 20, p 20

21 Latent Heat Consider a liguid-gas phase transition at constant pressure. he phase diagra doesn t tell the whole story. Look at how the syste changes as we add heat: p Solid Liquid Gas Heater: his all happens at the white dot. Plot teperature vs the aount of heat added: Reeber the 1 st law: Q = DU + pdv. Soe of the energy goes into breaking the bonds (DU), and soe goes into raising the weight (pdv). he p- diagra doesn t show either of these changes. liquid liquid +gas DH lg Heat of transforation Latent Heat gas Q he heat we need to add to effect the phase change is called latent heat. Lecture 20, p 21

22 Latent Heat (2) he heat required to convert the liquid entirely into gas is directly related to the entropy change of the aterial: Q 12 = 1 DS 12 = DU + pdv DH lg H = U + pv is defined as the Enthalpy ( heat content ) of a aterial. For phase transitions: Heat of vaporization = enthalpy change = latent heat : DH lg = H gas - H liquid = Latent Heat = Q L Latent heats are typically given in units of J/ol or J/gra Latent Heats of Evaporation Gas Boiling tep (K) DH (J/ol) DS (J/ol. K) Heliu Nitrogen 77 5, Oxygen 90 6, H 2 O , here is a siilar latent heat for the solid-liquid transition. For H 2 O at 1 at: Q L (ice-liquid) = 80 cal/g = 333 kj/kg Q L (liquid-stea) = 540 cal/g = 2256 kj/kg Q L (solid-liquid) Q L (liquid-gas) Q Lecture 20, p 22

23 Mass, (kg) D ie (s) Dt Method: easure rate of ass loss with heater off turn heater on, add Q (e.g. 150 W x 60 seconds) turn heater off, easure rate of ass loss calculate D latent heat is Q/D see Lab 4 for ore precision Lecture 20, p 23

Lecture 20. Phase Transitions. Phase diagrams. Latent heats. Phase-transition fun. Reading for this Lecture: Elements Ch 13.

Lecture 20. Phase Transitions. Phase diagrams. Latent heats. Phase-transition fun. Reading for this Lecture: Elements Ch 13. Lecture 20 Phase ransitions Phase diagrams Latent heats Phase-transition fun Reading for this Lecture: Elements Ch 13 Lecture 20, p 1 Solid-gas equilibrium: vapor pressure Consider solid-gas equilibrium