University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014

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Gervase Wade
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1 Lecture 12 7/25/14 ERD: Devoe: , , Unversty o Washngton Department o Chemstry Chemstry 452/456 Summer Quarter 2014 A. Free Energy and Changes n Composton: The Chemcal Potental We have thus ar consdered ree energy changes n sngle component systems,.e. ree energy changes accompanyng a changes n pressure o an deal gas. But many chemcal and bochemcal systems are naturally composed o a number o components and the composton o such systems may change as a result o chemcal reactons, physcal transport, etc. In such cases we must consder how the ree energy changes when compostons changes. Consder the reacton na A nb B. The ree energy change s gven by G G G G dg = dp dt + dna + dnb P Tn, A, n T B Pn, A, n n B A n PT,, n B (12.1) B PT,, na = VdP SdT + µ dn + µ dn In general The terms... A A B B dg = VdP SdT + µ dn (12.2) G µ = n P, Tn, j (12.3)...are called a chemcal potentals. The chemcal potental o a pure substance s smply the molar Gbbs energy. For a pure substance: G G µ = = = G (12.4) n PT, n We can relate partal molar quanttes usng the same dervatve equatons that we used or thermodynamc state unctons. Agan or a pure G µ V G µ S substance: = = = V and = = = S P P T T n T T P P n And the chemcal potental s an exact derental so that µ µ dµ = dt + dp = SdT + VdP (12.5) T P P T

2 B. The Crteron or Equlbrum Equlbra are dvded nto three categores: mechancal, thermal, and materal. We are prmarly concerned wth materal equlbra. Materal equlbra n turn are classed as ether reacton or phase equlbra. In the ormer substances (.e. reactants) are converted nto other substances (.e. products). In phase equlbra, substances are transported rom one phase to another. Consder a system composed o a sngle substance parttoned nto two phases α and β. The Gbbs ree energy expresson s α β dg = S dt S dt + V dp + V dp + µ dn + µ dn (12.6) α( orβ) α( orβ) G where the chemcal potental µ = α( orβ). n β( orα), T, P For ths closed system to be at equlbrum at constant T and P, P-V work only, we have α β dg = 0= µ dn + µ dn (12.7) Thereore the crteron or phase equlbrum s α β 0= µ dn + µ dn (12.8) (dt=dp=0, closed system, P-V work only). The crteron or equlbrum at constant V and T s also µ α dn α + µ β dn β = 0. Suppose an amount o the substance leaves phase α and enters phase β. α β Then dn = dn and dn = dn. Then α β β α µ dn + µ dn = µ dn + µ dn = ( µ µ ) dn = 0 (12.9) β α µ = µ The crteron or equlbrum s µ = µ, closed system, constant P and T, P-V work only. β α I the closed system has not yet reached equlbrum ( µ µ ) dn < 0. Then t also ollows that µ > µ. C. P-V-T Dagrams or One Component Systems For a sngle component system, the gas phase s the most stable phase at hgh temperatures and low pressures. As the temperature decreases and the pressure ncreases, lqud and sold phases appear. These acts are vsualzed n a P-V-T dagram, whch dsplays whch phases are most stable at a gven P-V-T. At rght s a dsplay or a substance whch

3 contracts n volume when t passes rom the lqud to the sold state. A common example o such a case s CO 2. A second case s represented by a surace where the densty o the sold s less than the densty o the lqud, so that the volume o the system ncreases n passng rom lqud to sold. Such a case s represented by water, where the densty o ce s less than the densty o lqud water. See below, rght. The lnes perpendcular to the temperature axs are sotherms. Dark lnes represent equlbrum state between the two phases n contact at a partcular P and T. The trple lne s a condton when three phases are n contact. The crtcal pont s a pont beyond whch gas-lqud equlbrum no longer occurs. Typcally occurrng at hgh pressures and temperatures there s a contnuous transton rom lqud to gas and vce versa. Three dmensonal phase dagrams are oten vewed as projectons, where the sold mage s collapsed onto the P-T or the P-V plane, as shown below or CO 2 and H 2 O. In the P-V projecton the gas phase s dened by sotherms that do not ntersect a phase equlbrum. At hgh T and low P the gas phase s the most stable phase. At lower T, a vapor s can condense to a lqud phase or at even lower T and P a vapor can rost drectly to a sold phase. The opposte o rostng...drect ormaton o a vapor rom a sold... s called sublmaton.

4 In a P-V dagram as we move along a vapor sotherm we eventually ntersect a vapor-lqud equlbrum lne. As the volume s urther decreased nto a regon where vapor and lqud coexst, the sotherms become straght lnes parallel to the V-T plane because the pressure has ceased to change. Wthn the lqud-vapor regon, as the volume decreases, vapor s converted to lqud so the pressure does not change. There sotherm/sobars are called te lnes. These te lnes represent the sothermal expansons/compressons experenced by the workng luds o heat engnes and rergerators. Once encounterng the lqud phase, even small decreases n volume requre very large pressure ncreases, so the sotherms rse steeply. From the P-T dagrams, the densty o the sold s greater than the densty o the lqud, ncreasng the pressure o the lqud wll eventually orm the sold. Ths s the case wth CO 2. I the densty o the lqud s greater than the densty o the sold, ncreasng the pressure on the lqud wll not orm the sold. Ths s the case wth water. D. Quantyng the Equlbrum Between Two Phases In P-T projectons, lnes represent equlbra between phases. From the slopes o these lnes we can obtan expresson or transton enthapes and entropes. The slopes o these lnes are quanted by the Clausus-Clapeyron equaton. Recall or equlbrum between two phases α and β at a gven P and T µ α ( = G α ) = µ β ( = G β ) (12.10) Now P and T are changed by a small amount, changng the chemcal potentals slghtly but otherwse mantanng the equlbrum we have µ α + dµ α = µ β + dµ β (12.11) Combnng and we get dµ = dµ (12.12). αβ, αβ, αβ, αβ, Now dg = dµ = S dt + V dp so α β S dt + V dp= S dt + V dp dp S S S (12.13) = = dt V V V The process o transportng a substances between phases at equlbrum s reversble so H dp S H S = and = = (12.14) T dt V T V Equaton s called the Clausus-Clapeyron equaton.

5 Example 1: Use the Clausus-Clapeyron equaton to measures the meltng temperature o water ce at P=400 atm. Soluton: A practcal use o the C.-C. equaton s to predct a meltng pont T at a pressure P above P =1 atm where T =273.15K...bascally tracng out the sold-lqud equlbrum lne n the P-T dagram. To do ths we ntegrate the C.-C. equaton: dp H dt V T V = = dp ln = ( P P) dt T V T H T H Now at T=273.15K and P=1 atm, the denstes o lqud water and ce are M ρlq = ml / gm and ρce = ml / gm. Note V = where the ρ specc volume s per mole and M s the molecular weght o water= 18gm/mole. The heat o uson or water s H = 6010 J / mole. Then 1 1 M T dt V ρ uson lq ρ ce = = ( P P) T H H T = K 1 1 ( 18 gm / mole) T gm / ml gm / ml ln = K 6010 J / mole ( 18gm)( mL) ( atm )( J mlatm ) = / = J T = K e = K ( ) uson ( atm atm) I one o the phases n equlbrum s a vapor and s treated as deal, the Clausus-Clapeyron Equaton can be smpled. There are two cases. I vapor s n equlbrum wth lqud we can a vaporzaton equlbrum. I vapor s n equlbrum wth sold we have a sublmaton equlbrum. dp H H 1 H = = dt T V T vap, subl Vvapor V lq, sold Tvap, sublvvapor From the deal gas law V RT dp H HP V = = = = 2 n P dt TV RT dln P dln P H = = 2 ( dt / T ) d( 1/ T) R Ths modcaton o the Clausus-Clapeyron equaton s used to calculate the change n the pressure o a vapor n equlbrum wth a

6 lqud or a sold as a uncton o temperature. The relevant equaton s obtaned by ntegratng both sdes o the equaton P H 1 dlnp = d 1/ T dlnp Hd 1/ T R = R ( ) ( ) P I the heat o uson or sublmaton s constant over the temperature range we can take t out o the temperature ntegraton P H 1 1 ln = P R T T T T

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,