Non-Idealty Through Fugacty and Actvty S. Patel Deartment of Chemstry and Bochemstry, Unversty of Delaware, Newark, Delaware 19716, USA Corresondng author. E-mal: saatel@udel.edu 1
I. FUGACITY In ths dscusson, we consder one way to ncororate non-dealty nto our hyscal model of fluds and solds. It s to be arecated that once we leave the realm of dealty, we are relegated to consderng models of hyscal realty. Thus, we need to always be careful of our nterretatons arsng from such models whch, deste our best ntentons, are always aroxmatons of actual hyscal realty. Moreover, we are requred to bear n mnd the assumtons, aroxmatons, and lmtatons assocated wth such models. Wth ths reface we now consder an alternate descrton of the chemcal otental of a ure seces and a seces n varous mxtures (lquds and gases. A. Pure Gases and Gas Mxtures 1. Pure Gas The total dfferental of Gbb s Free energy for a ure flud s gven by: dg V d SdT + µdn (1 Dvdng ths extensve roerty by the moles of ure flud, we have the total dfferental of the molar Gbb s Free Energy or the chemcal otental: dµ dḡ V d SdT (2 For constant temerature, ths reduces to: dµ dḡ V d (3 For an deal gas, the molar volume s: V RT. Ths leads to: dµ dḡ RT d(ln (4 We now take nsraton from Equaton 4 for an deal gas to ntroduce a new functon, the fugacty of ure seces whch wll contan all the non-dealty of the flud. 2
dµ dḡ RT d(ln f (5 Note that fugacty s not ressure Fugacty s a functon of ressure Fugacty s a drect measure of the chemcal otental of a real, non-deal flud! Fugacty s chemcal otental! Equvalence of a seces chemcal otentals n varous hases at equlbrum coexstence s the same as the equvalence of fugactes n varous hases at equlbrum coexstence. lmt( 0 f(. Thus, f g. Now, for an deal gas the chemcal otental at a artcular temerature, T, and ressure,, relatve to some reference state chemcal otental, µ ref (T, ref, at a reference state ressure of ref s: ( µ g (T, µ g,ref (T, ref + RT ln ref (6 We can comute the same for the real gas, rg: ( f (T,,ref (T, ref + RT ln f ref (7 Takng the dfference between Equatons 7 and 6 gves: ( f (T, µ g (T,,ref (T, ref µ g,ref (T, ref ref + RT ln f ref (8 If we take the reference state to be an deal gas state (low ressure reference state, the frst two terms on the rght hand sde of Equaton 8 cancel one another, and ref f ref ; ths leads to (T, µ g (T, RT ln ( f (9 Equaton 9 allows us to defne the fugacty coeffcent for a ure seces as: φ ure gas (T, f (10 3
Fugacty coeffcent of a ure seces at a temerature, T, and ressure, P s defned as φ ure gas (T, f We can thus rewrte Equaton 9 as: (T, µ g (T, + RT ln (φ ure gas (11 If we now reference the deal gas chemcal otental to a reference state wth ressure 1 bar ( 0 1bar, we can rewrte the revous equaton as : ( (T, µ g (T, 0 + RT ln 0 + RT ln (φ ure gas (12 Fnally, we can combne the second and thrd terms on the rght-hand sde of equaton 12 to obtan the followng relaton whch we wll return to later n our dscusson of actvty. ( (T, µ g (T, 0 + RT ln (φ ure gas 0 (13 2. Gas Mxtures We can follow an analogous route as we dd for the ure gas n our treatment of the chemcal seces of a seces n a mxture of real gases. For an deal gas seces n an deal gas mxture (mxture of deal gases at a temerature, T, and ressure, P, relatve to a ure gas reference state at ref and the same temerature, we can wrte the chemcal otental as: ( µ g (T, µg (T, ref, ure + RT ln ref (14 For a real (non-deal gas (suerscrted as rg n the followng seces n a non-deal gas mxture, we can wrte analogously, makng use of our defnton of fugacty as resented n the secton on ure gases, as: (T, (T, ref, ure + RT ln ( f ref (15 Here, s the fugacty of seces n the non-deal gas mxture. Agan, the reference state s a ure flud state; thus we omt the hat on the fugacty of seces n the denomnator of fracton of whch we take the natural log. 4
Takng the dfference of Equatons 15 and 14, we obtan: (T, µ g (T, µrg (T, ref, ure µ g (T, ref, ure + RT ln ( ref Takng the reference state to be an deal gas state, we can make the smlfcatons as for the ure gas case to obtan: f ref (16 ( (T, µ g (T, RT ln (17 Snce y Total y, ( (T, µ g (T, RT ln y (18 We now defne the fugacty coeffcent for seces n a gas mxture (non-deal as ˆφ y. Ths allows us to wrte Equaton 18 as: (T, µ g (T, + RT ln ( ˆφ (19 The chemcal otental of the deal gas seces at temerature T and total ressure can be wrtten n terms of the chemcal otental of the ure deal gas at a reference ressure 0 as we dd before: (T, µ g (T, 0 + RT ln ( 0 ( + RT ln ˆφ (20 whch can be wrtten as: ( ( (T, µ g (T, 0 + RT ln ˆφ 0 (21 Equaton 21 s analogous to Equaton 13; we wll return to Equatons 13 and 21 n our dscusson of actvty. B. Pure Lquds and Lqud Mxtures 1. Pure Lqud Unlke the ure deal gas, there s no lqud analogue, so we defne the ure lqud fugacty by analogy to the ure non-deal gas case: 5
ure lqud dµ RT d ln ( f ure lqud (22 We wll not say more about ths relaton, but wll use t later n our dscusson of general vaor-lqud equlbra. 2. Real Lqud Mxtures We have seen that the chemcal otental of a seces n an deal mxture at a temerature T and total ressure s: µ s (T, µ (T,, ure + RT ln(x (23 Usng Equaton 22, we can wrte an analogous exresson for a real soluton as: ( µ rs (T, µ (T,, ure + RT ln f (24 Takng the dfference between Equatons 24 and 23, we obtan: µ rs (T, µ s (T, + RT ln ( x f (25 We defne the actvty coeffcent as γ x f to allow us to wrte Equaton 25 as: µ rs (T, µs (T, + RT ln (γ (26 Insertng Equaton 23 for µ s (T,, we obtan: µ rs (T, µ (T,, ure + RT ln(x + RT ln (γ (27 whch can also be exressed as: µ rs (T, µ (T,, ure + RT ln (x γ (28 6
Equaton 28, along wth Equatons 13 and 21, wll be central to our dscusson of actvty further below. To summarze our dscusson u to ths ont: We have defned the fugacty for ure gases and lquds, as well as for seces n a gas and lqud mxture. Fugacty s a drect measure of the chemcal otental of a seces n a mxture. Fugacty coeffcent of a ure gas or lqud: φ ure gas (T, f Fugacty coeffcent of seces n non-deal gas mxture: ˆφ y Actvty coeffcent of seces n non-deal lqud mxture: γ x f II. VAPOR-LIQUID EQUILIBRIA Usng the defntons of the varous coeffcents n the last secton, we can generalze the dea of chemcal otental equalzaton at equlbrum to fugacty equalzaton at equlbrum. Thus, our exresson for vaor-lqud equlbrum usng fugactes s: l (T,, x v (T,, y (29 In Equaton 29, we have ncluded the temerature, ressure, and comoston deendence of the fugactes. Usng the defntons of actvty and fugacty coeffcents ( for the lqud and vaor fugactes, resectvely, we obtan: γ x f l (T, y ˆφ (T, (30 We now make use of Equaton 30 to derve a ractcal exresson for vaor-lqud equlbra at low to moderate ressures whch are common for most alcatons. To obtan the fugacty of ure lqud at temerature T and ressure, we use two stes. Frst, we consder the change from zero ressure to the saturaton ressure at the temerature of nterest. We then transton from the saturaton ressure to the ressure of nterest. For a ure gas, snce we can relate the fugacty to the fugacty coeffcent, we consder the fugacty coeffcent. d ( µ g RT d ln (φ ( V rg V g d (31 7
Recall that the comressblty factor, Z, s Z V rg V g. Thus, Equaton 31 becomes: RT d ln (φ RT (Z 1 d d ln (φ (Z 1 d (32 (33 We can use Equaton 33 to ntegrate from 0 to the saturaton ressure of the flud at the desred temerature. At 0, φ 1 and ln(φ 0. Thus, uon ntegratng we obtan: sat 0 d ln (φ sat 0 (Z 1 d (34 Thus, the fugacty coeffcent at the saturaton ressure, sat s gven by: ln ( φ sat sat 0 (Z 1 d (35 Once we know φ sat, we know the fugacty of the lqud at the saturaton ressure: f sat,lqud f sat,vaor φ sat sat (36 Now, we can ntegrate from the saturaton ressure to the desred ressure to obtan the lqud fugacty needed n Equaton 30. Usng Equaton 36, we obtan: dµ lq RT d ln(f lq V lq d (37 lq V ln(f lq ln(f sat,lqud d (38 RT ( sat V f lq f sat,lqud lq ( sat ex (39 RT ( V f lq φ sat sat lq ( sat ex RT (40 Equaton 30 can now be revsted, usng the exresson for the ure lqud fugacty at the temerature and ressure of nterest gven by Equaton 40: 8
( V γ x φ sat sat lq ( sat ex RT y ˆφ (T, (41 Fnally, we make two major aroxmatons: For low ressures u to 1 bar, the exonental can be taken as unty (equlbrum ressures are not that far from saturaton ressures of ure seces Takng the vaor to be deal gas, the fugacty coeffcents are unty Ths gves fnally, γ x sat y (42 Equaton 42 s sometmes referred to as a modfed Raoult s Law. The non-dealty of the lqud hase s totally contaned n the actvty coeffcent, whle the vaor s consdered to be deal. One can construct T-xy and P-xy hase dagrams usng ths exresson for lqud non-dealty f one s gven nformaton about the actvty coeffcents and ther temerature, ressure, and comoston deendence. A. Actvty Returnng to Equatons 13, 21, and 28, we note a smlarty n the form of each of the relatons: ( (T, µ g (T, 0 + RT ln (φ ure gas 0 (43 ( ( (T, µ g (T, 0 + RT ln ˆφ 0 (44 µ rs (T, µ (T,, ure + RT ln (x γ (45 Each of these exresson for the chemcal otental of a seces (Equaton 43 for a ure seces, and Equatons 44 and 45 for seces n gas and lqud mxtures, resectvely has the followng general form: 9
µ real (T, µ (reference state + RT ln (a (46 Ths defnes the actvty of seces, a wth resect to some reference, or standard, state. The actvty contans the non-dealty of the mxture (enthaly and entroy of mxng. Snce t s usually ntroduced n the context of lqud mxtures, we frst consder the condensed hase (lqud case. For a seces n a lqud mxture, the actvty s defned as: a γ x (47 a lqud x f x (48 lqud f (49 Equaton 49 s a general defnton of the actvty of a seces n a lqud mxture. To generalze to any mxture, we can wrte a mxture f (50 Fnally, we allow for the denomnator to reresent an arbtrary reference, or standard, state to obtan our fnal exresson defnng the actvty generally n terms of fugactes: a mxture f 0 (51 where the suerscrt 0 refers to the standard state. Equaton 51 s our general defnton of actvty. Wth ths defnton of actvty, we can now quckly wrte new exressons for the actvty coeffcents for lqud and gas mxtures: Lqud Mxture: γ lqud a x lqud x f 0 Gas Mxture: γ vaor a y vaor y f 0 vaor ˆφ y y f 0 vaor ˆφ f 0 For the gas mxture ntroduced n the revous lne, we can consder, for the moment, two reference (standard states. 10
Pure deal gas standard state at 0 : γ vaor vaor ˆφ 0 ˆφ vaor 0 Pure deal gas standard state at the ressure of nterest, : γ vaor vaor ˆφ ˆφ vaor vaor ˆφ Usng the relatons for the actvty, actvty coeffcent, and knowledge of the meanng of fugacty (.e, ts relaton to chemcal otental, we are equed to handle varous cases for treatng vaor-lqud equlbra. In effect, ths allows us to treat all cases n a general way. For examle, f we have an deal vaor above a non-deal lqud mxture, the actvty of a seces n the lqud mxture s: a lqud lqud f 0 (52 Here, we make use of the fact that lqud vaor and vaor ˆφ vaor y y (recall that ˆφ vaor 1 for deal vaor to wrte: a lqud lqud f 0 vaor f 0 y f 0 (53 If we take the standard state for the lqud as beng the ure lqud at ts saturaton (vaor ressure, then we have f 0 f sat,lqud f sat,vaor φ sat,vaor sat (1 sat, the last equalty beng snce the vaor s deal (for ure and when t s n the deal gas mxture. Ths gves us fnally for the actvty of seces n the lqud mxture: a lqud lqud f 0 vaor f 0 y sat (54 Makng use of the fact that y s the artal ressure of n the deal gas mxture above the non-deal soluton, we can wrte: a lqud lqud f 0 vaor f 0 sat (55 The last exresson s Equaton 9.48 n your text book, though n ths resentaton, we have formally derved t wthn the rgorous context of fugactes. Note that Equaton 9.48 n your text s resented based on the same aroxmatons and standard states we have used n the resent dscusson, thus both are equvalent. 11
III. ACKNOWLEDGEMENTS Ths document tyeset and formatted usng LaTex2e. 12