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, n addton to the two varables n each of the above equatons, one must also consder the varaton of the number of moles of each speces, n Therefore, for open systems, we wrte du = TdS pdv + 5 U n dh = TdS + Vdp + 5 H n da = SdT pdv + 5 A n dg = VdP SdT + 5 G n S,V,n j dn S,p,n j dn T,V,n j dn T,p,n j dn The partal dervatves wthn the summatons above are called chemcal potentals for the -th speces, and denoted by the letter µ Therefore, = U n = H S,V,n j n = A S,p,n j n = G T,V,n j n T,p,n j It may seem strange that the dervatves of all these thermodynamc propertes wth respect to composton are all equal One way to thnk about ths s to consder each property as a multdmensonal surface n the approprate space, Ie, the space of (S,V,n ) for U, (S,p,n ) for H, and so on Then the above relatonshp requres that the slopes of these surfaces n one partcular drecton, wth the other quanttes fxed, are the same Below, we defne certan quanttes called partal molar quanttes, whch are also dervatves of state propertes wth respect to composton, but always wth T,p fxed The fact that the partal molar free energy s also equal to the chemcal potental s noteworthy 1
Partal Molar Quanttes Partal Molar quanttes are requred to deal wth open systems, e, systems that permt mass transfer between themselves and surroundngs Consder an open system wth n 1 moles of component 1, n 2 moles of component 2, n 3 moles of component 3, etc We would wrte the free energy change dg for such a system as dg = G P dp + T,n1,n 2, G T dt + G P,n1,n 2, n 1 = VdP SdT + G 1 dn 1 + G 2 dn 2 + P,T,,n 2, dn 1 + In the second equalty, the quanttes G 1, G 2, etc are called partal molar free energes Smlarly, we may defne partal molar volumes, partal molar enthalpes, nternal energes, and entropes: V 1 = V n 1 ; H 1 = H P,T,,n 2, n ; etc 1 P,T,,n 2, Because of ther great mportance n the thermodynamcs of solutons, we dscuss partal molar volumes and partal molar free energes further Partal Molar Volume: The total volume of a soluton of, say, two mscble lquds s gven by V = n 1 V 1 + n 2 V 2 The unts of partal molar volumes are the same as molar volumes The relatonshp between the two, e, partal molar volume and the molar volume s a subtle but mportant one In the case of deal solutons, the partal molar volume of each component wll be dentcal to the molar volume of the pure substance n the absence of the other component However, n the case of non-deal solutons, the presence of the second component has a measurable nfluence on the molar volume of the frst component and vce versa Therefore, n general, V 1 V 1 and V 2 V 2 (1) (2) 2
The standard state for defnng partal molar quanttes s a 1 molal soluton, e, a soluton that contans 1 mol of the substance n 10 kg of solvent Physcal Interpretaton of partal molar quanttes: It may appear that there s somethng not qute rght about the followng two equatons: V = n 1 V 1 + n 2 V 2, where V 1 = V n 1, and V 2 = V P,T,n 2 n 2 P,T,n 1 Based on what we have seen so far, the frst equaton should be dv = V 1 dn 1 + V 2 dn 2, whch s smply another applcaton of the chan rule n partal dfferentaton However, Eq (2) can ndeed be justfed on physcal grounds as follows Consder a large volume of soluton contanng ethanol (E) and water (W) We now add a small amount of water, say, n W moles of water, to ths soluton We would want to express the new volume of the soluton as V new = V old +,n W V W,m where, V W,m s the molar volume of pure water However, ths wll gve us the fnal volume only n the case of an deal soluton In the ethanol-water soluton, the effectve molar volumes of both substances are dfferent from ther molar volumes n the absence of the other substance Desgnatng the actual molar volume of water n the presence of ethanol as V W,m, the change of volume of the soluton s,v =,n W V W,m Therefore, we get V W,m =,n,v W The partal molar volume of water, V W, s defned as the value of the fracton on the rght hand sde n the lmt of an nfntesmal change n the number of moles of water Mathematcally, we wrte V W =,nw Lm,V = @0,n dv W dn W Once we mpose the condtons that temperature, pressure and the number of moles of ethanol, n E, are to be held constant, the dervatve on the rght hand sde becomes dentcal to the defnton of the partal molar volumes used above and n Eq (1): V W = V n W T,P,n E 3
An example of the applcatons of Eq (2): Consder a 40% by mass ethanol soluton of ethanol n water at 25 C From the fgure of partal molar volumes of ethanol and water n the presence of each other, estmate the volume of 1000 g of the soluton Compare ths to the volume that would have resulted f the soluton was deal Densty of ethanol = 0785 g ml 1 and pure water = 0997 g ml 1, at ths temperature Therefore, usng Eq (533), we get In 1000 g of soluton, we have 400 g ethanol (E) and 600 g water (W) 400 g n E = 1 = 868 mol 4607 g mol 600 g n W = 1 = 3330 mol 1802 g mol x E = 0207; x W = 0793 From the fgure, we estmate that at these mole fractons, the partal molar volumes are V E = 555 ml mol 1 ; V W = 172 ml mol 1 V = n E V E + n W V W = 868 555 + 3330 172 = 10545 ml If the soluton was deal, we would use the molar volumes of the pure substances to obtan V = 868 mol = 1111 ml 4607 g mol 1 1802 g mo 1 + 3330 mol 0785 g ml 0997gmL Therefore, we see that the non-deal nature of the soluton s reflected n a contracton of volume 4
Another example of applyng Eq (2): Densty of a 50% by mass soluton of ethanol n water at 25 C s 0914 g ml 1 Gven that the partal molar volume of water at ths composton s 174 ml mol 1,what s the partal molar volume of ethanol? No of moles of ethanol n 100 g of soluton: 50 g/4607 g mol 1 = 1085 mol No of moles of water n 100 g of soluton: 50 g/1802 g mol 1 = 2775 mol Now, snce V = n E V E + n W V W, we get V E = V n W V W n E Partal molar Free Energy: = 100g/0914gmL 1 2775 mol 174 mlmol 1085 mol = 5633 ml mol 1 Partal molar free energy s commonly refered to as the chemcal potental, and denoted by the letter µ: A = G A = G n A B = G B = G n B P,T,n B P,T,n A The fgure to the left shows a plot of the total free energy G vs n A The slope of the plot of G vs n A at varous compostons yelds the partal molar free energy G A at that composton From ths nformaton, G B can be found as n the example above For a pure substance, partal molar free energy s smply a measure of the molar free energy snce, n ths case, G = ng m, and G n = G m T,p 5
Gbbs-Duhem Equaton We now derve an mportant equaton related to partal molar propertes Consder a general property Z Usng the defnton of partal molar quanttes above, dz = 5 Z dn (3) But, generalzng Eq (2), we get Z = n Z (4) At the same tme, the general dfferental of Z(n, Z ) (usng the chan rule) s dz = 5 n d Z + 5 Z dn (5) Comparng Eq (3) and (5), we conclude that 5 n d Z = 0 (6) Ths s called the Gbbs-Duhem equaton One applcaton of ths equaton s llustrated below Determnaton of partal molar quanttes It s often more convenent to express the property (and therefore, the correspondng partal molar propertes) n the form of an emprcal functon of concentraton, usually molalty Consder a bnary mxture, a soluton consstng of one solute and a solvent Once agan, usng volume as an example, we wrte 1 V = a + bm 1/2 + cm + dm 3/2 + em 2 where m s the molalty of the solute and (a, b, c, d, e) are emprcal constants Usually, the solvent s assgned the subscrpt 1 and the solute, the subscrpt 2 Therefore, V 2 = V n 2 T,p,n 1 = V m T,p,n1 = 1 2 bm 1/2 + c + 3 2 dm 1/2 + 2em Now, recallng the Gbbs-Duhem equaton (appled to volume), n 1 d V 1 + n 2 d V 2 = 0, 1 Ths model seems to work well for volumes 6
we conclude that n 1 d V 1 + m( 1 4 bm 3/2 + 3 4 dm 1/2 + 2e)dm = 0 Therefore, d V 1 = 1 n 1 1 4 bm 1/2 3 4 dm 1/2 2emdm Integratng both sdes, we get V 1 V1,m d V 1 = 1 n 1 0 m 1 4 bm 1/2 3 4 dm1/2 2em dm V 1 = V 1,m + n 1 b 1 m 8 1/2 d 2 m3/2 em 2 Thus, n a bnary mxture, f we have an expresson for the varaton of the partal molar volume wth concentraton for one of the components, we can fnd an expresson for the other Smlar treatments are generally possble for other partal molar propertes as well Thermodynamc relatonshps among partal molar quanttes We now show that the thermodynamc relatonshps derved earler for state propertes also apply to partal molar propertes For example, consder the thermodynamc equaton of state H p = V T V (7) T T p Dfferentatng both sdes wth respect to n, we get n H p T = V T,p,n j n T V T,p,n j n T p Snce the order of dfferentaton s mmateral for state propertes, T,p,n j p H n T,p,n j T H p T = V T T V n = V T V T p T,p,n j p or Smlarly, every equaton we have derved for state propertes can be appled to partal molar quanttes as well 7
Crtera for equlbrum The thermodynamc crteron for equlbrum n open systems s also formulated n terms of partal molar free energes, or chemcal potentals If A and B are present n two phases α and β, the phase equlbrum condton s: G A = = G A > (or A = = A > ) and G B = = G B > (or B= = B > ) For chemcal equlbra, the net chemcal potental of the reactants (weghted by the stochometrc coeffcents) must be equal to the net chemcal potental of the products Consder the equlbrum n A A + n B B n C C + n D D Then, the crteron for chemcal equlbrum s (n C C + n D D )=(n A A + n B B ) 8