Solution Thermodynamics

Transcription

1 Soluton hermodynamcs usng Wagner Notaton by Stanley. Howard Department of aterals and etallurgcal Engneerng South Dakota School of nes and echnology Rapd Cty, SD January 7, 001

2 Soluton hermodynamcs by S.. Howard hs treatment of soluton thermodynamcs descrbes common soluton parameters nvolvng volume, enthalpy, entropy and bbs energy. Furthermore, t contans the natural constrants placed on solutons by mathematcal consderatons. Soluton terms are ether ntegral or partal propertes. Integral propertes descrbe the propertes of the entre soluton whle partal s used to descrbe the propertes of any one component n the soluton. Before formal consderatons begn, t s useful to consder an example stuaton where there s a need for partal molar quanttes. Such a case s easly llustrated by the followng stuaton. he reacton between go and SO to form sold Forsterte s go (1) + SO (1) = g SO 4(s). (1) At equlbrum there s one lqud phase consstng of a soluton of go and SO and one sold phase consstng of pure g SO 4. If one needed to know what temperature-pressure loc are requred to mantan a fxed equlbrum dstrbuton of the consttuents n ths reacton, one would use the Clausus-Clapeyron equaton dp d ln ΔH = ΔV () he use of ths equaton requres knowng the volume and enthalpy changes for the reacton. he volume change s ΔV = Volume of 1 mole of pure, sold Forsterte Volume of moles of - go n the lqud soluton - of go and SO Volume of 1 mole of SO n the lqud soluton of go and SO he enthalpy change may be wrtten n a correspondng way. It s of nterest to note that t s possble to measure each of the partal molar volumes to obtan ΔV but t s not possble to ever measure the absolute partal molar enthalpes. However, t s a farly straghtforward mater to measure the change n the enthalpes, ΔH. A second process that llustrates the need for soluton propertes s the reacton that descrbes equlbrum n the Betterton Process where Pb s deznced usng PbCl. PbCl (1) + Zn (1) = ZnCl (1) + Pb (1) (3) At equlbrum there et two lqud phases: a lqud metallc phase of Pb and Zn, and a molten salt phase of ZnCl and PbCl. he volume change for the reacton n ths case s ΔV = Volume of 1 mole of ZnCl n the molten salt phase of equlbrum composton. Volume of 1 mole of PbCl n the + molten salt - phase of equlbrum composton Volume of 1 mole of Pb n the molten metallc phase of equlbrum composton. - Volume of 1 mole of Zn n the molten metallc phase of equlbrum composton. S.. Howard 001

3 Soluton hermodynamcs by S.. Howard 3 Absolute Values he volume that one mole of a consttuent occupes n soluton rarely equals the volume of the consttuent n the pure state because the nteracton forces n soluton are dfferent from those n the pure component. he volume of one mole of n soluton s denoted by the symbol V and may be determned expermentally by the methods descrbed below. easurement and Defnton of Absolute Partal olar Volume here are three methods of measurng V. 1. he absolute partal molar volume s equal to the ncrease n the total volume V of soluton when one mole of speces s added to a very large quantty of the ntal soluton. V = Δ V (4) Snce V vares wth composton, a large quantty of soluton s specfed so that the ntal soluton s concentraton s essentally unaffected by the addton of one mole of.. he second method for measurng V s only a mnor modfcaton of the frst method. he only dfference s that rather than addng one mole of to a large quantty of soluton, a smaller quantty of s added. Consequently, the volume change s for the smaller quantty of and to obtan the volume change per mole one must dvde by Δn whch s the moles of added to the soluton. ΔV V = Δn (5) hs procedure becomes more precse as Δn 0 because the change n soluton composton becomes smaller. herefore, the best value of V s n the lmt. V = n Lmo ΔV Δn V = n, P, n j (6) he subscrpts,p, and n j ndcate that V s measured whle, P, and the number of moles of other soluton components reman fxed. Snce V s defned as a partal dervatve t s easy to see why V s called the partal molar volume of speces n soluton. 3. he defnton n Eq. [6] mples another method whch may be used to determne the partal molar volume, V. If one starts wth n j moles of component j at temperature and pressure P and adds to t speces n several ncrements, a curve lke that shown n Fgure1 s obtaned. Accordng to Eq. [6], the slope of the lne n Fgure 1 equals the partal molar volume of speces at the composton selected. For example, the partal molar volume of at n = 0.06 s 38.9cm 3 mole. S.. Howard 001

4 Soluton hermodynamcs by S.. Howard 4 Fgure 1. Partal olar Volume of, V, from Volume vs. n bbs-duhem Equaton he bbs-duhem Equaton descrbes a constrant on the behavor of the partal molar quanttes n a soluton. he followng dervaton s n terms of volume but apples to H, S, and, as well. Constant temperature and pressure are assumed throughout the development. he total volume V of a bnary soluton composed of n 1 moles of component 1 and n moles of component s whch may be dvded by (n 1 + n ) to gve the ntegral molar volume where X 1 = the mole fracton of component 1 X = the mole fracton of component. V = n V 1 + n V (7) V = X 1 V 1 + X V (8) Consderaton of Fgure 1 shows that a bnary soluton s volume s dependent upon the number of moles of each component. hat s, V = f(n 1, n ) at constant and P. he total dfferental of V gves V dv = n 1, P, n j V dn 1 + n, P However, the followng expresson for dv s obtaned from Eq. [7]: Subtractng Eq. [9] from Eq. [10] gves, n j dn = V 1 dn 1 + V dn (9) dv = n 1 dv 1 + n dv + V 1 dn 1 + V dn (10) S.. Howard 001

5 Soluton hermodynamcs by S.. Howard 5 Dvdng by (n 1 + n ) gves n 1 dv 1 + n dv = 0 (11) X 1 dv 1 + X dv = 0 (1) hs mportant result s called the bbs-duhem Equaton. It shows that change n one partal molar volume s related to change n the partal molar volume of the other component. he Slope-Intercept ethod wo valuable relatonshps are obtaned by dfferentatng Eq. [8] wth respect to X and combnng the result wth Eq. [1] to gve V x = V - V 1 (13) Note that dx 1 = -dx snce X 1 + X = 1. Elmnaton of V 1 or V wth Eq. [8] yelds V 1 =V- V X (14) x V V = V + (1 - X ) x (15) hese two equatons are the bass of the angent-intercept ethod of graphcally determnng partal molar volumes from a plot of V versus X. As ndcated on Fgure, V 1 equals the ntercept at pure component 1 and V equals the tangent ntercept at pure component. V V 3 cm gmole X V V x V (1 X ) X V V 1 0 X 1 ole Fracton of X Fgure. he angent Intercept ethod of Fndng Partal olar Volumes S.. Howard 001

6 Soluton hermodynamcs by S.. Howard 6 Summary of Absolute Quanttes As stated prevously, the above development s drectly applcable to the thermodynamc propertes H, S, and. In general, the partal molar quantty s defned as Q Q = n, P,n j (16) he ntegral molar quantty for a bnary soluton s related to the partal molar quanttes as and the bbs-duhem Equaton requres that Q = X 1 Q I + X Q (17) X 1 Q I + X Q = 0 he angent-intercept ethod may be used to obtan the partal molar quanttes Q from the ntegral molar quantty Q. he quantty Q can be V, H, S, or. Absolute values are determnable for V and S, but not for H and. Snce only changes n H and are requred for thermochemcal calculatons, there s no need for the absolute values of H and. However, the changes n H and must be related to some standard (or reference) state. When ths s done the changes n H and are called the relatve enthalpy and relatve bbs energy. he next secton consders these relatve values. Relatve Values (or of xng ) o assgn numercal values to the thermodynamc soluton functons of enthalpy and bbs energy t s necessary to relate them to a reference state. Snce absolute heat energy cannot be measured, all tabulated energy and enthalpy data equals the change from some reference state. For solds and lquds ths state s the pure component at the pressure and temperature of nterest. he dfference between the partal molar volume n soluton V 1 and the molar volume of the pure speces s V called the relatve partal molar volume and s denoted by the symbol V In general, V V - V (18) Q Q - Q (19) he parameters on the left sde of Eqs. [18] and [19] are also called partal molar quanttes of mxng, because the term mxng mples a quantty relatve to the reference state whch s the pure component. he dscusson of relatve quanttes began by ctng the necessty of relatng the enthalpy and free energy quanttes to a reference state so that the change from the reference state could be measured. ethods of measurng these changes for H and are presented below. S.. Howard 001

7 Soluton hermodynamcs by S.. Howard 7 easurement of Relatve Partal olar Enthalpy and bbs Free Energy he partal molar heat of mxng of speces can be measured calormetrcally by droppng a small amount of lqud component at temperature nto a soluton at temperature. he transfer of heat requred to return the soluton to the orgnal temperature equals the heat of mxng for the small quantty Δ n. herefore, H ΔH = Lm n o Δ Δn, P, n other H = n, P, n other (0) Of course, the quantty Δ n. must be small enough so as not to change the soluton s composton, because H =f(x ). he partal free energy of mxng may be determned by measurng the partal pressure of speces above the soluton and above pure provdng deal behavor of the gas prevals. = = R ln a P = R ln o P (1) Here o P s the pressure of the pure component. A second method of measurng s by a galvanc cell. If the galvanc cell shown n Fgure 3 creates the potental E, then the partal molar bbs free energy of mxng s = nfe () where n = the number of equvalents for the cell Joules F = Faraday s constant: 96,55 volt * equvalent he relatve partal molar entropy of speces may be calculated from the equaton = H S (3) E, volts Pure Ionc Conductor wth + ons Soluton wth Fgure 3. alvanc Cell for Determnng Actvtes of Component S.. Howard 001

8 Soluton hermodynamcs by S.. Howard 8 Relatve Integral olar Quanttes he ntegral quanttes refer to one mole or soluton. Lke partal quanttes, ntegral quanttes are both absolute and relatve. For example, the absolute volume of one mole of a bnary soluton (two components) s gven by equaton V x1 V1 + xv = (4) and n general Q x1 Q1 + xq = (5) he relatve ntegral molar quantty Q may also be called the ntegral molar quantty of mxng. he angent-intercept ethod and Relatve Quanttes Relatve partal molar quanttes may be obtaned from the relatve ntegral molar quanttes by the angent-intercept ethod as llustrated n Fgure. he same procedure used to derve Eqs. [13] and [14] s followed to obtan Q Q = 1 Q x (6) x ( ) Q = Q + 1 x bbs-duhem Equaton Lkewse, the bbs-duhem Equaton for relatve quanttes has the general form 1 1 = Q (7) x X Q + X Q 0 (8) Excess Values he excess quanttes are defned merely for convenence. hey are smply the dfference between the actual value of the relatve quantty and the value of the relatve quantty were the soluton deal. Q, deal = Q Q (9) An deal soluton s a soluton n whch the nteracton among the atoms and molecules of unlke components s the same as the nteractons among lke atoms and molecules. In such case S deal 1 V = 0 (30) deal 1 H = 0 (31) deal 1 = R ln X (3) 1 deal = R ln X (33) herefore, S.. Howard 001

9 Soluton hermodynamcs by S.. Howard 9 sx V = V (34) H = H (35) a = R ln a R ln X = R ln = R ln γ X (36) S = S + R ln X (37) For the ntegral excess molar quanttes S V = V (38) m H = H (39) = S + R( X 1 ln X 1 + X ln X ) (40) ( X ln X + X X ) = R (41) 1 1 ln As wth all other partal quanttes, the bbs-duhem Equaton s vald. 1 1 = X Q + X Q 0 (4) Lkewse, the angent-intercept ethod may be used to obtan excess partal molar quanttes from the correspondng excess ntegral molar quantty. Summary he four thermodynamc quanttes V, H, S, and may refer to ether one mole of a soluton component (partal molar quantty) or to one mole of soluton (ntegral molar quantty). hese quanttes may be reported as the absolute value (except for H and ), relatve value, or excess value. he bbs-duhem Equaton apples to any of the partal molar quanttes and has the general form X Q + X Q 0 (43) 1 1 = where Q1 = any partal molar quantty for component 1 Q = any correspondng partal molar quantty for component Lkewse, the angent-intercept ethod may be used to fnd any partal molar quantty from the correspondng ntegral molar quantty Q Q 1 = Q X (44) X ( ) Q = Q + 1 X Q X (45) Any ntegral molar quantty may be found from the correspondng partal molar quanttes accordng to the followng summaton Q = X Q + X Q... (46) S.. Howard 001