I. Linearization Methods for Systems not at Chemical Equilibrium. II. Equilibrium Pressures and Phase Compositions for Non-Ideal Mixtures Citation Laidlaw, William George (1961) I. Linearization Methods for Systems not at Chemical Equilibrium. II. Equilibrium Pressures and Phase Compositions for Non-Ideal Mixtures. Master's thesis, California Institute of Technology. doi:10.7907/CF3P-3866. https://resolver.caltech.edu/CaltechTHESIS:05072010-075038096 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Chemistry) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1960 Record Number: CaltechTHESIS:05072010-075038096 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05072010-075038096 DOI: 10.7907/CF3P-3866 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5781 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 18 Aug 2010 15:31 Last Modified: 09 Nov 2023 19:39 Thesis Files Preview PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

I LINEARIZATION METHODS FOR SYSTEMS NOT AT CHEMICAL EQUILIBRIUM IT EQUILIBRIUM PRESSURES AND PHASE COMPOSITIONS FOR NON-IDEAL MIXTURES Thesis by William George Laidlaw In Partial Fulfillment of the Requirements For the Degree of Master of Science in Chemistry California Institute of Technology Pasadena, California 1960 PART IT. TABLE OF CONTENTS TITLE LINEARIZATION METHODS FOR SYSTEMS NOT AT CHEMICAL EQUILIBRIUM... . INTRODUCTION . 2. 6 2 6 oe eo SINGLE CHEMICAL REACTIONS .. Application and Numerical Calculations Discussion . .« « « «2 © SYSTEMS OF COMPETING REACTIONS OTHER ASPECTS . . 2. 2 0 2 oe REFERENCES o 2. se ee ee ee e ° EQUILIBRIUM PRESSURES AND PHASE COMPOSITIONS NON-IDEAL MIXTURES . ..... INTRODUCTION . 2 2 2 « © © © 0 DEVELOPMENT AND METHOD... APPLICATIONS AND DISCUSSION . Tables and Figures ...... REFERENCES SECTION II ...-. PAGE 18 19 23 2 26 2? 28 32 34 SUMMARY In Part I a method for calculating the effect of changes in the variables temperature and pressure on the reaction yields of chemical systems not at equilibrium is developed. The method circumvents the exact solution of the differential rate equation by expanding the time derivative of the progress variable about its equilibrium value and re- taining the linear terms. The equations are valid near equilibrium and comparisons with the results obtained by exact solution of the rate equation show good agreement. Much economy of algebraic labour is achieved at small cost in accuracy. The method is extended to systems of competing reactions but the results do not appear to be very useful. In addition the ease with which some other questions may be answered is illustrated, In Part II an equation for the calculation of equilibrium pres- sures and phase compositions for non-ideal systems is derived. Conformal solution theory is applied to the liquid phase and a second order virial equation is employed in the description of the gas phase. Calculated and experimental results for binary systems show good agreement in most cases. LINEARIZATION METHODS FOR SYSTEMS NOT AT CHEMICAL EQUILIBRIUM INTRODUCTION In investigating the effect of changes in termperature or pres- sure on chemically reacting systems a state of equilibrium is frequently assumed. Such an assumption allows the use of the relatively simple equilitrium equations in predicting the result. However, the assumption of the equilibrium state for systems not at equilibrium can lead to re- sults which are significantly in error. In fact Rastogi and Denbigh (1) have shown that, even for a system which has attained ninety percent of its equilibrium state, the results can be in error by a factor of ten. To investigate the effect of a change in temperature or pressure without assuming the equilibrium state generally requires that the rate equation for the system be solved. The rate equation can be solved for the concentration of one of the chemical species and the resulting equa- tion involves the concentration of one or more chemical species and the rate constants for the system. The appropriate derivative of this equa- tion yields the effect of a change in temperature or pressure. Rastogi and Denbigh have utilized this approach in the paper cited above but such a method becomes cumbersome for all but the simplest rate laws. The assumption of equilibrium can be avoided and the unwieldy equations, which result from direct solution and differentiation of the rate equation, can be replaced by the use of the Kirkwood and Crawford linearization (2). In this, Section I of the thesis, the Kirkwood and Crawford linearization will be applied in a completely general way to the problem of obtaining the effect of a change in temperature or pressure on a -~3- system not at equilibrium. Further, the method will be worked out in detail for several systems and the results compared with results obtained by the exact method of Rastogi and Denbigh. In the second part of this section the methods are extended in a completely general way to a system of competing reactions and in particular to a system of two competing re- actions, In the final part some other questions about the system are answered with the aid of the equations developed. SINGLE CHEMICAL REACTIONS Essentially the method of this work is the expansion of the time derivative of the progress variable about its equilibrium value of zero and the retention of the linear terms. The progress variable WA is defined as dng = vy ar> (1) In this definition ne is the concentration of species No in moles per liter and Vy is the stoichiometric coefficient of No in the reaction considered. The sign attached to Vy is such that dA is always posi- tive. It should be noted that is for convenience defined here in moles per liter but this does not detract from the generality of the methods developed. If we expand the time derivative d A/dt about its equilibrium value and retain only the first derivative term we obtain ada ada e ar ae af * 2% a e a e Here the subscript indicates evaluation of the derivative at equilibrium. Since dn /at is zero at equilibrium it follows from equation 1 that (d A /at) . is zero and we have aA. A =3 Gap / @) a a oe ~ he Integration of equation 1 yields ne - ne = vy (x - r°) where the super- scripts "t and e" indicate evaluation at time t and at equilibrium re- spectively. Insertion of this last statement into equation 2 gives , t e ar Az OA OE Mean, / (3) s Vu an / depends on A _ for the system, a constant for a given a a/e aA system, and is denoted by 6. Rewriting equation 3 we have dt _ »e = Bdt which has the solution t e In (AX - A) = Bt +InI where I is a constant of integration. Since for t = 0 A= QO then e t e Bt) ; In (- A“) = InI so we have AY = HX (1 -e''’, Replacing B by -1/ where T , designated as the relaxation time, is greater than zero we obtain finally t -t we wWa-et) (4) We may use equation 4 to obtain the effect of a change in tempera- ture or pressure. Letting X= P or T and taking the derivative d/dX of equation 4 we have aX A, ad® ax ~ 9& dX ~ Fa - 4) ln (1 - vy stn iis) Defining f, the fractional attainment of equlibrium as f = MY ye and equation 5 can be rewritten as e gh p48) Yaepnma-f din 1/t ax ax (6) dX From equation 1 we have dn /aX =v d A/aX so that we obtain for dn/ax the equation e Qos yy ¢p8$AXL yy Ma-fma-s ee (2) ax a” “ax a ax -5- This equation clearly gives the effect of a change in temperature or pressure in terms of the relaxation time T and the fractional attainment of equilibrium for any reaction near equilibrium. The effect of a change in temperature or pressure for the equili- brium state is given by equation 1 as d a n° ny = y a A ( 8 ) ax ax dn, /aX Defining r as the ratio r = one obtains on dividing equation 7 by dn€ /ax equation 8 dln 1/% fax (9) ain eax 9 r=f- (1 ~- f) ln (1 ~ f) Clearly the ratio r is a measure of the effect of a change in temperature or pressure for a non equilibrium state compared to the effect of a simi- lar change for the equilibrium state. As such, the magnitude of risa measure of the error which arises from assuming the equilibrium state compared to the result when equilibrium is not assumed. It is to be emphasized that the equations 4 for A and 9 for r are completely general and can be formulated for any system near equili- brium for which the relaxation time TV and the fractional attainment of equilibrium f are known. Application and Numerical Calculations The magnitude of r is a convenient and significant number by which the results of the linearization methods developed above can be compared to the exact results obtained by the direct solution method used by Rastogi and Denbigh. In order to make the comparison it is necessary to formulate r for the systems considered both by the linearization method -6- and by the direct solution method. For further comparison the system types chosen are of increasing complexity. The systems considered are: System A k, Ennead Kp Ny System B N, +N, aA kp System C N. —fi— 2 ggeeemnemnen ka N, where: No is the molecular formula of a chemical species a Nz, +N, : k, and k, are respectively the for- ward and backward rate constants Na +N, The kinetics for the systems are taken as; for A: unimolecular opposed by a unimolecular reaction so that the rate law is te ym - ym (10) for B: bimolecular opposed by a bimolecular reaction so that the rate law is dn; Jr = mm - kensm, (11) for C: unimolecular opposed by a bimolecular reaction so that the rate law is dn, at = 4m - kyngn,; (12) where in each case the a for a particular reaction are again mole/liter concentrations of species No at time t. For convenience the initial con- centrations of the products are taken as zero and the initial concentra- tions of all reactants are taken the same and equal to "a'' moles/l. Thus at any time t the following relations hold: for System A. m =a-h (13) for System B m = 2g =a — Ny3 My = ny (14) for System C m =a - Ds; m = ny (15) dng In each case the equilibrium condition i 7 O defines the equilibrium constant K as K = k,/k,. Writing ne as the equilibrium concentration of species Nu and making use of the equilibrium condition we have; for System A: e ak m =T.K (16) for System B: e ak” es (17) for System C: ny = 5 where 1+ K = @ (18) We may now formulate r for a change in termperature for the systems A, B and C. To obtain r by the method of linearization (i.e. by equation 9) we proceed as outlined below for each system. For system A we find the relaxation time T by noting that from equations 1 and 10 that ° A=kmy - = nm, so that from equation 4 A. kK +1) K and thus dinl/f adink _ 2a in K + K . din kK (19) aT ~ at aT 1+kK at ~ 8. The derivatives in equation 19 can be taken from thermodynamics as dink AH aT RT? (20) where MH is the enthalpy change of the reaction and dink, & (21) at RT2 where E is the activation energy of the forward reaction. Substituting equations 20 and 21 into equation 19 gives dini/t. AH 1 (K + 1)_E aT = Rr? ° art anid (22) To find ” for system A we make use of equation 1 and the conditions no = O and ON = O so that ns = 2. Then from equation 16 we have x =T oe which gives din aink _K_ dink (23) aT at "“1+K° aT Substitution of equations 19 and 23 into equation 9 yields the required formulation of r for this system as r=f-Q-)mQa-n[4a+n -1) (24) We proceed in an identical manner for the systems B and C and.obtain the formulas summarized in Table I. To obtain r by the method of exact solution and differentiation requires, for each system, that the differential rate equation be solved for one of the components and then the solution differentiated in order that dn, /aT in the ratio r can be formulated. For system A the differential rate equation to be solved is from equations 10 and 13 dn » wee = K arn at a[ 2 G28 | | (25) ~9- Since nz = O we may write the solution as (l1+kK)n 1+K zy - dn tA B27 2 in [ x 5 ] a Ka t (26) no _kK and from equation 16 we may substitute 3 *IeK and obtain ne K +1 In {1 - Fe] = -ki(“y~*) t (27) On solving this explicity for n, and differentiating with respect to temperature we have e See nf 1-32 we (hte.d ee i. ak) + B2- dn2 aT n9 at K aT ng aT Substitution for the derivatives from equations 20 and 21 allows one to write dng _ _ ng(1 - ™2/n5) in (1 = "ft [= (1 +K) _ 1] 4H, 22 ane aT 1+kK Au RT? n§ at Eliminating t with aid of equation 27 we obtain _ nf _ Maze e Gop ~ Be (= 8M) an 2 = B28) ¢ Ba + x) - 1 ABs Be M2 (28) aT 1+K Ou RT2 on dat For the derivative dn$/at we differentiate the expression for nS in equa- tion 16 and obtain dng aK = a An K aT (1 + K)? at Substituting for the derivative d ln K/aT from equation 20 we have dng _ = ak «~=—sC<AAC (29) aT (1 + K)? RT? Dividing equation 28 by equation 29 and replacing nS/n, by f yields the required formulation of r for this system as E Poact =f > (1 - IGG +k) -1Jma- sf) (30) In an exactly similar manner we obtain P exact for the systems B and C. The formulations for v oxact are summarized in Table I. It is to (F-T)+(F+T)9 Vy dM | HV) 1 9/e2-T-3) 2} y > [EDGED ot} 2° i a=2i+CI- 9 a) +t) = azeym fn + @N -T)UT(I-T)- xepTnoe Tow =— (Oye Eee) oy ¥9/ez-T-9 are: a =glts DELFT) $ e My peweddo by 297 eatt(T- )F] BZ (3-1) 22 eo -(1/ee-1-9)5 ee a zre[TNoeTowtun by {(( - T) peur] + i} a HY, cia! ] TeTNoeToUTG "Maen a aze—tnoetoutq ®y + tN HY aze[TnosaTomtun fo fi-(ut) SY VY }($-T)BL(F-T)~3 [TORT A | (s-T)Ut(s-T)-F -B 4a pasoddo "y= tN IeTNOsT owtun by UoTPeTpUeresFTp Aq paeMoTTos uotzenbs ayer ay} JO UOTYNTOS yoOexXe kg woTyezTrwaUTT FO poyzow oui Ag aky SOTJOUTY ° wa ysis einzeredms}, ut eZueyo e AZOy 2 FO VOT}eTNMIOY eanjededuey, UT eSueup @ OJ I roy suotzenbg yo Arewums T eTaeL ~ ll - be emphasized that although r andr are identical for system A that exact this is not the case for the systems B and C. This is due of course to the fact that the rate law for system A was linear to begin with while B and C do not have linear rate laws. For purposes of comparison,numerical calculations of r are carried out here for different values of f for each of the systems of types A, B and C, The chemical systems along with the calculated values of r and Poaxact 2re tabulated in Table II. Chemical systems were chosen having the characteristics of A, B and C,and for which the required data were available. In addition the systems were chosen,such that the magnitude of E/AH was in the range 10-20;for. as pointed out by Rastogi and Denbigh, r is principally a function of E/AH and is small,unless E/AH is of the order of 10. We next consider the formulation of r for a change in pressure for a system of the type B. The chemical system chosen was again HI + HI =~ H,+I1,. The reaction is taken to be in the gaseous phase, The gases are considered non-ideal and the mixture of gases is considered a perfect mixture. To obtain r,by the linearization method,we first note that it is expedient to express A as a function of the fugacities of the chemical species in the system rather than the concentrations. The Taylor expan- sion of equation 2 now reads Az & (f - #2) SA / a a a e where fh is the fugacity of species Ny in the reaction mixture. The Lewis and Randall approximation is employed to define the fe as ~12- Table II Effect of a Change in Temperature Comparison of r computed by present method [r (present)] with r computed by the exact method of Reference (1) [r (exact)]. Numbers in brackets are literature references, Reaction H,+1,m en on, a HCH , “cbocon 2 ace Kinetic Bimolecular- Unimolecular- Unimolecular - Type Bimolecular Unimolecular Bimolecular C4] E(Keal) Ay [3] 18 [8] 14.9 C7] H(Keal) 3 [3] 950 [8] 0890 [6] K .0022[3] 1.12 [8] 005 [5] T(°K) 764 620 298 (present) (exact) (present) (exact) (present) (exact) 0.99 2.49 2.86 2.8 2.8 2.64 2.88 f 0.95 5.80 70 6.8 6.8 6.45 - 0.90 8.40 10.6 9.9 9.9 9.3 10.8 ~13- & f' “a “Ea, a i where =e, n is the total number of moles in the mixture and where re is the fugacity of the pure species Ne at a given pressure. If An = 0 and only differential pressure changes are considered, then fl can safely be taken constant with respect to time. Since An = O then Zz; n, = C,a constant and fo can be written as Deft fh = =n ds where qs X~ £ (t) From equation 1 we can obtain m7 ne = vy (A- »*) and on substi- tuting for n, we have e e fo - fed vy, (A- A*) Thus we can now write the expansion as A= & 4, vi, (A-A*) GA = CD-R) B a at, / e where Be= 2 4, Vy ar / a a ‘’e and is again constant with respect to time. As before we have the solu- tion Az AP (1 - e t/t ) but where > = - ad vp aA / (31) t a a “O df, /e Since equation 31 is identical with equation 4,except for the definition of v,one can write directly the pressure form of equation 9 as 1 d_in ~/t /aP ain >¢/aP (32) r=f- (1 -f) In (1 - f) To obtain r for system B we note first that the rate law (11) writ- ten in terms of fugacities is which gives + = +k, [daf$ + dofS + “ (4,f% + 4,£5)] Using the conditions of equation 14 and the Lewis and Randall relation of ne and f. given above we may obtain the expression for 1/% as e dd n t 2k.d4d, [a-nS+ 3-2) <= dad, (33) ee €y2 . d4d, and for convenience write wy" K = o K dzd, so that on solving for ns we have a) af = 2 C2K (34) 3“ 14+ (o6kK)# Substituting from equation 34 for nS in equation 33 we have Ye Ye 1 (¢ K) (ek) - which reduces to i -e fit) Noticing that we can write oO = Fire 3 ft ft ce? and d,d, = and inserting these last two expressions into equation 35 one obtains finally 1 ' -~ = Z in th 2 in K in My = in + Ink, + $ Taking the derivative with respect to pressure we have din /t _i 5 ding aP - 2 = SP -15- d in f% Va ; But 2p «= RP where ve is here the molecular volume of species Ne at temperature and pressure T, P. Thus we can write lh ain l% 1 aP = ORE 2 Vu (36) To find d In ns /aP we note since nS = 0 that one has ns e YA So that from equation 34 Im A° = In n§ = Ina + + In oK-in{l+ (ok)?] Differentiating this with respect to pressure gives , dinA® 1 ding -LldinkK __d a (ok)” aP 2 aP 2 aP 1+ (oK)# aP which reduces to din AS 1 1 AV (37) aP “ 2 1+ (ek) RT Inserting equations 36 and 37 into 32 we obtain the required expression for ras ref+(i-f)[ in (1 ~ #)] [2 - (oxy) Zeta To obtain r,by exact solution of the rate equation,we note first that the rate equation 11 in terms of fugacities is dnz = k, f,f,-k, f; f, dt acd Inserting te = a from the definition given earlier,the rate law becomes dn ; K, 1 spt ce Eth thu, n,-gn,n, £5 £4) Noting that for this chemical system that f4 = f4, and making use of equa- tion 14,we can eliminate n,, nz and n, and obtain Geer Lat 2any + (1-6 )ng] wnere $= see This differential equation has the solution 2(1 - d)n - 2a + 2p” 2k,(£1)2 ath Bsn; oa abe = ag”’t +ini ln - 16 - The constant of integration I can be evaluated since for t = 0, n; = 0 then 4 1 + te =lintI in [-o” and the solution can be written as (2-0) (1 +0"), - a(1 +0") _ in i +9" 2 ZED? oo in (1 -O%)(1 +O%)n; - a(1l -h%) "TI OR c2 which reduces to (1 - on, -a _ 2&4 (#4)? ao” ‘ (1+ 6”)n,-a c? Taking the derivatives with respect to pressure we obtain after some algebra e dn 5 % dn, 0” en i) - 2(n;-a) ‘ailnf! aP 3 3 aP 2atK [2 nity ,24 at } (£1)2” (38) n;(1 -) - 2an; + a? o aP 2 Vv _ fh fh ding _ Av dint, Va But = fir 5° that “5 Rr and aD ar: | Also % %e dn; _ 230" Ay Gg g sms aP 2RT dP Inserting these last three equations into equation 38 and rearranging one can obtain dn, _n;(n;-a) AV Za . pBgQ- 9 )-2anzta?) rh, (1-9 )n 5-a aP a oer * arr | 2a ob” | err ore (39) To obtain dn 3 /aP we note that since ns = © then from equation 1 e e n;° = A® ana 3s. 2 ne Aes) dP e e dn n AV : ony fe 3 . From equation 37 “ap atl + (ok)*) RT % . e a( ok) and from equation 34 n; = f+ (ok) dn 5° a( ok)" AV so that 2. = (40) “ap. s=—“‘i(C GG CKO ORE -17 - On dividing equation 39 by equation 40 we can obtain r as _ n(n 5-a) (14( rK)”) . aYa “pase ¢ )-2an ;+a? 7} . (in (1- $”)n 5a a®( ok) Vv 2a? g (rk)? (1+¢ 2)n,-a from p fs ft, and oOo = (4)? (£1)? K ry fh we have the identity $= . Eliminating n, by the relation n, = fn;° and the use of equation 34 and inserting the identity o =o we obtain finally = £ (Con) *(1-t)at}— Bayer o x-1)-20( 0K) feoK) ade eK)" J r = exact % CK) - 21 (ens (AN l1-f ‘inf k For the system chosen, HI + HI = i + I,, the known data are 2 taken as;T = 764°K , ref. (3) K “2 2.2 x 10, ref. (3) P = 0.4 atmos., calculated from reference (3) data The molecular volume ve is considered given by the formula v= ti+ et (t) P] where Bo is for the pure gas of species a at a pressure P and temperature T. The Bt (T) were calculated with the aid of Lennard-Jones potential functions and for the species involved were found to be it for HIB! (T) = 7.76 x 107 * atmos?’ 4 "t for H, 8B (T) = 2.26 x 10° 34,1 x 107° 1" forI, 6 (T) So that -— 18 «. RE (he 2. = 5 (4e002) RT 4 QV =f x 10 x 20.8 Inserting the above data into the equations for r and Teoxact We obtain the data tabulated and compared in Table III. It should be pointed out that the magnitude of r is largely due to the pressure effectively increasing the concentration. Table III Effect of a Change in Pressure r by exact solution r by linearization of the rate law % Difference in f method followed by differen- r anil tiation (r ) exact 1,0 + L + L 0 0.999 ~ 12.2 inaccurate 0.99 - 87 - 97 10 0.90 - 4ho - 500 12 Discussion In comparing the algebra and manipulations involved,in obtaining r by the exact method and by the method of linearization,it should be quite clear that the method of linearization involves much less labour, Comparing the complexity of the equations summarized in Table I it is also evident that numerical evaluationswith the r obtained by lineariza-~ tion will be much less tedious than those required to obtain P oxact’ From the percent difference columns of Tables II and III it is clear that the values of r by the two methods are similar (within 10-20%) -19 - and that the difference in,r by the two methods, becomes more significant further from equilibrium. It is also apparent that the difference in r, by the two methods, increases somewhat for an increase in the “order" of the equation that has undergone linearization, Thus we may say,for systems not far from equilibrium,that it would appear thatruch saving of labour may be accomplished by the use of linear- ization methods and without significant loss in accuracy. SYSTEMS OF COMPETING REACTIONS We now extend the concept of linearization to a system where we have competing reactions. For the sake of clarity and simplicity a system of two competing reactions has been chosen. The methods of solution and all manipulations are, however, quite general and would apply to a system of n competing reactions. Let the progress variable of the two reactions be denoted by A, and ,, and define the "lag" of a reaction as x. = A - AS. If we expand rg as in equation e we have YN =2e(A) + Ein -n®) 2As / (42) 8 se a a a’ an a where the subscript on A refers to the progress variable of a particular reaction s. For a system of reactions we must write QW = Sy anv a 6 a Ss where the Vv . is the Sterivivmetric coefficient of species No in reaction s and for a system of two reactions _ » 1 2 dn = vg aA + ve AA, so that n -n° = vi (A, - AG) + ve (A, - ™) a - 20 - This allows equation 42 to be written for a system of two reactions as aA, Ar = Elvt (Ay - Ai) + V2 (A, - AQ) + an, if e Defining Bas as dAr B = vs / (43) e u rs a a on, we have , = =. Bs (A, - r*) s in general and for a system of two reactions we have A, = Bar (AG - AD + Bao ( A, - AQ). Since As = O then XK Bay X4 + Baz Xo (44) il Similarly Xo = Bay Xq + Bao X2 (45) Taking the Laplace transforms of equations 44 and 45 we obtain @w u AGT AGT TAG (4?) s Foi non trivial solutions we have the determinant Bar ~ P Baa Ba, Baa2-P with roots Pi and PL where P = unas [(B44 - Bool? + 4842 Baal” (48) + So we can write Bay -P Bae Bay Bao -P . 1 . 1 : 1 (PP, )(P-P_) ~ (P|-P_)(P-P_) © (P_+P_)(P-P_) For which the inverse Laplace transforms are -21- Pyt Plt Pt — 2 ana e _._—€ P -pP P -P ~"~p -P °* + - - + + - For the solution of the "set" (equations 46 and 47)we have -x,(0) Bap ~X, (9) Bao -x,) Baa Z,- [%(°) Bar -P -x, (9) By, = Py -x, 0) 62. - PL Ba, - P Bao (Pp, - P)(P-P) (P,- P_)(P- P) Bas Bo2 - P So that -x, ©) Bap -x, (0) Bae _ -X, (O. Baa - P, Pat -X,(0) By, - PL Pot (P, - P_) (P/- P) Replacing X, by A, - Ay and since X,(0) = - AY we obtain A, = Ay + P = {C(Byi-P,) AGB at Je" **-£(B 4 4-P_) N-B as Ase" -*K49) +e and in a similar fashion we obtain Az = AGvptpe (C81 P) MBarAPJe CBr -PAGBar Arle} 0) + = Equation 48 can be expanded as Baa + Bao Bar — Bag Ba2 Baa P = 1 i a ee eeee i rr a en CO Pe So that Baa Baa PL Bar + 2(Ba4 - Baa) Baz Baa P = #8 - ™ 22 2(Baa - Bae) And we can then write, by making use of these approximations for Po and Pos 1 ~ 1 Baz Bas _ Po - P ” Bay - Baz [a (Baa - Boo)? } (51) - 22 - And if we take Biz Bay 1 LK (Ba, - Bo2)? then equation 51 becomes 1 _ 1 PL - Po _ Baa _ Boo (52) On substituting for Pi ’ PL and (PL ~ PL ) into equation 49 we have Pit, Az 812 (ePtt_gPaty AGB Pyt_ Pat = r~e(qeert*y. SA E42 (QF ya tPrzher + If Bie Bei is taken as small,compared to the first two terms,we (Baa - Boz)? have A% & A= MN - ett) - rr (est ~ eft) (54) and similarly for A, we have Ac A= AG (a= Pty - pA Rat (Pet _ Pty (55) Equations 54 and 55 give the extent of each reaction and could be used to determine the concentration of any component in the system at any time t not far from equilibrium. For any chemical species No we may write dn = vidaA,+ vi adaA; a a a so that n= 2 + vu Ay, + ve A, (56) Suppose we were interested in knowing the effect of a change in tempera- ture on the yield at time t. We insert equations 54 and 55 into equation 56 and on differentiating one obtains dn. = yi aks + v2 dA, a at aT aT - 23 - e e = cnePat A - tert? AS St dP _ Az Bre Pa [ te Pyt aPe te -* &- 7 | at “aT Baan Bee “aT at ) (57) +(e Pit oP t Tr? BrsBeo FB a2 Ae Bre (=e _ GB ae Bip ar *(Baa-Ba2)? aT (Ba4-B a2)? aD / Baa~Bag at Equation 57 appears rather awkward and further substitutions and manipulations have failed to improve it. Thus it appears that even with the approximations made in leading up to equations 54 and 55,that their use in calculating the effect of a change in a parameter,such as tempera- ture,leads to rather unwieldy equations such as equation 57. The equa- tions are workable but still so cumbersome that the extension of lineari- zation methods,to the problem of a change in a variable for a system of competing reactions, has not led to readily applicable equations. OTHER ASPECTS In this section we point out that certain other questions, not considered in the first two sections,may be answered about the system with the aid of the methods already developed. Suppose we are interested in knowing the effect a change in temperature would have on the time required for a system to reach ninety percent of its equilibrium state. From equation 4 we have t = - tin(1- yy n*) but for these considerations we have fixed ” rn = 0.9 asa constant. Taking the derivative we have the answer sought as at e, at _ e dint see-in(l- YR) Se2-an 1 - WAP) SERS ky For the reaction N, + N, = N, in a closed vessel,we can 2 quite easily obtain the effect that a change in initial concentration of N, = N, = a would have on the yield, near for example ninety percent of equilibrium, compared to the effect of a similar change on the equilibrium - 2h. yield. In a manner similar to that employed for equation 19,one can ob- k tain = = — for this reaction, where e° = 1 + 4aK, and subse- 1 quently din ft = = . Similarly we can obtain dinA® = a and on insertion of these equations into NPS equation 9 we obtain eKa 1 dn;/da f(y. ey, - ~e te 7 i and /aa f-(1-¢f)[ mn (1 IJ / i 1 These two examples should demonstrate the ease with which various other questions concerning a chemical system may be answered when lineari- zation methods are employed. - 25 - REFERENCES SECTION I R. P. Rastogi and K. G. Denbigh, Chem. Eng. Sci. 1957, 7, 261-8. J. G. Kirkwood and B. Crawford, Jr., J. Phys. Chem. 1952, 52, 1048-50. A. H. Taylor and R. H. Crist, J. Am. Chem. Soc. 1941, 63, 1377-84. K. Laidler, Chemical Kinetics, McGraw-Hill Book Co., New York, 1950. O. Knoblausch, Z. Phys. Chem. 1897, 22, 268-76. I. Wads8, Acta Chem. Scand. 1958, 12, 630-40. H. A. Smith and H. S. Levenson, J. Am. Chem. Soc. 1939, 61, 1172-75. Kistiakowsky, Ruhof, Smith and Vaughan, J. Am. Chem. Soc. 1936, 58, - 6- IT EQUILIBRIUM PRESSURES AND PHASE COMPOSITIONS FOR NON-IDEAL MIXTURES - 27 - INTRODUCTION A. good deal of effort has been expended in the past few years in developing equations equal to the task of describing the equilibrium pressure, temperature and composition relations for non-ideal mixtures. Contributors prominent in this field have been Clark, whose equations relate liquid and vapour composition; Hirata, whose equations express the liquid-vapour relation as three straight line segments; and Redlich and Kister, whose equations describe the ratios of activity coefficients. Such equations have been successful in the sense that they predict cor- rect results with varying degrees of accuracy. However, all have in- volved the determination of a number of empirical constants and conse~- quently have required much experimental data on the mixtures. Since the necessity for determining much experimental data in part nullifies the usefulness of these equations it would be worthwhile to develop a method employing only the readily available pure component data and fewer empirical parameters. Such an equation will not be ex- pected to be as exact but since all the required data are available or readily determined analytically it should prove useful. in this section an expression is derived for the equilibrium total pressure over a liquid mixture. In addition partial pressures and phase compositions may be calculated from the equation. To obtain the equation ,conformal solution theory is applied to the liquid phase and a second order virial equation is employed in the description of the gas phase. - 28 DEVELOPMENT AND METHOD We let the equation of state for the gas phase be given by PV P et ttele 2 y y 35] ) i jij where: P is the total pressure over the liquid phase;y, , i... n;, is the mole fraction of species Ne in the gas phase, Bi,is the second virial coefficient for the interaction of chemical species N, and Nj in the gas phase. From equation 1 we have the partial molar volume V,. of species Ny given by Vo = - ‘ RT + Z é, ¥53; [B. By + Bad (2) Taking Qin fy V.. 37° -s (3) one obtains (g)P _ P fi. = y, P exp [ RT 2 2, ¥i¥; (Bey. - Bs + Bad (4) in which s(6)P is the fugacity of species Ny. in the mixture of gases at a total pressure of P and mole fraction Vee For the liquid phase,we express the fugacity fe (1) of species N, in the liquid mixture by the relation k (2) = x, S. gclref. (5) where: xX, = mole fraction of N,. in the liquid phase. : p(tret. = the fugacity of species Ny in some reference state, ri k 7 is the activity coefficient. For the vapour and liquid in equilibrium,at the same temperature - 29 - and pressure,we may equate the fugacities as (1) _ ,(g) fo = oF (6) Substituting equations 4 and 5 into equation 6 we have (1) ref. PTs Inx, +1n % +1nf -inyt oe 4 yyy; (BiB +B) (7) We select as the reference state, for pret. the pure liquid of species Ny. at the same temperature and pressure as the mixture so that fi. == f, (8) And " o(1) P GP in 30D Pk Ve f es ns rae aF " Pe . o(1)P, " in which fi. represents the fugacity of species Ny at a pressure Pio where Pe is the vapour pressure of pure species at the same temperature as the mixture. ve?) is the average molar volume of Ny over the inter- val P to Pie Since we know o(1)Py o(g)Py fi = fi and o(g) PK tt Pi. \ fy = Py exp ( ar Be o(1) PB vV, (P - P, (l)ref. tt k *k k x ) in f = In P. +Rr ot EP (9) To obtain an expression for % x we resort to conformal solution theory (1,2), and use the expression from reference 2 E ) = — - Ss d.. 10 In 8 k RT [ 4 54a. i= *5x5955 J (10) - 30 ~ Insertion of equations 9 and 10 into equation 7 leads to the required equation " ou) -2 2 Ey yp, yp DERBY, a " RT i oj Vir gj ijkl. ;: PY TPM [e He Eo oT ( 2 xd, )- iz j 57545 | | (11) e In equation 11 we have used the following notation: Ai5 7 Bag? “til (12) Dige = Six 7 Sig * Ope = Bi 7 Fig t Bue 7 Fla (13) 4; = 2f5 4 “(£55 + f5;) (14) ds, = 2fiy ~(£55 + f5;) (15) The significance of the first two parameters,which are defined by equations l2 and 13,is clear and we see that the term in D.. attempts to correct jk for the mixing imperfections of the gas phase. Although a more complete description of the significance of the parameter d,, is available (1,2) we can, for the purposes of this section, consider a as performing the same function for the liquid phase as Ds ik - figurational energy of the reference species used in defining as 5 For did for the gas. BE, is the con- convenience,we may think of the first and last exponential terms as ac- counting for imperfections due to mixing of the liquid and gas phases respectively. The middle term is not due to mixing but rather to the pure components. For binary systems equation 11 simplifies considerably to (1) ve TERI)? A yg gp CPP) Bee) | | By (Lem) dae P=Pl! —k [ e RT He RT || ext | (16) Ik To apply equation 16 to any system we require values for A,2, Pia Boo - 31 - gO) k? Ey and d,2 and in line with the ideas stated earlier,we have tried to obtain the required data primarily from the more readily available pure component data. The proposed sources and methods for obtaining the data are outlined below. The vapour pressure P' of most compounds is tabulated at various k temperatures in appropriate handbooks or in reference tables such as Jordan's tables (3). In order to obtain the average molar volume ve) we follow the procedure outlined by Fried and Vilim (7). This procedure, based on theory of corresponding states as modified by Watson (5,6), is valid for T< 0.8 and is reasonably accurate. The virial coefficients By. may be obtained from the literature. However, since this limits the applicability of equation 16 to the temperature ranges covered by the ex- periment,we have used a Lennard-Jones potential and the force constants tabulated in Tables IA and IB, pp. 1110 in reference (8). Similarly in calculating the B,, for A,,,we have used the Lennard-Jones potentials and have assumed applicability of the combining rules (8) u oom rT + of ) ig 2 i " ; %e é (€5€ 5) ij The conformal solution interaction parameter a. may be found tabulated in the literature (1,2) or calculated from experimental data as described by reference (1). In the calculations herein,only systems for which a. is tabulated are considered. The configurational energy EY may be ob- tained by the equation (4) lL BE, = Ve ° y RT? ooo ~ AH, +P v's) - vi)) (17) where the A refers to the change of the reference species from liquid to vapour. This equation can easily be derived (4) with the aid of - 32 - thermodynamics and the statements g = Eo) . g(8) 3° fo} -O5 io) ° OE =AH =- PAV eo} ce] [e) and the virial equation of state for the pure gas given earlier, The data so obtained is given in Tables I and II. APPLICATIONS AND DISCUSSION In equation 16 we have already pointed out that the exponential is the sum of three different terms namely the corrections to the liquid, to the gas and to the pure components. At lower temperatures and pres- sures (i.e. at the boiling temperatures near one atmosphere) the terms in 4,, and for the pure components may be small in comparison with the term in d,,. In such a case equation 16 can be approximated by Vy RT This equation,which corrects only for the liquid imperfections,is readily solved (i.e. it is not a transcendental equation when x, and T are the independent variables). A less drastic approximation is valid when A,, is much smaller than either d,, or the Poynting term for the pure com- ponents. In this case we may write Pe Py Xk + exp [ Bo (1 - X, 2 dy, - (P- Pk) (Bi - ne) | (19) Vx RT This latter approximation is valid for certain systems at higher tempera- tures and pressures and we may consider it as adequately correcting for both the gas and liquid imperfections. The systems indicated in Table II have been selected to test the accuracy of the results predicted by equations 18 and 19, The selection - 33 - of these systems is based on the availability of experimental data with which to compare the calculated results. The choice is limited at higher temperatures and pressures because of the lack of experimental data. Calculated and experimental values are tabulated and compared in the Tables III through VII and the figures I through VI. From the Tables III and IV it is seen that the maximum deviation of the calculated from the experimental is about 5%. Also it is evident that of the order of 85% of the deviation from ideality is predicted by equation 18. Although it was known that conformal solution theory is not too satisfactory for the Benzene-Cyclohexane system,the system was never- theless selected since experimental data was available. Table V thus only bears out the known fact that the conformal solution parameter d42 = ~ 0.065 does not give satisfactory results. If d,, is adjusted to give a best fit of the experimental data,as in Table VI, then one can use this adjusted d,. at other temperatures and pressures. In Table VII we have the pressures calculated by equations 18 and 19 using the different values of d,2,. It is clear from Table VII and the accompanying figures V and VI that if the conformal solution is adjusted at a given tempera- ture then the parameter applies equally well at another temperature. It is also evident that the vapour phase contributions are significant being about 25% of the total correction. Again experimental and calculated re- sults agree to within about 5%. - 34 - Pure Substance Data Table I " OH Ep °(1) Symbol OK atnos. Koal/ Keal/ 1 /nole "oy ata. Liters CS, 308.4 0.674 (CH) CO 308.4 0.453 7.41 6,80 CHCl, 293.2 0.215 7.498 6.917 CS, 293.2 0.399 CC gHy2 313.0 0.243 0.770 554.2 40.4 0.115 Qu 313.0 0.239 7.06 7.14 0.850 561.7 41.7 0.095 CC gHy2 475.5 13.4 0.355 554.2 40.4 0.143 GH 475.5 14.4 5.325 4.754 0.408 561.7 47.7 0.115 System Data Table II Numbers in parentheses are literature references Temp -Biz S42 Ga (are -dyz, adjusted Jf, System °K lfmole l/mole from lit.) for fit at (T,) °K CS./(CH;),CO 308.4 0.130 (2) CS,/CHC1; 293.2 0.054 (2) GH/CyC gy 2 313.0 0.065 (1) 0.0426 313 GH/CyC Hy. 475.5 0.388 0.007 0.065 (1) 0.0426 313 - 25 Calculated Results Compared to Experimental Results System: Carbon disulphide/Acetone Table III Temp: 308,4°K d42? = 0.130 - Exptl. data: Ref. (14) Total Vapour Compo~- Pressure Partial Pressures (atmos. ) sition,Mole Mole fraction atmos. Fraction CS, CS, in liquid CS, Acetone Exptl. Calc. Exptl. Calc. Exptl. Calc. Exptl. Calc. re) 0.453 0.453 6) ¢) 0.453 0.453 fe) 0) 0.2 0.750 0.725 0.368 0.341 0.382 0.384 0.50 0.469 0.4 0.833 0.797 0.497 0.454 0.335 0.343 0.59 0.569 0.5 0.862 0.809 0.55 0.484 0.325 0.598 0.6 0.862 0.815 0.559 0.510 0.303 0.305 0.62 0.625 0.8 0.857 0.800 0.605 0.571 0.250 0.229 0.70 0.714 1.0 0.674 0.674 0.674 0.674 0) 0 1.0 1.0 - 36 - System: Carbon disulphide/Chloroform Table IV Temp: 293.2°K ayo: -0.054 Exptl. Data: Ref. (15) Total Vapour Compo- Pressure Partial Pressures (atmos.) sition, Mole Mole fraction atmos. Fraction CS, CS, in liquid Exptl. Calc. CHCI 5 CS 2 Exptl. Calc. Exptl. Cale. Exptl. Calc, ©) 0.215 0.215 0.215 0.215 0 ) 0 @) 0.2 0.285 0.297 0.180 0.176 0,103 0.120 0.42 0.41 0.4 0.332 0.344 0.143 0.143 0.190 0.201 0.60 0.59 0.5 0.350 0.361 0.125 0.126 0.225 0.234 0.66 0.65 0.6 0.365 0.371 0.103 0.108 0.260 0.265 0.072 0.71 0.7 0.375 0.384 0.083 0.088 0.293 0.296 0.78 0.77 0.8 0.385 0.392 0.060 0.065 0.325 0.327 0.83 0.83 1.0 0.399 0.399 oO 0 0.399 0.399 1 a - 37 - System: Benzene/Cyclohexane Table V Temp: 313° G42: - 0.065 Exptl. Data: Ref. (13) Total Vapour Compo- Pressure Partial Pressures (atmos.) sition,Mole Mole fraction (atmos. ) Fraction CS, Benzene on Benzene Cyclohexane +qua Exptl. Calc. Exptl. Calc. Exptl. Calc. Exptl. Calc. 0 0.243 0.243 6) 0.243 fe) 0 0.2 0.262 0.277 0,077 0.200 0.245 0.277 0.4 0.271 0,290 0.125 0.164 0.430 0.432 0.5 0.272 0.291 0.144 0.146 0.500 0.497 0.6 0.271 0.298 0,162 0.127. 0.570 0.560 0.8 0.262 0.276 0.197 0.078 0.740 0.715 1,0 0.239 0.239 0.239 ce) 1.0 1.0 - 38 - System: Benzene/Cyclohexane Table VI Temp: 313° Exptl. Data: Ref. (13) Total Vapour Compo- Pressure Partial Pressures (atmos. ) sition,Mole (atmos. ) Fraction Mole fraction Benzene Benen? Benzene Cyclohexane 4 Exptl. Calc. Exptl. Calc. Exptl. Cale. Exptl. Calc. ce) 0.243 0.243 6) 0.243 O 0 0.2 0.262 0.262 0.067 0.196 0.245 0.250 0.4 0.271 0.272 0.114 0.158 0.430 0.420 0.5 0.272 0.273 0.135 0.137. 0.500 0.496 0.6 0.271 0.2715 0.155 0.116 0.570 0.572 0.8 0.262 0.262 0.193 0.066 0.740 0.744 1.0 0.239 0.239 8) 1.0 1 - 49 = System: Benzene/Cyclohexane Table VII Temp: 475.5°K dag: - 0.065 (literature ave.) and d42 = ~ 0.0426 (adjusted) Exptl. Data: Ref. (12) Calculated Total Calculated Total Pressure for Pressure for : Exptl. Total (atmos. ) (atmos.) Fraction Benzene Pressure in Liquid (atmos.) Corrected Corrected Corrected Corrected anand for liquid for gas and for liquid for gas and imper~ liquid im- imper- liquid im- fections perfections fections perfections (@) 13.40 13.40 13.40 13.40 13.40 0.2 14.20 14.41 14.59 14,12 14,22 0.4 14.70 14.91 15.17 14.545 14.71 0.5 14.80 15.04 14.67 14,85 0.6 14.96 15.14 15.42 14.73 14.90 0.7 15.00 15.10 14.745 14.90 0.8 14.97 14,98 15.18 14.70 14,82 1.0 14,40 14,40 14.40 14.40 14,40 OSSUuUrPG Dae en raf © . oo 0.4 O.2 L a) experimental Pressures ahOn Figure 1 T Ae ee . _o ™ 2 ~ a“ Total Pressures RN ‘ et Z oe ”,. + a a" _ 7 _ —_— ra —_— atl “ a“ 4 ay, 4 Partial Pressures Z. 7 / if sS N VA a N\ > aN N\ aN + + + —} OW 0.4 0.6 0.8 . 1,0 Mole Fraction Carbon disulphide in Liquid Phase Total Pressure and Partial Pressures over a Liguid Mixture of Carbon disulphide and Acetone emperature 303°K Calculated Pressures — eee Raoults Law (ideal) Pressure eres) z pn ressure Citmos t i Figure 2 0.4 4 a) We, 0.2 + Partial Pressures i i i i ¥ Li v r o 0.2 0.4 0.6 0.8 1,0 Mole Fraction Carbon disulphide in Liquid Phase Total Pressure and Partial Pressures over a Liquid Mixture of Carbon disulphide and Chloroform Legend Temperature °92,2°K “xperimental Pressures Calculated Pressures a ee heros) rs (Atrios on Presser whom 1.0 7 Fisure 3 0.8 0.6 C.& vapour 0.2 P System B 4, 4 rl i v ¥ ¥ 7 o 0.2 0.4 0.6 0. Mole Fraction Carbon disulphide Composition of Liquid and Vapour Phases for a Given Temperature and Pressure . Legend System A (upper) Carbon disulphide and Acetone nonpenature 3 308, 4K System B (lower) Carbon disulvhide and Chlorcform Temperature 295.2°K Experimental Pressure alculated Pressure —_- — — — — \ / Pressure (Atmospheres Q.29 4 9.28 -43 0.27 T 0.26 0.25 0.24 T 0 0.2 ok 0.6 0.8 1.0 Mole Fraction Benzene in Liquid Phase Total Pressure over a Liguid Mixture of Benzene and Cyclohexane Legend Temperature 313°K axperimental results Corrections for liquid with d,,_ Corrections for liquid with d,, — eee Whe Figure 5 15.5 T 14.5 14.0 F i i =a v —_ 7 T 0 O.2 O.4 0.45 0.8 1.0 Mole Fraction Benzene in Liquid Phase Total Pressure over a Liquid Mixture of Benzene and Cyclohexane Legend Temperature 475.5°K dio =z = 0.065 Experimental Corrections for liquid only Be ee ee Corrections for both gas and liquid —-—-—--——-—- =— (Atmospheres) ye EL Pressurs 15.0 14.5 14.0 13.0 ~45= Figure 6 I 1 i i 7 ss ¥ as Mole Fraction Benzene in Liquid Phase Total Pressure over a Liguid Mixture of Benzene and Cyclohexane Legend Temperature 475.5°K Experimental results Corrections for liquid only —-—-- - Corrections for both gas and liquid l. 9. 10. ll. le. 13. 14, REFERENCES SECTION II I. Prigogine, Molecular Theory of Solution, Chapt. IV; Interscience Publishers Inc., New York, 1957. H. C. Longuet-Higgins, Proc. Roy. Soc. A(1951), 205, 247-69. T. E. Jordan, Vapour Pressures of Organic Compounds, Interscience Publishers Inc., New York, 1954. D. Cook and H. C. Longuet-Higgins, Proc. Roy. Soc. A (1951), 209, 28-38. K. M. Watson, Ind. Eng. Chem. 1931, 23, 360-4. K, M. Watson, Ind. Eng. Chem. 1943, 35, 398-406, Fried, Vilim, Vapour-Liguid Equilibrium, pp. 107, Pergamon Press, New York (1958). Hirschfelder, Curtis and Bird, Molecular Theory of Gases and Liguids, John Wiley and Sons Inc., New York (1954). Hougen and Watson, Chemical Process Principles, Part II, Chapt. XII; John Wiley and Sons Inc., New York (1947). B. H. Sage and W. N. Lacey, Ind. Eng. Chem. 1940, 32, 992-6. G. Haselden et_al., Proc. Roy. Soc. A(1951), 209, 1-14, V. N. Kumarkrishna, Amer. Inst. Chem. Eng. Jr. 1957, 3, 191-97. J. Zawidzki, Z. Physik Chem. 1900, 35, 129-203. J. Hirshberg, Bull. Soc. Chim. Belg. 1932, 41, 163-95.