Phase equilibria of strong electrolytes in aqueous solutions from total pressure measurements

Fluid Phase Equilibria, 6 (1981) 149-168 Elsevier Scientific Publishing Company, Amsterdam

149 - Printed

in The Netherlands

PHASE EQUILIBRIA OF STRONG ELECTROLYTES IN AQUEOUS SOLUTIONS FROM TOTAL PRESSURE MEASUREMENTS *

G. WOZNY Uniuersit&I Siegen - Gesamthochschule,

Fachbereich

11, 5900 Siegen (F.R.G.)

H. CREMER GVC, VDI-Gesellschaft Verfahrenstechnik (F. R.G.) (Received

August

29th, 1980; accepted

und Chemieingenieurwesen,

in revised form February

4000 Diisseldorf

6th, 1981)

ABSTRACT Wozny, G. and Cremer, H., 1981. Phase equilibria of strong electrolytes in aqueous tions from total pressure measurements. Fluid Phase Equilibria, 6: 149-168.

solu-

Some methods are presented for calculating the phase equilibria of strong electrolytes; existing experimental data are treated by means of different calculation procedures resulting in a 2-parameter formulation for the phase equilibrium of HCl/H,O. Initially these methods have been developed for the estimation of phase equilibria from total pressure measurements of hydrocarbons. They have been expanded for strong electrolytes and their efficiency is shown with reference to the system HCl/H,O. The choice of the system is arbitrary, the methods have been used as well for HBr/H,O and H2SO,JH20 with comparable effectiveness. Fortunately a lot of experimental data is available for the system HCl/HgO making possible a detailed comparison between calculated and experimental data. The critical temperature of HCl is about 51.4’C. For phase equilibria to be presented between 0 and 200°C an unsymmetrical standardization is necessary for temperatures higher than the critical temperature. On the other hand a uniform and consistent representation of the phase equilibrium is required e.g. for the mathematical simulation of separation processes to avoid problems of convergence. Therefore the azeotropic point has been used as a reference state for the activity coefficients of the electrolyte within the whole temperature range quoted. The influence which is introduced by this choice on the representation of the phase equilibrium is discussed in detail. An advantage of the methods presented is that they do not need the difficult experimental measurement of vapor mole fractions. The tot,al pressure of the azeotropic solution which is required for the standardization is estimated from the total pressure curve.

* A comprehensive 19’78, Miinster.

version was presented

0378-3812/31/0000-0000/$02.50

to the Thermodynamik-Kolloquium,

@ 1981 Elsevier Scientific

Publishing

Oct. 2-3,

Company

150 INTRODUCTION. SYSTEM HCl/H20

Phase equilibria data for the system HCI/H,O have been published in the temperature range from 0 to 110°C and for total pressures up to 1 bar (Schmidt, 1953; Fritz and Fuget, 1956; Othmer and Napthali, 1956; Haase et al., 1963; Perry and Chilton, 1972). Total pressure data from 110 to 280°C and for total pressures from 2.7 to 68 bar have been measured by Staples and Procopio (1970). Virial coefficients have been stated by Kao (1970) and Funk (1974); recently Rau et al. (1978) published a modified Redlich-Kwong equation - the matching was made on the basis of P, V, T measurement,s in ranges of pressures and temperatures from 1 to 150 bar and from 20 to 5OO”C, respectively. Formulations to estimate ionic activity coefficients have been given by Greenly et al. (1960) for molalities m < 1, by Akerliif and Teare (1937) for 3 < 172< 16 and temperatures between 0 and 50” C, by Pitzer and Mayorga (1973) and Bromley (1973) for m < 6. Hala (1963,1967) as well as Chen and McGuire (1970) derived a Debye-Hiickel formulation expanded by a Margules term. Vega and Vera (1976) estimated the phase equilibrium of the system HCl/HzO from 0 to 100°C using an empirical 4-parameter equation. Cruz and Renon (1978) published a new thermodynamic representation of binary electrolytic solutions including the vapor-liquid equilibrium of the hydrochloric acid/water system at 298.15 K using the known dissociation constant and 6 parameters. Chen et al. (1979) also published the phase equilibria of aqueous electrolyte systems enclosing HCl/H,O. They used a 5-parameter formulation for the excess Gibbs energy and the Henry constant for the standard state only at 25°C for the HCl/HzO system. This publication tries a 2-parameter formulation and a representa.tion of the phase equilibrium from 0 to 200°C from total pressure data whicln is possible because of the standardization to the azeotropic point for the activity coefficients. The effectiveness of the formulations of H&la (1963 and 1967), Chen and McGuire (1970) and of Vega and Vera (1’976) has been evaluated and compared by Wozny (1979). The 2-parameter formulation of Hala (1967) showed very good results; however, H&la (1967) restricted his calculations to the relative volatility, This procedure avoids the explicit calculation of the activity coefficients and the determination of the reference fugacity. The methods of Barker (1953), Tao (1961), Minh et al. (1970) and Mixon et al. (1965) for the calculation of phase equilibria from total pressure data were expanded. Primarily these methods were developed for hydrocarbon mixtures, they are now used for electrolytes. BASIC ASSUMPTIONS

The basic assumptions of this method are that (a) the reference fugacity of water is equated to the product of the vapor pressure and the fugacity coefficient of pure water at the system temperature; and (b) the reference

151

fugacity of HCl is equated to the adjusted total pressure of the azeotropic solution at the system temperature. Consequently, the activity coefficient of water is normalized to unity for pure water and the activity coefficient of HCl is normalized for the fluid with azeotropic composition. The vaporliquid equilibrium calculation for the system HCl/H,O is complicated by the fact that HCl does not exist as a pure liquid at the temperature of the system for t > 54.1”C. Thus one cannot know the vapor pressure or, more fundamentally, the fugacity of the pure liquid upon which the activity is usually based. The definition of a hypothetical pure liquid does not allow the experimental determination of the standard state. The unsymmetrical normalization which leads to Henry’s law is not chosen because the Henry constant is not known for the temperature range under investigation, from 0 to 200” C. Additionally Van Ness and Abbot (1979) showed that there is no inherent advantage in using the standard states based on Henry’s law. It is well known that the choice of the standard state is arbitrary but has to be done at the temperature of the system. The standard state has been chosen to be close to conditions of interest in phase equilibria. The azeotropic point which is chosen for the system HCl/H,O is almost in the middle of the concentration range. Since in the literature no phase equilibria data are published at temperatures > 110” C, but total pressure measurements are published by Staples (1970), the azeotropic point was chosen for the arbitrary standard state. Only for this point is it possible to calculate the phase equilibrium conditions from total pressure data. EQUATIONS

The thermodynamic description of electrolytes based on the following basic equations

in aqueous solutions is

mass balance m,, = mmol+ 0.5 (m, + m_) electron

(1)

balance

m+ =mchemical equilibrium

(2) between dissociated

and undissociated

electrolyte

K = (a+a-)l%l,l

(3)

phase equilibrium puv = /$ If association within the gaseous phase is negligible and the fluid is completely dissociated it is

(4)

152 (P2y2p

= f%2*Y2

(6)

with the indices 1 = HCl, 2 = H20, and the ionic molar fraction is as derived by Hala (1963,1967) xq, = X2/(2X,

+ X2)2

(7)

x2* = Xzi(2Xr

+ X,)

(8)

The reference fugacity e is derived as for a pure condensed phase following the fact that the azeotropic solution vaporizes as a pure component (Prausnitz, 1969).

With the chosen normalization it follows from eqn. (5) for the azeotropic point with X1 + X,,; Y, + YaZ= X,,; P + P,, and r;+ = 1, that after some rearrangements KN

= [2x,,

+ (1 -&d2/xaz

(10)

With eqns. (7) to (10) it follows from eqns. (5) and (6) that q1

y,p

(2x” +P -x,,>>” ,,,,~,

=

az rp2y2p

= 2xx; 1

~2Po2~0

x

)~$+P,,IJT,,

exp u’Z(\~paZ)

(11)

2

2 exp

uw ___I_

2

-.Po2)

RT

(12)

*

The partial molar volume is replaced by the molar volume and the pressure dependence is omitted. The formulation of H&la et al. (1967‘) holds for the excess Gibbs energy GE

= C13’2 - AIXl + Ar2X1X2

(nl+.303RT

With regard to the standardization tropic composition (see Appendix log &

= C(-0.5X:.5

-C(-0.5X;,5

+ l.5g.5)

+ l.5ei5)

log y2 = -0.5CX;.5

to an HC1/H20 solution with azeoA): + A12(X2 - X1X2)

-A,,[(1

+ A,,X1(l

(13)

--X,,)

-X,,(l

-X,,)l

(14)

-X2)

(15)

The calculation of the phase equilibrium is based on the following equations for the temperature dependence of the virial coefficients BHCI

=*11

&r20 = *22 *HCI-H~O

= = =*,2

77.43 X [l.O - 0.704 exp(400/T)] 49.85 X [l.O - 0.328 exp(128&57/T)] =

0.085827

X [l.O - 2i30.110 exp(lOOO/T)]

(Funk, 1974) (Kao, 1970) (Wiister,

1979) (16)

153 TABLE 1 Mean difference (%) of fluid molar fraction X,, and total pressure P,, of the azeotropic solution according eqns. (17) and (18) compared with literature data Range

Mean difference

X a2

o-loo”c o-210°c

0.44% 1.4%

P az

o-loo”c o-210°c

1.28% 2.22%

For the temperature dependence of X,, and Paz the following relations were determined by non-linear regression from literature data [Schmidt, 1953; Fritz and Fuget, 1956; Othmer and Naptha-h, 1956; Haase et al., 1963; Staples and Procopio, 1970; Perry and Chilton, 19721 X,, = 1.9935 X 1O-1o p - 2.9323 X lo-’

T3 + 1.5788 X 1O-4 T2

- 3.7305 X 1O-2 T + 3.4218

log Paz= 8.552104

(17)

1979.558 15.8989 t + 241.0547 + (t + 25)2

(18)

where T is in K, t is in “C and Pa, is in Torr. Table 1 shows the mean difference in percent between eqns. (17) and (18) and the literature data. The molar fluid volume of the azeotropic solution was determined according to u,“~= [(1.01205

+ X,,O.86118)

- (0.000362

(m3 kmole-‘)

+ X,,O.O01513)t]

M,,/lOOO (19)

with the molecular weight of the azeotropic solution M,, and temperature t in “C. The molar fluid volume V: and the vapor pressure of water were calculated according to Schmidt (1963). DISCUSSION

OF CALCULATION

PROCEDURES

The methods of Barker (1953), Tao (1961), Minh et al. (1970) and Mixon et al. (1965) which had been developed initially for hydrocarbon mixtures were expanded by Wozny (1979) by introducing the relations for the phase equilibrium according to eqns. (11) to (19). The influences of this formulation on estimated phase equilibria of the system HCl/H,O are discussed by Wozny (1979) in detail. Some results which were obtained using the extended method of Barker and the method of Tao are discussed here. The basic equation of the method

154

of Barker (1953) is

where M = number of measuring points with the calculated from eqns. (5) and (6)

pressure Peal

(21) The optimization procedures of Powell (1965) were used to determine the parameters C and Al2 of the excess Gibbs energy in eqns. (14) and (15) after introducing them-into eqn. (21). The initial equation for the iteration of the phase equilibrium from total pressure measurements by means of the numerical integration is the GibbsDuhem equation

HE gdP--dT RT2

= Xi d(ln ~9~) + X2 d(ln y2)

from which a non-linear equation is derived (Tao, 1961; Wozny, 1979) to determine the following ratio of the activity coefficients depending on several parameters

Po2,

(~02,

~2,

VE,

HE,

Y&C,

Y%Y~,

~1

(23)

The following chapters comprise some sensitivity analyses to determine the influence of different variables on the representation of the phase equilibrium according to eqns. (20), (21) and (22), (23). INFLUENCE OF THE NORMALIZATION

The correlations for X,, and P,, result in mean differences shown in Table 1. In order to analyze the influence of X,, on the representation of the phase equilibrium the value of X,, was reduced by arbitrarily 3% - P,, being constant - and the activity coefficients and the phase diagram were calculated by numerical integration of eqns. (22) and (23) respectively (and by direct minimization with respect to the total pressure, eqns. (20) and (21)). Figure 1 shows the shape of the activity coefficients and Fig. 2 the McCabe-Thiele diagram. The full line in Fig. 2 represents the experimental phase equilibrium. It is shown that the optional difference of X,, results in comparatively small differences around the azeotropic point. The differences between the experimental data and the results from the estimation of the phase equilibrium by numerical integration are smaller than those obtained by minimization with respect to the total pressure. The variation of Pa, - which is neces-

155

sary for the normalization as well -has a similar influence as it is shown by Wozny (1979). After the stepwise integration a regression of the values of ln(&/rz) is necessary to determine the parameters C and Al2 of eqn. (13). It is advantageous to operate with a vanishing weighting in the range of the azeotropic point to ensure a good estimation of the phase equilibrium. The equations for calculation are shown in detail by Tao (1961) for hydrocarbons and in analogy by Wozny (1979) for electrolytes. INFLUENCE

OF THE BOUNDARY

ACTIVITY COEFFICIENTS

The calculation of the phase equilibrium by stepwise integration of the Gibbs-Duhem equation (22) according to Tao (1961) requires the determination of the boundary activity coefficient rgl+. This coefficient results from eqn. (11) by means of a limit procedure. For a constant temperature for example it is -yol =l’/P,,(l + (U/L+X),l/P) for hydrocarbons (Gautreaux and Coates, 1955). The gradient aP/aX cannot be determined with sufficient accuracy from the P-T-X data available in the literature for the HC1/H20 system. Therefore Wozny (1979) developed a method derived from the Gibbs-Duhem equation to calculate the boundary activity coefficient for azeotropic solutions and the given standard state. 100,o

Y2 l?

v, 10.0

o.ood 0

O.OL

0.08

032

0,20 0.22 016 XtiCl

Fig. 1. System HCl-H,O at 5O’C. Influence of standard state 02” activity coefficients: YPcal; miniindex 1, HCl; index 2, Hz0 (Pa, = +O%;XaZ2= -3%). 9Yl*cal, . -, mization with respect to total pressure. X, ‘ylTcal; 0, ~2~~1; stepwise integration of the Gibbs-Duhem eqn.

156 1D ViCl :-3%

0.6

Ot6

0.4

0,2

t

--

5

l b

0,08

012

036

ct20022

XHCI

Fig. 2. System HCI-Hz0 at 5O’C. Influence of standard state on phase equilibrium (Pa, = fO%; X,, = -3%). X, Y~cl,~~l, minimization with respect to total pressure; m, eqn. Y HCl,exp; 0, Y~cl,~~l, stepwise integration of the Gibbs-Duhem

The influence of the calculated boundary activity coefficient is determined by a sensitivity analysis. Figure 3 shows the courses of the calculated phase equilibrium at 50” C. The calculation was made with a boundary activity coefficient y& = 0.024 calculated with the Gibbs-Duhem equation; this result is compared with similar curves derived from arbitrary changes of the boundary activity coefficients by *8% (i.e. y&+ = 0.022 and y&+ = 0.026). These changes are in accordance with the differences calculated using the method published by Gautreaux and Coates (1955). The boundary activity coefficient has an effect in the range of the azeotropic point; the calculation of the phase equilibrium is inaccurate in the vicinity of the total pressure minimum if the boundary activity coefficient does not match, A good representation of the phase equilibrium was obtained using the extended method of Barker (1953). Comprehensive sensitivity analyses have been made by Wozny (1979) with respect to the methods of Barker (1953), Tao (1961), Minh (1970) and Mixon (1.965). These analyses showed that Barker’s method can also be operated without numerical problems for electrolytic systems and temperatures higher than the critical temperature.

157

1.oo YHCI 0.80

0.60

OL' 0

-

0.08

0,OL YHC,

exp

0.12

0.16

yHClco,

v&o.22

x x YHClcol

V&O,2L

’

o

q

o YHClca,

0.20 X HCI

Y,,;, =0.26

Fig. 3. System HCl-Hz0 at 5O’C. Influence of the boundary phase equilibrium; index 1, HCl; index 2, HzO.

activity coefficients

on the

RESULTS

Direct minimization with respect to the total pressure according to Barker (1953) has been applied to calculate the phase equilibrium of the system HCl/H,O in the temperature range from 0 to 200” C and for pressures from 0.1 tcr 20 bar. Figures 4and 5 show the calculated phase equilibrium and the courses of the activity coefficients at 60” C each. This calculation was done on the basis of the total pressure data of Fritz (1956), Staples and Procopio (1970), Perry and Chilton (1972) and Wiister (1979). A comparison between calculated and experimental data of the total pressure and the partial pressure of HCl at temperatures of 50,70 and 90” C is shown in Figs. 6 and 7. There is good agreement between estimated values and experimental data. Figure 8 emphasizes the wide confidence interval of the parameters A, 2 and C of eqn. (13) which confirm the good matching to the experimental data. The values of Fig. 8 are for pressures in Torr according to Z(P,,, -P&

(

m

-

m Y,exp

Fig. 5. Activity

coefficients

0.12

O.l6

0.22

index

0.04

to total pressure:

0

with respect to total pressure.

with respect minimization

minimization

XHCI

1

02

at 60°C;

at 6O’C;

of HCl-Hz0

HCl-Hz0

0.08

x, cd

0.0-i

Fig. 4. Phase equilibrium

0.2

0.4

06

08

1.0 YHCI

0.12

1, HCI; index

0.08

0.16

2, H20.

)A,,

0.2 X HCI

:

0.22

gm.

temperature

A: 5o"c

.-

0

600.

0.01

0.0s

x

0.16

0.20

HCl

Loo.

F n

003

O.l6

0.20

mole fraction XHCl

032

experimental data: Perry, Chem.Eng. Handbook

calculated-experimental.

0

MO.

ki

I

G

600.

700.

800.

;

::

P -

Fig. 7. HCl - partial pressure over aqueous hydrochloric acid; comparison: calculated-experimental.

at 50, 70 and 9O’C; comparison:

mole fraction X

q.12

Fig. 6. Total pressure of HCl-H20

c,

Ff0 'O" experimental data: g 700.Perry, Chem.Eng.

n

L

a,

ui

*P I

160

I

A12

-10.1 system:

I

HCl - Hz0

temperature:

80°C

minimization with to total pressure

respect

-11

-11.:

IO

-12

Fig. 8. Confidence interval of parameters -Peal)?; pressures in Torr. we,,

10

F = 200 G = 220

Al 2 and C of eqn. (13); contour

lines of

The temperature dependence of the parameters Al2 and C is given in Figs. 9 and 10. For the calculation of the fugacity coefficients the virial equation, the Redlich-Kwong equation and a modification of it (Rau et al., 1978) have been used. Table 2 gives the constants for the calculation of the phase equilibrium of the system HC1/H20 in the temperature range from 0 to 200°C. The temperature dependence of the parameters Al2 and C is given by a fourth order polynomial. Figure 11 shows the results of boiling point calculations at 1 bar, 10 bar, 100°C and 150°C. There is maximum difference of about 1.5 mole % at a total pressure of 10 bar between the results of the virial equation and the ideal gas phase representation. The maximum difference between the Redlich-Kwong and virial equations is about 1 mole %. A comparison of experimental and calculated ion activity coefficients is given in Fig. 12 which indicates the dependence of the ion activity coeffi-

I 0

2 -

0

I

40

LIE 2.303RT

=

P

80

C I

- A,

I Xnu

I 120

Xm,+

I 160

I

TPCI

Au

1 200

I

I 220

1

Fig. 10. Temperature dependence of parameter C of eqn. (13) eqn.;-----, cp = l;---, modified Redlich-Kwong eqn.

eqn.,

Fig. 9. Temperature dependence of parameter A12 of eqn. (13) .___ eqn. 9cp= 1; - - -, modified Redlich-Kwong

-16

-

A12 -

(compare

(compare

1 0

with Table

with Table

-5

2). -,

2). -,

I 40

I 120

viriai eqn.;

-,

virial eqn. ;-,

I 80

T[OCl

I 200

I 220

Redlich-Kwong

Redlich-Kwong

160

162

TABLE

2

Constants

for the phase equilibrium RT = C1312 -A,X,,,

gE/2.303

equation

C

a4

C

-1.48492 x lo+”

-3.96790 1.19896 -1.24742 3.73691

a3

Viriaf equation

-412

-2.29736

a2

.

+a4t4

Redlich-Kwong

a0

at 0-200°C

+ XHC1XH20AIS

C;A r2=ao+alt+a2t2+a’3t3

01

of HCI/H20

x x x x

1O-2 1O-3 1O-5 lo-*

2.00763 6.51592 -6.21026 1.88407

x x x x

A12

-2.30705 -3.95522 1.17126 -1.22669 3.69791

lo-” lolo--‘; 10-a

x x x x

10-2 1O-3 lO-5 lO--s

-1.48572 x lO+l 2.01501 x 1O-2 6.28848 x lO@ 4.04695 x lo-+ 1.85639 x 1O-8

cients from the molality (Wozny, 1979). The activity coefficients calculated from P-T-X data are compared with experimental data from measurements of electromotive forces. For temperatures of 70 and 90°C the data of Akerlijf and Teare (1937) have been extrapolated. AY

0.001

lbar

l---i xfg

0.0

A x

A

A

- 0.001

AAXx x x

A

0.1

0.0

d,

x

0.0

x

0.2 0.3 XHCI

2.3 XHCI

_I_;:: 1;pJ::; 1 0.0

0.'

0.2 x-

0.3

0.0

0.1

Cl.2

0.3

X-HCI

HCI

Fig. 11. System HCl-H20. Calculation of boiling points at 1 and 10 bar, 100 and 15O’C; difference of vapor mole fraction according to Redlich-Kwong eqn. and virial eqn. compared with ideal gas phase. A, Redlich-Kwong eqn.; AY = Y~cl(,+,=l) Y HCl(Red.-Kw.eqn.).

X, virial

ew.;

AY

=

YHC~(~=I)

-

YHCl(virialeqn.).

163

Modified

Redlich-Kwong

C 2.26316 -4.31824 1.26980 -1.31482 3.91244

x x x x

1O-2 1O-3 10” 10-a

equation

cp= 1.0

A12

c

-1.48111 x lO+l 1.58923 x 1O-2 7.39200 x 10-4 -7.02427 x 1O--e 2.09154 x 10-a

-2.23237 -4.57145 1.35267 -1.36563 4.00958

-412

x x x x

10-Z 1O-3 1O-5 lo-*

-1.47819 x lO+l 1.36627 x 1O-2 8.11755 x 1O-4 -7.45075 x 10-6 2.16866 x 1O-8

3 lnyt

0

molality I

gE -=-ax RT

Inx+ 2bxi-2cx$dx2+Kx

3 K.a*aln~~-~~-(~-2d)~-(b-3cix~-cx,,-dx,Z, Fig, 12. System HCl-H20. Comparison of ion activity coefficients of HCl; 0, experimental, electromotive measurements (G. Akerliif and J.W. Teare); +, calculated from P-T-X data, 4-parameter formulation (Vega) macroscopic description; A, calculated from P-T-X data, 2-parameter formulation (Hala) microscopic description.

164

mass fraction g,,, Fig. 13. System HCl-H,O. Liquid enthalpy: -, experimental data of Van Nuys.

X, calculated

from activity

coefficients;

Using eqns. (24) and (25) the excess enthalpies of Fig. 13 have been determined from the temperature dependence of the activity coefficients. H= = X,R: a In 7fr/aT

+ X2@

= --HF/(RT2);

(24)

a In T2/aT

q

= -FlF/(RP)

The calculated excess enthalpies are compared Van Nuys (1943) using his reference points.

with the experimental

(25) data of

CONCLUSION

The calculation of the phase equilibrium of strong electrolytes from total pressure data has been discussed in detail extending several methods which were developed primarily for hydrocarbon mixtures. Particular attention is required for the evaluation of the reference state to ensure accurate results. There are two methods for the calculatSion of the boundary activity coefficients which require accurate data for the total pressure either at high dilution or at the azeotropic point. The data routine for the system HCl/H,O is reliable in ranges from 0 to 200” C and from 0.1 to 20 bar. The described method has been applied to the system HBr-H,O by Wiister et al. (1979) successfully.

165 LIST OF SYMBOLS a

a+ B,B11,Brz,B22 C,

AI,

$ &?&

G HE BE HL I KN m : POi P az R t T VE VL r 21, x2* xi XX, yi

Yi

701 Y* Pi V CpOi Pi cp az

;

Al2

activity ion activity second virial coefficient parameters of formulation for excess Gibbs energy (system HCl/H,O) fugacity of substance i reference fugacity of substance i molar excess Gibbs energy total Gibbs energy molar excess enthalpy partial molar excess enthalpy molar enthalpy of solution ionic strength ( =Xnc,) normalization coefficient molality number of moles pressure vapor pressure of pure substance i total pressure of azeotropic solution gas constant temperature (” C) temperaure (K) excess volume molar fluid volume of azeotropic solution molar volume of pure fluid i ion molar fraction (refers to component HCl) mean molar fraction of H,O fluid molar fraction of substance i fluid molar fraction of azeotropic solution (X,, = XHC1) vapor molar fraction of substance i activity coefficient of substance i boundary activity coefficient ion activity coefficient chemical potential of substance i number of electrons fugacity coefficient of pure substance i fugacity coefficient of substance i fugacity coefficient of azeotropic mixture step size mass fraction

166

Indices az E

azeotropic composition excess value calculated experimental molecular ion stoichiometric substance i reference state fluid phase vapor phase cation anion

Cd ew

mol ion st 0

L V + APPENDIX

A

From eq. (13) with = u1 RT In ylk

(W

= RT In y2 it follows for HCl with u1 = 2 log $+ = C(-0.5X:.5

+ 1.5X9

+ A12(.X2 -X1X2)

-Al

(A3)

and for H,O log y2 = -(C/2)

Xi.” + A12Xl(l

With the chosen normalization for X1 = X,,

and

r:t

-X2) the parameter A 1 is eliminated

=1

A, = C(-0.5X,1,5 + 1.5X:,5) +A12[(1 --X,,) Introducing

(A4)

-X,,(l

-X,,)]

(A5)

eqn. (A5) into (A3) gives eqn. (14).

REFERENCES Akerlaf, G. and Teare, J.W., 1937. Thermodynamics of concentrated aqueous solutions of hydrochloric acid. J. Am. Chem. Sot., 59: 1855-1868. Barker, J.A., 1953. Determination of activity coefficients from total pressure measurements. Austr. J. Chem., 6: 207-210. Bromley, L.A., 1973; Thermodynamic properties of strong electrolytes in aqueous solutions. AIChE J., 3: 313-320.

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