1
Derek Jaques Chance Technicol College Smethwick Stoffordshire, England
How to test the
Thermodynamic Consistency of Liquid-vapor Equilibrium Data
T h e basic mincides of liauid-vaoor equilibrium (1) and its appiicatioi t o theoretical calculations for fractional distillation (8) have recently been described. I n practice considerable weight is attached to the equilibrium data and hence its reliahilitv must he established. This Daver deals with a muchused testing- procedure. A
For a binary liquid mixture we can write for a change in an extensive property such as the free energy (G)
+ VdP + wdn, + mdnr
(1)
and pz denote chemical potentials and nl and nz the numbers of moles. At constant temperature and pressure we have the general equation (3) PI
G
=
+
n t p ~ nwz
(2)
(3)
Equating eqns. (1) and (3) Dividing throughout by (nl+nz)dxl and putting SM = S/(nl n2) and VM = V/(nl n2) gives
+
+
z ~ d p d d z , zdp2ldz1 = -SMdTldz,
z in z2 ~ +~n
l in ?, ~ +~n
2in ?, ~
~
(10)
For a11 ideal solution the free enernv of mixinn iA0,) is
+ VMdP/dz,
and therefore AGB = AG
- AGi
=
ntRT ln
YI
+ lhRT In y*
(12)
here AGE is the excess free energy change per mole of mixture, i,e,, nl + nz = 1, H~~~~we have AGE/2.303RT = z1 log
7,
+ 21 log y2
=
Q
(13)
This is the Redlich-Kister Q function. Differentiating eqn. (13) with respect to xl gives d Q l d z ~= z d d log r,/dzJ
+ log Y I + z d d log yddzJ
- log ~2
(14)
Rearranging eqn. (14)
(15)
By eqn. (a), with the introduction of a change of base for the logarithms, we have
Suhstituting eqn. (16) into (15) yields (5)
where SMis the integral entropy of mixing per mole of mixture, VMis the integral volume of mixing per mole of mixture, and XI is the mole fraction of component 1 in the liquid phase. Now pl = plo R T in rlxl and by differentiation with respect to XI
+
here 71 is the activity coefficient of component 1. Similarly for component 2 dp2/dzl = RT(d in m l d z , )
+~n
dQldzt = log Y , / Y ~
+ n d w + rrdn, + w d m
+
, in z, ~
+ zdd log r t l d z ~+) z d d log v d d z d
Differentiation of eqn. (2) yields dG = nldw
AG = n
.
The Gibbs-Duhem Equation
dG = -SdT
and
- RTIz.
(7)
Suhstituting eqns. (6) and (7) into (5)
This is a form of the Gihhs-Duhem equation when the temperature and pressure are varied (4). The Redlich-Kister Q Function and Its Application to Thermodynamic Consistency (5)
If nl moles of liquid 1 and n2 moles of liquid 2 are mixed, we have for the free energy change
+
dQ/dx, = log y r / y p - (SM/2.303RT)dT/dzl (VM/2.303RT)dP/Lzt (17)
Integrating equation (17) gives
+
x2logp; when XI = But by eqn. (13), Q = x,logr, 1, 71 = 1, and w = 0 and so Qz = 0; when XI = 0, x2 = 1, and 7 2 = 1, and so Q1 = 0. Thus AQ = 0 and the right-hand side of eqn. (18) equals zero. Hence if the expression in square brackets is plotted against the liquid mole fraction, the area under the curve should he zero, i.e., the ratio of the areas should he unity (Fig. 1). If the deviation from unity is appreciable, after suitable allowance for normal experimental errors, the data is not thermodynamically consistent. There are two cases to be considered. At constant temperature. The integral volume of mixing is often less than 0.5 ml/mole (6) and no serious error is introduced by neglecting the expression, (VM/2.303RT) ( d P / d ~ , )and ~ thus equation (18) hecomes (19) Volume 42, Number 12, December 1965
/
651
This is the condition for thermodynamic consistency of isothermal L/V equilibrium data. At constant pressure. Under isobaric conditions we have SM = HM/T where H M is the integral heat of mixing per mole of mixture and T is the boiling point of the mixture. Hence equation (18) becomes
This is the condition for thermodynamic consistency of isobaric L/V equilibrium data. However, if the boiling point difference of the components is less than 10°C, eqn. (19) can be used for isobaric data (6).
Table 2.
L/V Equilibrium Data for Ethanol and Ethyl Benzene at 1 atm (8)
Temp
("Cl
21
v,
Redlish-Kirter test of thermodynornic consistency.
The following exercise is designed to demonstrate how eqns. (19) and (20) are used to test experimental data.
Table 3.
here z and y are the mole fractions in the liquid and vapor phase respectively, po is the saturated vapor pressure a t 45%, and I1 is the total pressure. Hence calculate the logarithm of the activity coefficient ratio for each concentration (Table 1). Plot log (yl/-y,) against XI (Fig. 2) and estimate the positive area (A+) and the negative area (A-). The ratio of the areas (A+/A-) is 1.01which shows that the data are thermodynamically consistent. Isobaric data. Experimental L/V equilibrium data for ethanol (component 1) and ethyl benzene (comL/V Equilibrium Data for lxlpropanol ond Benzene at 45.00°C (7) p,O
=
136.05 mm Hg; p," = 223.66 mm He:
Figure 2.
652 / Journal o f Chemical Education
Z,
yn/zzp0
(22)
Calculated Data for Ethanol and Ethyl Benzene a t 1 atm
Isothermal data. Experimental L/V equilibrium data for isopropanol (component 1) and benzene (component 2) a t 45'C are given in Table 1 (7). Calculate the activity coefficients of both components a t each concentration from the equation
1.
ZI
The inadequacy of eqn. (19) in the present case will be demonstrated first and then the correct equation used. (a) Use of equation (13). Calculate the activity coefficients for the components using eqn. (22). At each concentration find the logarithm of the activity coefficient ratio (Table 3). Plot the latter term against ethanol concentration (Fig. 3) and evaluate the positive and negative areas. In the present example we find that A+/A- is 0.840. This value is too far from unity to indicate consistency but the data cannot be
Exercise
Table
PP (at=)
ponent 2) a t one atmosphere pressure are given in Table 2 (8). For a large boiling point difference a correction factor must be applied for the nonideality of the vapor. It is convenient to use the correction factor Z which can be evaluated from Scheibel's nomograph (3) and hence eqn. (21) becomes =
Figure 1.
m0 (ktrn)
Iroproponol-benzene.
Table 4.
Calculated Data for Ethanol and Ethyl Benzene at 1 atm mu
Figure 3. Effect of the corrective foctor v2.303 upon log ?I/y. for the othonol-ethyl benzene syltem. The solid line reprelent, log yr/>l, and >~ i12.303. tho dotted ICne represents log y ~ / -
J
0
condemned because a 58°C boiling point difference requires the use of eqn. (20). (b) Use of equation ($0). The heat of mixing of ethanol is not available so the data for the similar system, ethanol and toluene (7) will be used (Fig. 4). The required values are given in Table 4. Read off the values of (dT/hl), from a graph of boiling point versus
0.2
0.4
0.6
0.8
1.0
Mole Frmtion EtOH Figure 5. benzene.
Experimental boiling point-composition curve for ethanol-ethyl
composition (Fig. 5) and hence calculate g/2.303, where = (HM/RTZ)(dT/dxl)p. The corrected activity coefficient ratio gives A+/A- = 0.904 (Fig. 3). Allowing for the experimental errors associated with the very wide boiling range, this ratio indicates consistency. Literature Cited
0
0.2
0.4
0.6
0.8
1.0
Mole Fraction EtOH Figure
4.
Heat of mixlng of ethmol and toluene a t
(1) .IAQIJES, D.,Scl,ool Sri. Kw., 46 (IS')), BSd(1064). (2)JAQUES, I)., School Sri. Rw.,46 (160),5!U (1961). (:I)1 . ~ ~ 1 C 6 ., S., .\Nn I~SD.
