Fluid Phase Equilibria, 14 (1983) 383-392 Elsevier Science Publishers B.V., Amsterdam - Printed in Tbe Netherlands

383

THERMODYNAMIC CONSISTENCY TESTING OF PTx-DATA VIA THE GIBBS-HELMHOLTZ EQUATION

JAMES

Union

South

D. OLSON

Carbide Corporation, Research and Development Department, P. 0. Box 8361,

Charleston, West Virginia 25303 (U.S.A.)

ABSTRACT

Although the Gibbs-Duhem slope and area consistency tests cannot be used on

PTx vapor-liquid equilibrium data, the Gibbs-Helmholtz equation can be used to

test the consistency of PTx data measured at several temperatures. This is

done by comparison to calorimetrically-determined excess enthalpy (HE) data.

Results are presented for Gibbs-Helmholtz tests on ebulliometrically-determined

PTx data for the systems acetone + methyl acetate, propylene glycol + ethylene

glycol and ethanol + water. In addition, results on the effect of random

errors on Gibbs-Helmholtz testing of computer-generated PTx data are given.

These studies on actual and simulated PTx data indicate that, on the one hand,

random errors in pressure measurement usually claimed (0.01 to 0.1 kPa) should

not cause the Gibbs-Helmholtz test to fail, and, on the other hand, random

errors in pressure measurement large enough to give Gibbs-Helmholtz inconsis-

tency may not be large enough to cause unacceptable errors in the calculated

vapor compositions. This indicates that the Gibbs-Helmholtz test is stringent

and data that fail may still have practical value for chemical process design.

INTRODUCTION

PTxy (pressure-temperature-liquid mole fraction-vapor mole fraction) vapor-

liquid equilibrium data are redundant according to the phase rule. Hence,

differential and integral thermodynamic consistency tests can be constructed

from the Gibbs-Duhem equation (Prausnitz, 1969). More recently, a PTxy

consistency test has been developed in which y(exper) vs. y(calc) are examined

for systematic errors (Van Ness et al., 1973). In contrast, Px or TX (PTx)

data are the minimum necessary to specify VLE so that the Gibbs-Duhem tests are

not possible. However, GE data derived from PTx data measured at several

0378-3812/83/$03.00 @ 1983 Elsevier Science Publishers B.V.

384

temperatures can be compared to calorimetric HE data with the Gibbs-Helmholtz

equation,

-RT+ (GE/~~)/2 T], = HE (1)

This comparison can be used as a thermodynamic consistency test for PTx data.

Examples have appeared (Harris and Prausnitz, 1968; Bae et al., 1980; Van Ness

and Smith, 1981; Olson, 1981; Rubio et al., 1982). The discussion here is

limited to low-pressure VLE (up to ~1.5 MPa) where volumetric effects can

usually be neglected.

The principal objective of thermodynamic consistency tests is to show the

likely absence of systematic errors. The practical uses are (i) to referee

between different data sets on the same system, (ii) to justify confidence in

the chemical process equipment designed from the data, and (iii) to ensure

data of the highest quality for experimental standards and for testing and

extension of theory. The failure of PTx data to pass the Gibbs-Helmholtz test

is indirect evidence that the experimental measurements and/or the details of

the data reduction procedure contain systematic errors. This also suggests

that the vapor compositions calculated from the PTx data contain systematic

errors.

The purpose of this paper is to discuss “how bad” PTx data have to be to

fail the Gibbs-Helmholtz test and whether y values predicted from PTx data

that fail can still have any practical value for design. This is done by

examining Gibbs-Helmholtz tests on three systems measured at Union Carbide and

on computer-generated data that contain successively larger random errors in

pressure.

01 SCUSSION

Gibbs-Helmholtz tests on measured PTx data

Experimental PTx data suitable for Gibbs-Helmholtz testing have been

determined at Union Carbide for three systems: acetone + methyl acetate,

propylene glycol + ethylene glycol, and ethanol + water. These PTx data were

measured by ebulliometry (Olson, 1982) and GE data were derived by Barker’s

method (for a discussion of PTx data reduction methods, see Van Ness and

Abbott, 1982, Chapter 6). The GE model fitted to the PTx data was the

Redlich-Kister equation. The vapor-phase fugacity coefficients were

calculated from the pressure-explicit second virial equation except for

propylene glycol + ethylene glycol where the ideal gas model was found to be

sufficient. Equimolar GE/RT data were plotted against temperature to obtain

385

an equimolar HE(PTx) from a graphical slope. This HE(PTx) was then

compared to an equimolar HE(exper) value from heat-of-mixing experiments to

perform the Gibbs-Helmholtz test. In addition, plots of GE, HE, and TSE

vs. composition were constructed from the equation:

GE = HE - TSE (2)

TABLE 1

Comparison of calorimetric equimolar HE(exper) with equimolar HE(PTx) derived

from Gibbs-Helmholtz analysis of PTx data

System (KT)

HE T;;oPyr) 5

"5 F'F'y) 5 4 ErrSr p

Acetone (1) + Methyl Acetate (2)

323.

Propylene Glycol (1) + 408. Ethylene Glycol (2)

Ethanol (1) + 323. Water (2) 343.

5 86.5a a8b 1.7 0.05-0.07

5" -g: -175d 42 0.05-0.12 118d 11

aH. C. Van Ness and M. M. Abbott, personal communication, 1980. bOlson, 1981. cJ. J. Christensen, personal communication, 1980. dUnpublished Union Carbide data. ePemberton and Mash, 1978. HE(exper) = HE measured directly by calorimetry. HE(PTx) = HE derived from Gibbs-Helmholtz analysis of PTx data. Error P = Standard deviation in pressure from Barker's method fit.

Table 1 gives a summary of the Gibbs-Helmholtz tests. If we use the

criterion,

0.5 HE(exper)sHE(PTx)s1.5 HE[exper), x1=x2=0.5 (3)

the systems acetone + methyl acetate and ethanol + water pass the Gibbs-

Helmholtz test while the system propylene glycol + ethylene glycol fails. Note

that the level of pressure uncertainty in these data is low (0.05 to 0.22% in

pressure equal to absolute errors of 0.01 to 0.08 kPa).

Acetone (1) + methyl acetate (2). The experimental data for this system

have been published (Olson, 1981). Figure l(a) indicates the high degree of

Gibbs-Helmholtz consistency shown by the derived GE for this nearly ideal

system.

(b)

L . 1 . ’ * ’ ’ ’ . 290 300 310 320 330

TEMPERATURE 1 K I

0 0.2 0.4 0.6 0.6

Xl

Fig. 1.

!',b, P!Z) equimolar (-)I

Gibbs-Helmholtz analysis of PTx data for acetone (1) + methyl acetate Equimolar GF/RT vs. temperature. Points (0) and ( -_) derived

data; (------) is the sl . HE.

ope at 323.15 K calculated from calorimetric (b) Excess thermodynamic functions at 323.15 K. Curves:

derived from PTx data; (------) derived from calorimetric HE.

In the published analysis, the degree of agreement of HE(PTx) and HE(exper) was

shown to be sensitive to the method of estimating virial coefficients for

calculation of vapor-phase fugacity coefficients. Use of a different method

gave a 323.15 K value of the equimolar HE(PTx) = 0.0 and hence Gibbs-

Helmholtz inconsistency. This shows that Gibbs-Helmholtz testing is sensitive

not only to the experimental data but also to the details of data reduction.

However, values of calculated vapor composition differed <0.0002 between the

two methods because of cancellation between changes in the fugacity

coefficients and changes in the activity coefficients. This sensitivity of

HE(PTx) to fugacity coefficients is found only in nearly ideal systems where

the activity coefficients and fugacity coefficients have similar magnitudes.

Figure l(b) indicates that, although the magnitude of HE(PTx) vs. x is

correct, the second-order skewness is not. This suggests a composition-

dependent systematic error.

Propylene qlycol (1) + ethylene glycol (2). Figure 2(a) shows that the

derived equimolar GE data have only qualitatively correct temperature

dependence; the equimolar HE(PTx) value is more than seven times too large.

(a)

387

(b)

I , I I . I .

390 400 410 420 TEMPERATURE ( K 1

-250 -

Xl

Fig. 2. Gibbs-Helmholtz ~thyle;ed~;~;~; I;;, Pi;?)

analysis of PTx data for propylene glycol (1) + Equimolar GE/RT vs. temperature. Points (0) and

data* (------) is the slope at 408.15 K calculated from calorimetric equimolar Ht. (b) Excess thermodynamic functions at 408.15 K. Curves: calorimetric HE.

(- ) derived from PTx data; (------) derived from

Data for this system were not determined directly on isotherms as were data

for the other systems. Instead, isobaric data at 6.67, 13.33 and 26.66 kPa

were measured and isotherms were constructed by interpolation. Note also that

the slope disagreement in Fig. Z(a) is worse at lower temperatures (if we

assume dHE/dT to be small). This suggests a temperature-dependent

systematic error in the measurement. For example, hydrogen-bonding materials

boil differently and less consistently at lower pressures. Finally,

relative-volatility data derived from these PTx measurements agree well with a

previous PTxy study (Sokolov et al., 1972) when actually measured vapor

pressures are used to analyze the 1972 data.

Figure Z(b) shows that the HE and TSE functions obtained from HE(exper)

are of the same magnitude as GE.

Ethanol (1) + water (2). This system is an excellent choice for testing

VLE measurement methods because of the extensive and consistent data published

by Larkin and Pemberton (1976) and Pemberton and Mash (1978). Figure 3(a)

shows that the sign reversal in the equimolar HE is correctly predicted and

that Gibbs-Helmholtz consistency is achieved at the +10-40X error level in

equimolar HE(PTx).

388

(a) (b)

306

;i d

“Cd

:: 305

?s. 0

8 -

F” 304 (r \

%

303 I . 4 , I

320 330 340 350

TEMPERATURE (K 1

600

yi -400

E -600

-600

0 0.2 0.4 0.6 0.6 I.0

Gibbs-Helmholtz analysis of PTx data for ethanol (1) + water (2). fig. 3. (a) Equimol data; (----

ar Gk/RT vs. temperature. Points (0) and (- ) derived from PTx --) is the slope calculated at 323.15 K and at 343.15 K from

calorimetric equimolar HF. (b) Excess thermodynamic functions at 343.15 K. Curves: ( -) derived from PTx data; (------) excess functions from Pemberton and Mash (1978).

It is significant that again the slope disagreement is worse at lower

temperatures which suggests that the ethanol + water system boils more smoothly

and consistently at higher pressures. Absence of this type of systematic

dependence on temperature can be used as a criterion of the suitability of

ebulliometry as a route to VLE data.

For this system, we can compare all three excess functions with an

independent study. Figure 3(b) shows that the GE function at 343.15 K

agrees with the NPL study well within experimental error. Hence, the derived

vapor compositions would be more than sufficiently accurate for design

purposes (error in yCO.0005 mole fraction). There is a systematic deviation

between HE(PTx) and HE(exper) in the ethanol-rich mixtures which again

suggests small composition-dependent changes in the boiling characteristics of

the system.

Gibbs-Helmholtz tests on simulated PTx data

It appears from the ethanol + water data discussed in the preceding section

that even if the level of random errors in the measured PTx data is low (0.05

to 0.12% scatter in the Barker's method pressure residuals), errors in

HE(PTx) can be much larger, say 10 to 50%. This is due to the temperature

389

differentiation of the GE data which magnifies experimental errors. As

noted in the case of propylene glycol + ethylene glycol and ethanol + water, a

particular level of random error may be large enough to cause Gibbs-Helmholtz

inconsistency but not large enough to introduce significant errors into the

calculated vapor compositions. This can also be true of certain types of

systematic errors as was discussed for acetone + methyl acetate in relation to

the effect of fugacity-coefficient calculation method. Hence, data that fail

the Gibbs-Helmholtz test could still be used for chemical process design.

To investigate further, Gibbs-Helmholtz testing was done on computer-

generated PTx data to which known

added. Details on the simulation

TABLE 2

levels of random errors in pressure were

are given in Table 2.

Details on simulation of PTx data for Gibbs-Helmholtz testing shown in Table 3

Antoine Constants (LogTO, Wa. K) Component 1 Component 2

A 7.242673 6.173133

: 1580.92 -53.54 1294.40 -72.15

Temperatures (K) = 303.15, 308.15, 313.15, 318.45, 323.15, 328.15, 333.15, 338.15, 343.15

$(T) = (GE(323.15K)/323.15 + HE((323.15-T)/(T*323.,5)))(T)

GE 1 AxTx2 l-5 GE(323'~~Kki = Ax,2/RT J

Py i = t .P.sati j = 1 2 ix11 3 9

Data were generated

The derived HE(PTx) equimolar GE:

for xl = 0.0, 0.05, 0.10, 0.15, 0.2, 0.3, 0.4, 0.5 0.6, 0.7, 0.8, 0.85, 0.9, 0.95, 1.0

is from a least-squares fit of Barker-method derived

GE/RT = HE/RT - SE R Fit parameters = H L and SE

The equimolar GE values were obtained from a Barker's method fit of the

corrupted PTx data. A one-parameter Redlich-Kister equation and ideal-gas

vapor phase were used in the data generation and fitting. Three levels of

nonideality were studied: equimolar GE(323.15K) values of 100, 500, and

1000 J/mol. Results of the Gibb-Helmholtz tests are shown in Table 3.

The results in Table 3 show that while the actual estimate'of equimolar

HE(PTx) fluctuates widely as the error in pressure increases, the uncertainty

in the estimated HE(PTx) increases monotonically. Hence, the level of

390

scatter in HE(PTx), which leads to Gibbs-Helmholtz inconsistency, increases

with the level of random error in pressure as expected.

TABLE 3

Effect of random errors in pressure on equimolar HE derived from simulated

PTx data via the Gibbs-Helmholtz equation

x1 = x2 = 0.5

GE(323.15K) HE Error P HE (PTx) Error HE <IY-YcalcI> J/mol J/mol % J/mol

W;;;Tx)) % mol %

100 150 0.01 150.2 0.8 0.05 151 5.6 0.25 148 15

500

0.50 1.00 1.00 1.75 2.50 5.00

750 0.5 1 .o 2.5 5.0 7.5

183 205 410

-374 188

-790

743 853 738 606 433

1000 1500 1.0 1428 5.0 2005

10.0 743

32

1:: 150 200 380

200

30 65

260 420 42

23240 436

0.13 0.7 1.3

22 37

170 350 25

630

0.9 14 .2 19

5 0.073 33 0.52 50 1.2

0.0011 0.0081 0.024 0.056 0.12 0.22 0.39 0.29 0.66

0.056 0.13 0.30 0.47 0.96

HE(PTx) = Equimolar excess enthalpy from least-squares fit of GE/RT vs. T SD = Standard deviation estimate from least-squares fit Error P Error HE

= Level of random error in pressure added to simulated data = (/HE-HE(PTx)l/HE) x 100

<(y-ycalcI>= Average mean deviation of vapor composition for liquid compositions 0.055x10.40

For the case where the equimolar GE(323.15K) value is 100 J/mol, a random

error level in pressure of 0.50 to 1.00% leads to Gibbs-Helmholtz inconsistency

as defined by eqn. (3). However, this level of random error introduces

uncertainties only of 0.0006 to 0.0012 into the calculated vapor mole

fractions. PTxy data thus derived could be used for chemical process design.

Results for the more non-ideal systems, equimolar GE(323.15K) values at 500

and 1000 J/mol, show that although higher levels of random errors in pressure

could be tolerated before Gibbs-Helmholtz inconsistency occurred, the effect

on derived vapor compositions remained constant.

391

Two final notes on this analysis of simulated data: (i) Any number of

artificial systems could have been studied although the present constraint

that equimolar HE = 1.5 GE(323.15K) increases the level of random pressure

errors that are needed to produce Gibbs-Helmholtz inconsistency and thus-

represents a conservative case. (ii) More often in practice, it is systematic

data errors that are important and no consistency test can "undo" systematic

measurement errors (Van Ness and Abbott, 1982, pp. 326-327). However, these

simulated data tests indicate that a modest level of random error may vitiate

the Gibbs-Helmholtz consistency test before it can be used to detect

systematic errors, particularly for nearly ideal systems.

CONCLUSIONS

The Gibbs-Helmholtz thermodynamic consistency test should be used to judge

the overall integrity of PTx data measured at several temperatures rather than

to test whether derived vapor compositions are suitable for chemical process

design. The weakest link in producing accurate vapor compositions from PTx

data often is the vapor-phase fugacity model (Abbott, 1977). The Gibbs-

Helmholtz analysis tests primarily for correct temperature dependence, usually

a second-order effect in VLE for chemical process design.

Data of the highest quality should pass the Gibbs-Helmholtz test; therefore,

it should be used to analyze measurements reported as reference data. In this

context, equimolar HE(PTx) values within +30% of HE(exper) can be regarded

as thermodynamically consistent. Failure to pass the Gibbs-Helmholtz test

suggests the presence of:

levels of random error higher than can be tolerated in reference

data (for example,>O.Ol to 0.13 kPa errors in pressure),

temperature-dependent systematic errors related to the experimental

apparatus or procedure,

systematic errors in either the data-reduction process or in

the required ancillary thermophysical property data.

REFERENCES

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Bae, K. H., Nagahama, K. ani Hirata, M., 1980. Evaluation and correlation of vapor-liquid equilibria in the ternary system nitrogen-argon-oxygen, Fluid Phase Equilibria, 4: 45-60.

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Larkin, J. A. and Pemberton, R. C., 1976. Thermodynamic properties of water + ethanol between 298.15 and 383.15 K, NPL Report Chem.

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