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Fluid Phase Equilibria 356 (2013) 90– 95

Contents lists available at ScienceDirect

Fluid Phase Equilibria

j our na l ho me pa ge: www.elsev ier .com/ locate / f lu id

urface tension of ethane–methane solutions: 1. Experiment andhermodynamic analysis of the results

ladimir G. Baidakov ∗, Aleksey M. Kaverin, Maria N. Khotienkovahe Institute of Thermophysics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg 620016, Russia

r t i c l e i n f o

rticle history:eceived 19 April 2013eceived in revised form 3 July 2013ccepted 5 July 2013vailable online 17 July 2013

a b s t r a c t

The paper presents the results of measuring the capillary constant a2 and calculating the surface tension� of ethane–methane solutions. Experimental data have been obtained in the temperature range from93.15 to 283.15 K at pressures from that of saturated vapors of pure ethane to 4 MPa. Equations thatapproximate the temperature, pressure and concentration dependences of a2 and � are suggested. Theconcentration dependence of the surface tension is described in the framework of models of an ideal and

eywords:apillary constanturface tensionthane–methane solutionsethod of capillary rise

a regular solution. The relative adsorption has been calculated.© 2013 Elsevier B.V. All rights reserved.

dsorption

. Introduction

An important problem which has not yet been solved to com-letion is determining the character of the temperature, pressurend concentration dependence of the surface tension � of solu-ions. The statistical approach to the calculation of � based on directllowance for interparticle interactions does not make it possibles yet to determine the explicit form of the dependence of sur-ace tension on thermodynamic state parameters with an accuracyequired for practical applications. In this connection experimentalnvestigations of � are topical.

The present paper continues a series of publications [1,2] onhe properties of liquid-vapor interfaces of binary solutions of

ethane–series hydrocarbons. In present experimental data on theapillary constant a2 and the surface tension of ethane–methaneolutions obtained in the temperature range from 93.15 to 283.15 Kt pressures from that of saturated vapors of pure ethane to �4 MPa.he capillary constant was measured by the differential variation ofhe method of capillary rise [3]. The surface tension was calculatedrom the values of a2 obtained and literature data on the orthobaricensities of the liquid and vapor phases of the solution.

The paper consists of an introduction, three sections, which

escribe the experiment, discuss and interpret the data obtained,nd a conclusion.

∗ Corresponding author. Tel.: +7 3432678801.E-mail address: baidakov@itp.uran.ru (V.G. Baidakov).

378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.fluid.2013.07.008

2. Experiment

The capillary constant was measured on the setup describedin detail in Refs. [3,4]. The measuring cell contained an assem-bly of three glass capillaries with internal radii: r1 = 0.6393 mm,r2 = 0.2297 mm and r3 = 0.09607 mm. The cell was secured in a cop-per thermostating unit, placed in a glass jacket and immersed intoa cryostat. In the temperature range from 90 to 200 K cooling wasrealized by liquid nitrogen. At higher temperatures the cryostat wasfilled with alcohol, a copper coil was located in it, and cooling wasrealized by pumping alcohol from a low-temperature thermostatthrough the coil.

The temperature in the unit was measured by a platinum resis-tance thermometer with an error of no more than ±0.02 K. Atpressures higher than atmospheric the pressure in the solution wasmeasured by a spring-type pressure gauge (±0.007 MPa), at lowerpressures measurements were performed by a digital vacuum gageof accuracy 0.15 kPa.

To prepare mixtures, the gases whose purities are presented inTable 1 were used. A gas mixture was prepared by the volumetricmethod in a measuring vessel and condensed into the measuringcell.

The difference of the liquid rise heights in two standing capillar-ies �h = h2 − h1 was measured by a cathetometer with an error of±0.03 mm. The capillary constant was calculated by the formula:

a2 = �h

1/b1 − 1/b2, (1)

V.G. Baidakov et al. / Fluid Phase Equilibria 356 (2013) 90– 95 91

a molar surfacea2 capillary constantb radius of curvature of the meniscus vertexg free fall accelerationh height of liquid rise in a capillaryn number of monolayerNA Avogadro numberp pressurer capillary internal radiusR universal gas constantt reduced temperatureT temperatureW mixing energyx mole fraction of the second component (methane)z coordination number

Greek symbols� surface tension� difference of values� energy parameter� density� molar volume� adsorption� activity of the adsorbed component

SubscriptsL liquidV vaporc critical point� surface layer

wiLt

f

�

wp

uoteAgamr

es

TP

01 pure substance (ethane)02 pure substance (methane)

here b1 and b2 are the radii of curvature of menisci in the first andn the second capillary. The radii b1 and b2 were calculated from theane equation [5] in an approximation of complete wettability ofhe inner walls by the solution.

The capillary constant and the surface tension are related asollows:

= 12

ga2(�L − �V ), (2)

here �L and �V are the orthobaric densities of the liquid and vaporhases, and g is the free fall acceleration (g = 9.8162 m/s2).

A modified equation of state of Lemmon and Jacobsen [6] wassed for determining the orthobaric densities and concentrationsf methane in solutions. The modification consisted in replacinghe equations of state for pure substances used in Ref. [6] by thequations of state for ethane and methane from monographs [7,8].s shown by our calculations, the equations of state of Refs. [7,8]ive a physically consistent description of the properties of liquidnd vapor not only in the region of stable, but also in the region ofetastable state. We will need this fact later for interpreting the

esults of measurements.A composition between the phase-equilibrium parameters of

thane–methane solutions calculated by the modified equation oftate and the experimental data of Refs. [9–17] has shown that

able 1urities of components used in experiment.

Chemical name Initial mole fraction purity Purification method

Ethane 0.9975 noneMethane 0.9997 none

Fig. 1. Pressure dependence of the capillary constant of solution at a constant tem-perature: 1 – T = 93.15 K; 2 – 133.15; 3 – 173.15; 4 – 193.15; 5 – 213.15; 6 – 233.15;7 – 253.15; 8 – 263.15; 9 – 273.15; 10 – 283.15. Dashed line–pure ethane.

in the liquid phase at temperatures above 200 K the root-mean-square deviation in concentration is ±0.003 mole fraction andincreases to ±0.015 at the lower boundary of the temperature rangeunder investigation. In the vapor phase the discrepancy betweenthe values of xV obtained from the modified equation of state andexperimental data [9–16] at T < 240 K is ±0.002 mole fraction andincreases to ±0.014 at temperatures of 240 − 285 K. Data on thedensity of the liquid and vapor phases of ethane–methane solu-tions in the state of phase equilibrium [15,16] are described by themodified equation of state with an uncertainty that does not exceed±0.8% (liquid phase) and 3% (vapor phase). The values of deviationsmentioned are within the uncertainty of the density measurementestimated by the authors [15,16].

3. Results

The capillary constant of ethane–methane solutions has beenmeasured in the temperature range from 93.15 to 283.15 K alongten isotherms. Measurements began from the line of phase equi-librium of pure ethane. At temperature from 93.15 to 173.15 K theexperimental data embrace the whole concentration range (0−1),at higher temperatures (T ≥ 193.15 K) the maximum value of themethane concentration xL in the solution liquid phase was limitedby the pressure value 4 MPa.

The results of measuring the capillary constant and calculat-ing the surface tension are presented in Table 2 and shown inFigs. 1 and 2. An increase in the pressure results in decreasing a2

and �. The dependences a2(p) and �(xL) straighten with increasingtemperature.

The baric and temperature dependences of a2 and � have beenapproximated by equations of the form (p and p01 are expressed inMPa):

a2 = a201 + B1(p − p01) + C1(p − p01)2, (3)

� = �01 + B2(p − p01) + C2(p − p01)2 (4)

where a 201, �01 and p01 are the capillary constant, the surface

tension and the pressure of saturated vapors of pure ethane. Thecoefficients B and C, and also a 2

01, �01, p01 are functions of thetemperature.

The pressure of saturated vapors of pure ethane was calculatedby the equation of Ref. [18]:

ln p01 = a0 + a1q + a2s + a3s2 + a4s3 + a5s(1 − s)˛, (5)

92 V.G. Baidakov et al. / Fluid Phase Equilibria 356 (2013) 90– 95

Table 2Capillary constant a2, methane concentration in liquid phase xL , orthobaric densities of liquid �L and vapor �V phases and surface tension � of an ethane–methane solutiona.

T (K) p (MPa) a2 (mm2) xLb �L (kg/m3)b �V (kg/m3)b � (mN/m)

93.15 – 10.32 0.000 648.3 9.21 × 10−5 32.900.002 9.74 0.141 – – –0.004 8.95 0.272 – – –0.008 8.39 0.465 – – –0.011 7.96 0.699 – – –0.016 7.60 1.000 447.7 0.34 16.69

133.15 0.004 8.53 0.000 604.9 5.07 × 10−2 25.320.106 7.34 0.194 577.6 1.79 20.740.213 6.24 0.414 536.9 3.31 16.340.261 5.78 0.527 513.4 3.95 14.450.372 5.04 0.821 441.3 5.34 10.780.438 4.69 1.000 388.7 7.16 8.78

173.15 0.064 6.68 0.000 558.1 1.12 18.260.389 5.94 0.133 539.2 4.39 15.590.406 5.87 0.139 538.0 4.58 15.370.706 5.16 0.256 516.9 8.74 12.871.114 4.27 0.421 482.6 15.1 9.801.131 4.20 0.429 481.0 15.4 9.601.204 4.04 0.459 474.0 16.5 9.071.439 3.65 0.565 448.4 20.3 7.671.502 3.45 0.587 442.8 21.2 7.141.724 3.11 0.684 416.4 24.8 5.981.742 3.03 0.691 414.2 25.1 5.791.982 2.60 0.793 383.0 29.2 4.512.045 2.54 0.818 374.6 30.3 4.292.053 2.53 0.821 373.5 30.5 4.252.600 1.59 1.000 301.0 45.0 2.00

193.15 0.164 5.77 0.000 533.2 3.11 15.010.717 4.96 0.134 512.7 9.05 12.260.826 4.76 0.160 508.0 10.6 11.621.354 4.08 0.284 483.5 17.3 9.341.369 3.97 0.287 482.8 17.5 9.071.823 3.43 0.395 459.1 23.3 7.342.267 2.76 0.501 433.0 29.5 5.472.985 2.06 0.672 383.7 38.7 3.493.044 1.94 0.685 379.2 39.5 3.23

213.15 0.397 4.81 0.000 506.5 7.06 11.790.525 4.61 0.024 504.6 8.52 11.221.084 4.12 0.116 487.8 14.5 9.571.420 3.82 0.170 477.1 18.4 8.601.845 3.38 0.238 462.9 23.9 7.283.178 2.26 0.447 411.8 41.2 4.113.775 1.65 0.539 384.4 49.9 2.71

233.15 0.791 3.86 0.000 477.0 14.1 8.771.398 3.30 0.075 464.5 20.7 7.191.953 2.92 0.143 450.3 27.2 6.062.036 2.86 0.153 448.1 28.3 5.892.551 2.52 0.214 434.0 35.3 4.932.664 2.44 0.228 430.8 38.5 4.703.212 2.00 0.292 414.7 45.1 3.633.257 1.97 0.297 413.3 45.8 3.553.820 1.64 0.362 395.4 54.7 2.743.884 1.56 0.369 393.3 55.7 2.58

253.15 1.439 2.82 0.000 443.1 26.0 5.771.905 2.60 0.048 436.9 31.9 5.172.413 2.32 0.099 424.8 38.3 4.402.882 2.06 0.145 413.1 44.6 3.733.410 1.75 0.196 399.3 52.4 2.983.889 1.46 0.241 386.0 60.1 2.343.903 1.43 0.242 385.7 60.3 2.28

263.15 1.874 2.35 0.000 423.4 34.7 4.482.391 2.08 0.048 416.1 41.8 3.822.947 1.82 0.098 402.7 49.7 3.153.434 1.58 0.142 390.3 57.2 2.583.920 1.33 0.185 377.1 65.4 2.03

273.15 2.404 1.84 0.000 400.9 46.2 3.202.823 1.64 0.035 392.9 52.9 2.763.408 1.40 0.083 379.3 62.5 2.193.871 1.20 0.124 368.2 70.9 1.75

283.15 3.032 1.28 0.000 374.2 62.1 1.963.256 1.21 0.017 372.8 66.6 1.823.619 1.07 0.045 363.0 73.6 1.503.865 0.98 0.065 356.0 78.8 1.333.882 0.95 0.067 355.5 79.2 1.29

a Standard uncertainties are u(T) = 0.02 K, u(p) = 0.007 MPa. The combined standard uncertainties of a2 and � does not exceed 1.5% and 2.5% respectivelyb These parameters are calculated. The method of calculation is described in the text of the article.

V.G. Baidakov et al. / Fluid Phase Equilibria 356 (2013) 90– 95 93

Table 3Coefficients of Eqs. (8) and (9).

j bj1 cj1 bj2 cj2

1 0.671128 × 101 0.2255000 × 101 0.5713 × 100 −0.48770 × 100

2 −0.652283 × 102 −0.2121265 × 102 −0.2139 × 101 0.22475 × 101

3 0.2416774 × 103 0.7679189 × 102 0.2006 × 101 −0.32725 × 101

3 × 103 0 0.14875 × 101

103 0 0 × 102 0 0

wacavrbt

e

a

w

�

w

B

C

Tfg

cc

a

�

Ft

Table 4Coefficients of Eqs. (8) and (9).

i 1 2

b0i −0.06551 −0.36062ˇi 3.411 2.5242tbi 0.77533 0.70953c0i 0.2812 × 10−4 0.6356 × 10−2

� i 8.82 3.5425tci 0.80037 0.66657

Table 5Coefficients of Eqs. (11) and (12).

j dj fj

1 −0.11287005 × 105 −0.20932765 × 103

2 0.10270331 × 106 0.20408614 × 104

3 −0.38487115 × 106 −0.81919566 × 104

4 0.75930767 × 106 0.17344185 × 105

6 5

4 −0.4250684 × 10 −0.13457455 0.3541423 × 103 0.1151574 ×6 −0.1123125 × 103 −0.3875485

here q = (1 − Tt/T)/(1 − Tt/Tc), s = (T − Tt)/(Tc − Tt), Tt = 90.348 Knd Tc = 305.33 K are the temperatures at the triple andritical points, a0 = −11.38996, a1 = 18.84523, a2 = −7.635415,3 = 5.428443, a4 = −1.362327, a5 = 0.7692493, � = 1.3. In Eq. (5) thealue of p01 is expressed in bars (105 Pa). In the whole temperatureange from the triple to the critical point the values of p01 calculatedy Eq. (5) within 0.1% agree with the data of our measurements andhe results of calculations by the equation of state of Ref. [7].

The temperature dependence of the capillary constant for purethane is presented as follows [4]:201 = a2

0∗tn (6)

here t = (Tc − T)/Tc, a20∗ = 14.59 mm2, n = 0.929.

For the surface tension of pure ethane according to [2] we have:

01 = �0∗t�(

1 + e1t + e2t6)

(7)

here �0∗ = 57.63 mN/m, e1 = −0.1685, e2 = 0.249, � = 1.279.The functions Bi and Ci are written as follows:

i = b0i

(tbi − t)ˇi+

6∑j=1

bji(1 − t)j−1, (8)

i = c0i

(tci − t)�i+

6∑j=1

cji(1 − t)j−1, (9)

he values of the coefficients of Eqs (8)–(9) have been determinedrom experimental data by the method of least squares and areiven in Tables 3 and 4.

Fig. 3 shows deviations of experimental values of a2 from thosealculated by Eq. (3). For pure ethane (solid circles) the discrepan-

ies do not exceed 2.5%.

The concentration dependence of the surface tension has beenpproximated by the equation:

= �01 + DxL + Fx2L , (10)

ig. 2. Concentration dependence of the surface tension of solution at a constantemperature (for designations see Fig. 1).

5 −0.83235780 × 10 −0.20439636 × 106 0.48096664 × 106 0.12719107 × 105

7 −0.11448309 × 106 −0.32668261 × 104

where the coefficients D and F are functions of the temperature andare presented as follows:

D =7∑

j=1

dj(1 − t)j (11)

F =7∑

j=1

fj(1 − t)j (12)

The values of the coefficients of Eqs. (11)–(12) are given in Table 5.

4. Thermodynamic description of surface tension

In an ideal solution there is no primary interaction of any kindof molecules, any pair combinations are energetically equivalent.

Fig. 3. Deviations of experimental values of the capillary constant of solution fromEq. (3) (for designations see Fig. 1). Solid circles – pure ethane.

94 V.G. Baidakov et al. / Fluid Phase Equilibria 356 (2013) 90– 95

F1E

Tl

�

wuoa

k

Tf

a

w

a��ttb

sWti

�

w

M

W

W�tdea

ig. 4. Concentration dependence of the surface tension of solution at temperatures33.15 and 173.15 K. Dots – experimental data; dashed lines – data calculated byq. (13); dash – dot lines–Eq. (16), solid lines – additive approximation.

he isotherm of the surface tension of an ideal solution will lookike [19]:

= �01 − RT

a02ln [1 + (k − 1)xL] , (13)

here a02 is the molar area of the surface of pure methane, R is theniversal gas constant. The value of k is the quantitative measuref difference between methane concentrations in the liquid phasend in the surface layer

= exp[

(�01 − �02) a02

RT

]. (14)

he molar surface a02 is related to the molar volume v� of the sur-ace layer of pure methane containing n monolayers as follows:

02 = (��)2/3N1/3A

n, (15)

here NA is the Avogadro number.The results of calculating surface tension by Eqs. (13)–(15)

t T = 133.15 K (a02 = 10.08 × 104 m2/mol, �01 = 25.32 mN/m,02 = 8.78 mN/m) and 173.15 K (a02 = 11.95 × 104 m2/mol,01 = 18.26 mN/m, �02 = 2.00 mN/m) are presented in Fig. 4. In

he whole concentration range the approximation of an ideal solu-ion underestimates surface tension. The maximum discrepancyetween the data of Eq. (13) and experiment is 6%.

The simplest model that takes into account deviations of theolution properties from ideality is the model of a regular solution.

hen the bulk phase may be considered a regular solution andhe surface layer an ideal one, the equation of the surface tensionsotherm will look like [19]:

= �01 − RT

a02ln (1 + (kM − 1)xL) − W

a02x2

L (16)

here

= exp[

W

RT(1 − 2xL)

], (17)

= NAzε12 − (ε11 + ε22)

2, (18)

is the mixing energy, z is the coordination number, �11, �22 and12 are the potential energies of pair interactions of molecules of

he mixture components per mole. The value of the coefficient k isetermined by Eq. (14). For ethane–methane solutions the mixingnergy calculated from the data on �11 and �22 of Ref. [20] on thessumption that ε12 = (ε11 · ε22)1/2 and z = 12 is W= − 310.8 J/mol.

Fig. 5. Concentration dependence of the relative adsorption of methane in the sur-face layer of solution (for designations see Fig. 1).

The sign of the mixing energy is determines by the character ofdeviation of the surface tension isotherm of a regular solution fromthat of an ideal one. With a primary interaction of heterogeneousmolecules (W < 0) the surface tension of a regular solution exceedsthat of an ideal one, and with a primary interaction of homogeneousmolecules (W > 0) just the contrary. As is evident from Fig. 4, theapproximation of a regular solution adequately describes experi-mental data on the surface tension of an ethane–methane solution,at least in the region of low temperatures.

With a given concentration dependence of the surface tensionof a solution the Gibbs equation of adsorption makes it possible tocalculate the relative adsorption of the component separated:

�2(1) = −xL�2

RT

(∂�

∂�2

)T

, (19)

where �2 is the activity of the adsorbed component in solution.For an ideal solution Eq. (19) will take the form:

�2(1) = −xL (1 − xL)RT

(∂�

∂xL

)T

. (20)

Experimental data on � make it possible to calculate the rela-tive adsorption of methane in the surface layer of a solution. Theresults of calculating � 2(1) in the approximation of an ideal solution(Eqs. (10) and (20)) are presented in Fig. 5. The relative adsorp-tion increases with increasing differences in the values of surfacetension of the pure components that form the solution.

5. Conclusion

For the fist time experimental data on the capillary constanthave been obtained and the surface tension of ethane–methanesolutions has been calculated in a wide temperature and concentra-tion range. The results of experiments and calculations have beenapproximated by equations determining the temperature, baricand concentration dependences of a2 and �.

A methane–ethane solution is a weakly nonideal system. Takingthe noniodeality into account in the framework of the model of aregular solution allows describing data on the surface tension inthe region of low temperature with an error close to experimental.

Today the statistical methods are actively developed for calcu-

lating the surface tension of solutions [21]. The second part of thiswork will be devoted to describing the properties of the interface ofethane–methane solutions in the framework of the van der Waalsgradient theory.

hase E

A

t9tCR

R

[[

[

[

[

[[[

[

[

V.G. Baidakov et al. / Fluid P

cknowledgements

The work has been performed with a financial support ofhe Russian Foundation for Basic Research (projects No. 13-08-6039-r ural a and No. 12-08-31261-mol a), the Government ofhe Sverdlovsk region, and the Ural Branch of RAS (project No 12--2-1001 of joint investigations of Ural and Siberian Branches ofAS).

eferences

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[

[

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