Augmented GibbsâTolman Model for Surface Tension - Langmuir

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Augmented Gibbs-Tolman Model for Surface Tension Sukesh K Tumram, Kaza Kesava Rao, and M. S. Ananth Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02041 • Publication Date (Web): 01 Sep 2017 Downloaded from http://pubs.acs.org on September 4, 2017

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Augmented Gibbs-Tolman Model for Surface Tension Sukesh Tumram,

†

K. Kesava Rao,

†

and M. S. Ananth

∗, ‡

†Department

of Chemical Engineering, Indian Institute of Science, Bangalore, India ‡Former Director, Indian Institute of Technology, Madras, India E-mail: [email protected] Phone: +919008020743

Abstract Gibbs developed the thermodynamics of a liquid-vapour system by introducing the idea of a `dividing surface' a hypothetical surface that separates the system into two homogeneous phases. The area and curvatures of a conveniently chosen dividing surface, the `surface of tension', are used to account for the eects of the smooth variation of properties across the actual transition layer between the phases. Tolman (1948, 1949) considered a more detailed model of the interfacial region and obtained expressions for surface tension (σ) and the location of the surface of tension. Based on qualitative arguments, Tolman's model introduced a surface of tension, such that the pressure (P ) increases from its saturation value (Psat ) to a maximum value (Pmax ) as the surface is approached from the vapour side and decreases from (Psat ) to its minimum value (Pmin ) as the surface is approached from the liquid side. Assuming an exponential decay of (P ) away from the surface, Tolman obtained an explicit expression for (σ) in terms of Psat ,

Pmax , Pmin , and two length scales. In the this work, the Gibbs-Tolman (GT) model is used along with the Lee and Kesler (1975) equation of state. The model is augmented

1

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to take into account the eect of the density gradient in the transition zone and a 4parameter augmented model (AGT model) is proposed. The GT and AGT models are shown to t the data for 152 pure liquids with an absolute average deviation (AAD) of 4.91 % and 2.02 % respectively. The corresponding AAD values for 57 liquid mixtures are 4.2 % and 3.0 % respectively. Arguments are also presented to counter some of the fundamental concerns that have been raised about the GT approach. Although the model correlates the data very well one of the length parameters turns out to be persistently negative and the reason for this behaviour is not clear.

Introduction As early as 1876 Gibbs

1

developed a theory for surface tension introducing several new con-

cepts. Gibbs envisaged a two phase system in equilibrium with each phase having substantially uniform distribution of matter throughout their interiors meeting in a thin transition layer of inhomogeneity. To simplify the picture Gibbs treated the system as separated into two parts by an imaginary geometrical surface located in the transition zone and `passing through points in the layer which are similarly located with respect to the condition of neighbouring matter'. Such a surface is called a Gibbs dividing surface (GDS). There are clearly many possible choices of the GDS all of which can be thought of as being parallel to the tangible physical surface of discontinuity.

The Gibbs-Tolman Model For each choice of the GDS, the properties of whole transition layer are determined in the Gibbs model by the surface area and the sum and dierence of the local curvatures of the chosen GDS. Gibbs chose a spherical interface ( c1 =

(c1 + c2 )

c2

=

c)

and set the the coecient of

to be zero in order to locate the surface of tension- a ctitious surface whose

tension represents the surface tension of the liquid. Tolman

2

worked out the implementation

2


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of what we hereafter refer to as the Gibbs-Tolman (GT) model using the van der Waals (vdW) equation of state (EoS).

Figure 1: The transition zone.

Figure 1 shows the typical variation in pressure (at a temperature below the critical temperature) in a two-phase system with a planar interface as calculated using an equation of state (EoS). The planar physical surface of discontinuity, plane of the paper,

y =0

is perpendicular to the

x = 0 is the `surface of tension' and is located at y = yt . x is measured in

a direction opposite to

y

and is equal to

(yt − y).

Since the density

ρ(y) varies monotonically

from the vapour density to the liquid density along the x-axis, this isotherm also plots (not to scale) the pressure as a function of distance from the planar interface. The corresponding pressure,

P (ρ(y), T ),

is given by the EoS. Both the planes AA ( y

= 0)

and BB

(y = yt )

are

parallel to one another and lie in the transition zone. The region between the planes AA and BB is the unstable region ( (∂p/∂ρ)

< 0)

that is not physically realisable.

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Figure 2: The Gibbs -Tolman model.

In the GT model (Figure 2), the planes AA and BB are merged in order to eliminate the thermodynamically unstable region. The resulting plane

(y = 0)

ities in the density and the pressure. It may be noted that

and

Pmax

Pmin

ρv,sat < ρmax < ρmin < ρl,sat .

Pmin < Psat < Pmax .

The corresponding pressures are ordered dierently: prets

is a GDS with discontinu-

Tolman

2

inter-

as the maximum pressure beyond which the substance cannot exist as a vapour,

(usually negative) as the minimum pressure (maximum tension) beyond which the

substance cannot exist as a liquid. Tolman's expressions

Z

2

for the surface tension and the location of surface of tension are:

0

Z

2

(Psat − P )(1 + cx) dx +

σ= −a

Z

Z (Psat − P )(1 + cx)x dx +

−a

P

approaches

(Psat − P )(1 + cx)2 dx

(1)

(Psat − P )(1 + cx)x dx

(2)

0

0

0= We note that

b

b

0

Psat as we move away from the dividing surface in either direction. 4


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Figure 1 shows the location of the surface of tension vis-a-vis the surface of discontinuity in the density and the pressure. Tolman

3

assumes that the local pressure in the transition zone

varies as follows:

y P − Psat = (Pmax − Psat ) exp λ v −y P − Psat = (Pmin − Psat ) exp λl

y≤ 0

(3)

y≥ 0

(4)

The exact mathematical form in the above equations is of the no consequence with a denite integral over

y

providing two correlating parameters. While there is no a priori reason why

these parameters should be independent of temperature, they are so treated in this work. Substituting these proles into equations (1) and (2) gives the correlating equations for and

yt

σ

in dimensionless form:

σratio = −λ∗v + λ∗l Pratio yt∗ =

∗2

(5)

∗2

λl Pratio + λv λ∗l Pratio − λ∗v

(6)

where,

σ

σ∗ = σratio Pratio

c 1/3 ) Pc ( kT Pc ∗ σ = ∗ ∗ Pmax − Psat ∗ ∗ Psat − Pmin = ∗ ∗ Pmax − Psat

The pressures are non-dimensionalised using critical pressure ( Pc ) and the lengths scaled using

c 1/3 ) , ( kT Pc

paprameters

λl

where and

k

λv

is Boltzmann constant and

Tc

is the critical temperature.

The

are characteristic lengths representing the thicknesses of transition

zone in the liquid and the vapour phases respectively. It is worth noting that the GT model is the only classical thermodynamic model that permits the prediction of surface tension from a bulk EoS. Preliminary work on the Gibbs-Tolman model, done using the vdW EoS in

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a series of undergraduate projects

46

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gave promising results. The work was set aside because

of reservations about the GT model raised by Hill

7,8

and Rowlinson and Widom

9

which are

re-examined in the next section.

Reservations about the GT model The GT model assumes that the GDS is indeed a surface of discontinuity. Hill's criticism is that this is physically unsatisfactory.

8

His own calculations, taking care of the non-local po-

tential energy contributions to the chemical potential, produce a smooth but steep variation in the density - a change in reduced density of about one over just three molecular diameters! A step change in density is indeed a good mathematical approximation to this result. Secondly Hill

7

observes that the local pressure should strictly be replaced by

ρ(µ − a).

The

two are indeed the same if we assume a plane interface and local equilibrium. Hill himself points out that the GT model is `exact for plane surfaces'. Rowlinson and Widom

9

7

present an insghtful discussion of the theories of surface tension.

They argue, using a thought-experiment, that the assumption of `point thermodynamics' (ie. `local equilibrium') leads to the prediction that

σ = 0.

This claim however is disputable

as shown below. The following expression of Rowlinson and Widom is a good starting point for this discussion:

Z

+∞

(Psat − PT (y))dy

σ=

(7)

−∞ The pressure is strictly a tensor in the transition zone with the transverse component in the interfacial plane is replaced at

T

PT

P (ρ(y), T )

diering from the normal component

ρ(y)

ignoring the dependence of

in the above equation is clearly a functional of

T, V , m

minimised. Since

and interfacial area

A

In the GT model,

PT (y)

assuming local equilibrium: the local pressure is given by the EoS

and the local density

constant

P.

S,

ρ(y).

P

on the density gradient.

σ

Rowlinson and Widom argue that at

the Helmholtz free energy

A

is minimised if

is completely determined by these constraints ( T, V, m and

6


S)

σ

is

from a

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thermodynamic point of view, it can be minimised only by minimising

σ S

Ideally the thought experiment should minimise the interfacial area

a material property. for a given

σ

rather

than the other way around.

σ

Further, since

in the above equation is a functional of

ρ(y)

its extrema are given by

the Euler Lagrange equation

∂p δσ =− =0 δρ ∂ρ In the GT model, constant

P

is given by the EoS and this equation has typically two real roots at

T (< Tc ): ρmin

and

ρmax .

This implies that the extremum in

in the vapour half of the transition zone and

Z

P = Pmin

0

Z (Psat − Pmax )dy +

σ= −a

where

a

and

b

(8)

σ

is given by

P = Pmax

in the liquid half. Hence we can write

+b

(Psat − Pmin )dy

(9)

0

symbolise the thicknesses of the transition zone on the vapour and liquid

side respectively. The integrand is zero outside this region. The mathematical form of the correlating equation for constants

P

λv

equal to

σ=0

and

Psat

λl

σ

remains the same as that of the GT model except that the decay

are replaced by

a

and

b.

Since

∂p/∂ρ 6= 0

at

ρ = ρv,sat

or

ρl,sat ,

setting

is not an admissible solution. The conclusion that local equilibrium leads to

is therefore not tenable.

The GT model assumes local equilibrium, an assumption has been successfully used by chemical engineers in the quantitative description of transport phenomena.

10

The fact

that the density gradient is very steep in the transition layer does make the validity of the assumption questionable in this context. Rowlinson and Widom

9

Following the suggestion of van der Waals,

point out the importance of augmenting the integrand in equation

(7) with a term proportional to the square of the density gradient. This is to take into account the eect of the inhomogeneity in the local environment on the pressure. This augmentation accounting for the eect of the density gradient on the pressure is incorporated into the GT model in this paper resulting in signicant improvement in its agreement with experiment.

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Based on the promising results obtained using the vdW EoS (Figure 4) and the arguments above that counter the reservations about the GT model, we revisit the model using a more realistic EoS, namely the Lee and Kesler (LK) EoS. thesis

12

11

The present work is part of a Master's

and makes use of the extensive data now available for pure liquids as well as liquid

mixtures.

The Augmented GT model The GT model provides the following expression for the surface tension (see equation (5))

∗ ∗ ∗ ∗ σ ∗ = −λ∗v (Pmax − Psat ) + λ∗l (Psat − Pmin )

Given an EoS and values for the constants

λ∗v

and

λ∗l ,

(10)

equation (10) permits the

a priori

∗ calculation of the reduced surface tension ( σ ). The Augmented GT model replaces the assumption of local equilibrium to take into account the eect of the density gradient on the tangential component of the pressure.

PT (y) = P (T (y), ρ(y), ρ0 (y))

where

0

ρ ≡

h i

∂ρ . The expression above can be expanded as a Taylor series in ∂y

∂P PT (y) ' P (T (y), ρ(y), 0) + ∂ρ0

1 ∂ 2P ρ (y) + ρ0 (y)2 02 2 ∂ρ 0 0

The rst term on the right represents the value of

ρ0

0

PT

under the assumption of local equilib-

rium. The second term on the right vanishes in the case of isotropic systems. Terms of order greater than

ρ0 (y)2

in the Taylor expansion have been neglected. The tangential component

8


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of the pressure can then be approximated as follows:

1 ∂ 2P PT (y) ' P (T (y), ρ(y)) + ρ0 (y)2 2 ∂ρ02 0 1 ≡ P (T (y), ρ(y)) + m ρ0 (y)2 2 where

m≡

h

∂2P ∂ρ02

i

. This is the same as the expression proposed by van der Waals empirically

0

and endorsed by Rowlinson and Widom. This paper explicitly identies what m stands for. It follows that

σ

in the AGT model is given by

+∞

Z

[(Psat − P (y)) +

σ= −∞

Z

+∞

σ= −∞ If

κ (=

"

m ∂ρ(y) 2 ( ) ] dy 2 ∂y

m (Psat − P (y)) + 2

∂ρ(y) ∂y

(11)

2 # dy

(12)

∂P ) is assumed to be approximately constant in each phase of the transition layer the ∂ρ

density dierences can be approximated by the corresponding pressure dierences divided by

κ.

The density prole and its gradient in the GT model are then given by:

(Pmax − Psat ) λy e v for − ∞ < y < 0 κv (Psat − Pmin ) − λy ρl,sat − ρ = e l for 0 < y < ∞ κl 2 2 2y ∂ρ (Psat − Pmax ) = e λv for − ∞ < y < 0 ∂y κv λv 2 (Psat − Pmin ) − 2y = e λl for 0 < y < ∞ κl λl

ρ − ρv,sat =

The augmentation due to vdW yields

m (Psat − Pmax )2 (Psat − Pmin )2 + 2 2κ2v λv 2κ2l λl

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The AGT model then gives the following equation for

σ:

σ = −λv (Pmax − Psat ) + λl (Psat − Pmin ) +αv (Pmax − Psat )2 + αl (Psat − Pmin )2

(13)

σratio = −λ∗v + λ∗l Pratio ∗ ∗ ∗ ∗ +αv∗ (Pmax − Psat ) + αl∗ Pratio (Psat − Pmin )

where

α=

mPc . There are four constants 4κ2 λ

λv , λ l , α v

and

αl

(14)

to be tted and this model is

hereinafter referred to as the 4-parameter Augmented Gibbs-Tolman (AGT) model. At the critical point

Psat = Pmax = Pmin

and

σ = 0.

The ratios are of the form zero divided by

zero and therefore indeterminate. However, in principle, they can be evaluated close to the critical point and extrapolated to yield the critical ratios. This information can be used to eliminate one of the parameters

λ∗v .

At the critical point we note that

c c σratio = −λ∗v + λ∗l Pratio

Eliminating

λ∗v

(15)

from equation (14) using equation (15) and re-arranging the terms we get:

c c ∗ ∗ ∗ ∗ (σratio − σratio ) = λ∗l (Pratio − Pratio ) + αv∗ (Pmax − Psat ) + αl∗ Pratio (Psat − Pmin )

(16)

This last equation has only three independent parameters and is labelled the AGT3 model. In practice, however the critical ratios are hard to get especially because experimental values for

σ

close to the critical point are both rare and somewhat inaccurate. For both Argon and

Octane, the

c Pratio

and the

c σratio

are estimated to be 1 and 0 respectively. From the rst of

the two equations above, this implies that

λv = λl .

The parameters obtained and the AAD

for Argon and Octane using the LK EoS are also reported in the next section.

However

considering the uncertainty in the estimates of the limiting values of the two ratios at the critical point, extensive calculations as in the case of the GT and the AGT model are not

10


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carried out for the AGT3 model.

Lee and Kesler equation of state The Lee and Kesler (1975) interpolative model for the EoS is well-known. It is a reliable EoS over a wide range of conditions for non-polar and slightly polar substances and their mixtures. The authors identify a simple and a reference uid, which are taken as Argon and Octane, respectively. The Pitzer acentric factor

ω

for Argon is nearly

Octane. Most uids and uid mixtures are characterised by

ω

0 while it is 0.3978 for

between 0 and 0.3978. The

compressibility factor is given by

Z = Z (0) +

where the superscripts

(0)

and

(r)

ω (0) (r) (0) Z − Z ω (r)

(17)

denote simple and reference uids respectively.

The

reduced compressibility factor is given by

γ B C γ D c4 P ∗V ∗ = 1 + ∗ + ∗2 + ∗3 + ∗3 ∗2 β + ∗2 exp − ∗2 Z = T∗ V V V T V V V ∗

where

B, C

and

Kesler (1975).

D

are functions of reduced temperature,

∗ ∗ , Pmin Pmax

in the Table 1 for range of

and

∗ Psat

c4 , β

and

γ

(18)

are given in Lee and

are calculated numerically and typical values are given

T ∗ = 0.55 − 0.95.

are provided to facilitate the calculation of

The following equations (19)-(20) and (21)-(22)

∗ ∗ Pmax − Psat

and

the reference uids respectively in the LK model.

11


∗ ∗ Psat − Pmin

for the simple and

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Table 1: Reduced pressure data for simple and reference uids Simple uid

T∗

Reference uid

∗ Pmax

∗ Pmin

∗ Psat

∗ Pmax

∗ Pmin

∗ Psat

0.95

0.802

0.514

0.737

0.751

0.273

0.663

0.90

0.659

-0.147

0.534

0.585

-0.742

0.429

0.85

0.543

-0.955

0.375

0.459

-2.015

0.266

0.80

0.445

-1.918

0.254

0.358

-3.564

0.155

0.75

0.363

-3.657

0.163

0.278

-5.421

0.083

0.70

0.293

-4.408

0.099

0.213

-7.626

0.040

0.65

0.234

-6.025

0.053

0.161

-10.232

0.017

0.60

0.182

-7.991

0.028

0.119

-13.295

0.006

0.55

0.141

-10.434

0.012

0.085

-16.873

0.0017

3

2

∗ ∗ Pmax − Psat = −3.5568 T ∗ + 5.3959 T ∗ − 2.1128 T ∗ + 0.2492 3

(19)

2

∗ ∗ Psat − Pmin = −61.066 T ∗ + 189.55 T ∗ − 204.4 T ∗ + 75.636 3

(20)

2

∗ ∗ Pmax − Psat = −7.8577 T ∗ + 14.964 T ∗ − 8.864 T ∗ + 1.741 3

(21)

2

∗ ∗ Psat − Pmin = −51.615 T ∗ + 193.28 T ∗ − 242.13 T ∗ + 100.15 (0)

σ∗ = σ∗

+

(22)

ω (0) ∗(r) ∗(0) σ − σ ω (r)

(23)

Most the information required to deal with pure liquids as well as mixtures are given in ref. 11. Where

Tc , Pc

and

ω

data are not given in Lee and Kesler,

11

they are taken from refs.

1315.

Results and discussion Figure 3 gives an idea about the various contributions to

σratio

as a function of

T ∗.

They

have been scaled so as to be between 0 and 1. For Argon the plotted curves correspond to

∗ ∗ σratio /30, 5(Pmax − Psat ), Pratio /80

and

∗ ∗ Pratio (Psat − Pmin )/800

∗ ∗ ∗ ∗ 5(Pmax −Psat ), Pratio /200 and Pratio (Psat −Pmin )/800.

and for Octane to

σratio /65,

The curves look broadly similar except

for the contribution due to the density gradient on the vapour side which has a maximum.

12


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(a)

(b)

Scaled contributions to σratio ; (a) Argon and (b), Octane: data (*), eect of density gradient in the vapour sidel - - - -, ; eect of density gradient in the liquid side -.-.-.-; eect of Pratio . Figure 3:

For the Gibbs-Tolman (GT) model, equation (5) implies that

Pratio .

σratio

varies linearly with

Using the van der Waals EoS for pure liquids (Ar, Br, CO, C 2 H6 , C2 H4 , Kr, CH4 ,

N2 , O2 , and Xe), it is found that the data agrees reasonably well with equation (5), except that the dimensionless length scale

λ∗v

turns out to be negative (Figure 4). This point will

be discussed later.

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Figure 4:

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σratio vs Pratio for Van der Waals liquids: data from refs. 16 and 17; linear t .

To quantify the performance of equation (5), it is helpful to introduce the absolute average deviation (AAD), dened by

N N ∗ ∗ − σexpt,i 1 X σratio,cal,i − σratio,expt,i 1 X σcal,i AAD = = ∗ N N i=1 σratio,expt,i σ expt,i i=1

(24)

where the reduced surface tension is dened by

σ∗ =

σ Pc (k Tc /Pc )1/3

The subscripts cal and expt denote calculated and experimental values, respectively, and denotes the

ith

i

data point.

For the Lee and Kesler (LK) EoS, Argon and Octane are used as the simple and reference uids, respectively. The four constants

λv , λl , αv

and

αl may be estimated by tting equation

(14) to data on surface tension using multilinear regression. It is seen from Table 2 and Figure

14


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5 that the GT model provides a reasonable t to the data. Table 2: Parameters of the GT, AGT and AGT3 models Argon

Constants GT

AGT

AGT3

GT

AGT

AGT3

λ∗v λ∗l αv∗ αl∗

-3.1511

-1.7954

0.4134

-4.6176

-0.4374

0.4261

0.3274

0.4516

0.4134

0.3051

0.4243

0.4261

-

2.1039

14.6198

-

11.2997

13.4808

-

-0.0119

-0.00776

-

-0.00664

-0.00667

AAD (%)

4.6

1.05

3.74

7.00

1.48

1.51

For both Argon and Octane, latter

αl

Octane

and

αv

λv

is negative for the GT and AGT models. Further, in the

should be of the same sign, but are of dierent signs for Argon and Octane

in the AGT model. The reasons for this behaviour are not clear. Attempts to apply classical thermodynamics to the interfacial region, whose thickness is of molecular dimensions, may be one of the causes. In the AGT3 model sign. In the AGT3 model, assuming

λv

and

σratio = 0

λl

are positive but

and

Pratio = 1

αv

and

implies

αl

are of opposite

λv = λl .

The AGT3

model is not discussed further as reiterated earlier because of the uncertainty in the limiting value of

σratio

and

calculated value of

Pratio . σ ∗ σ∗

vs.

is a function of the reduced temperature

∗ , σexpt

the measured value of

σ∗

T ∗.

Plotting

for dierent values of

Tr ,

∗ , σcal

the

it is seen

that the AGT model ts the data for Argon and Octane much better than the GT model (Figure 6).

15


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(a)

Figure 5:

Page 16 of 32

(b)

σratio vs Pratio for (a) Argon , (b) Octane using the GT model: data (*) linear t .

(a)

Figure 6:

(b)

∗ ∗ for (a) Argon and (b) Octane: GT model (◦); AGT model (∗). σexpt vs. σcal

For 152 pure liquids (Table 3 - 5) AAD values for the GT model range from 0.8 to 20.2 % and for the AGT model from 0.1 to 7.9 %, and for 57 liquid mixtures (Table 6 - 7), from 0.4

16


Page 17 of 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

to 20.6 % for the GT model and from 0.4 to 8.3 % for the AGT model. The better performance of the AGT model over the GT model may be simply because the former has additional adjustable parameters. pure liquids listed in Lee and Kesler

Figure 7:

11

A similar behaviour is observed for the

(Figure 7).

∗ ∗ for pure liquids listed in Lee and Kesler: 11 GT model (◦); AGT model (∗). σexpt vs. σcal

As shown in Table 3, the AAD varies from 2.7 - 13.3 % for the GT model and 0.5 - 3.7 % for the AGT model. The AAD values obtained using the model of Escobedo and Mansoori

17


18

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

and the correlation of Mulero et al.

19

Page 18 of 32

vary from 0.3 - 4.4 % and 0.2 - 2.5 %, respectively,

which are roughly comparable to the values obtained with the AGT model. Table 3: Comparison of the calculated AAD (%) for pure liquids listed in Lee and Kesler. Here

ω

11

is calculated using equation (19) in ref. 11, the superscript indicates the data source

and the number of data points are shown in square brackets

Liquid Benzene 16,17 [28]

Butane 17 [50] Cyclohexane 16,17 [45] Cuclopentane 17 [5] Ethane 17 [43] Hexane 17 [31] Heptane 17 [50] Hydrogen 17 [19] Hydrogensulde 17 [3] Methane 17 [27] Methylcyclohexane 16 [30] Methyl cyclopentane 17 [5] Octane 17 [34] Pentane 16,17 [16] Propane 17 [25] 1-Butene 17 [6] 1-Pentene 17 [4]

Pc (P a) 48.9 37.96 40.7 45.08 48.72 30.25 27.4 12.93 89.63

Tc (K) 562.2 425.12 553.5 511.6 305.32 507.6 540.2 32.98 373.4

Tb (K) 353.25 272.66 353.85 322.38 184.6 341.88 371.57 20.35 212.84

45.99 34.71

190.56 572.19

37.8 24.9 33.7 42.48 40.2 35.6

ω

AAD (%)

0.207 0.197 0.208 0.192 0.099 0.298 0.349 -0.211 0.094

T (K) 313.15-533.15 235.15-400.15 305.15-525.15 283.15-323.15 170.15-289.15 283.15-463.15 303.15-513.15 19.15-29.15 210.15-313.15

GT 8.58 6.28 9.57 3.8 7.88 6.47 8.99 13.3 2.11

AGT 2.5 1.86 2.57 0.95 1.43 1.22 2.78 3.72 1.72

ref. 18 0.55 1.86 0.3 3.66 3.18 1.55 3.87 -

ref. 19 1.32 1.41 2.6 2.33 2.54 0.19

11.6 374.09

0.011 0.233

110.15-181.15 293.15-423.15

8.23 2.66

4.02 1.74

3.39 1.27

1.55

532.7

344.98

0.228

293.15-333.15

2.99

1.36

0.31

-

568.7 469.7 369.83 419.5 464.8

398.8 309.22 231.02 266.92 303.11

0.3978 0.248 0.149 0.188 0.233

313.15-523.15 262.15-433.15 213.15-351.15 233.15-293.15 263.15-298.15

6.96 6.32 9.52 6.38 4.38

1.42 0.89 4.65 1.77 0.53

2.86 2.3 4.38 -

1.45 3.91 -

For pure liquids not listed in Lee and Kesler,

11

once again the AGT model performs better

than the GT model (Figure 8). Table 4 shows that the AAD is in the range 0.8 - 9.2 % for the GT model and 0.1 - 6.9 % for the AGT model. For some of the liquids, the correlation of Mulero et al.

19

has a signicantly lower AAD, in the range 0.3 - 1.8 %.

18


Page 19 of 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Figure 8:

∗ ∗ for pure liquids: GT model (◦); AGT model (∗). σexpt vs σcal

19


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 32

Table 4: Calculated AAD (%) for pure liquids not listed in Lee and Kesler

11

). Superscripts

indicate the source of the data, and the number of data points is shown in square brackets

Pc (P a)

Liquid Argon

16,17

[73]

Benzonitrile

16

Bromine

16

[4]

[7]

Bromo benzene Chloro benzene

16 16

16

CCl4

16,17

CHCl3

Cumene

Dimethyl ether

17

Ethylene

[3]

Ethylbenzene

Krypton

16,17 16

m-xylene Nitrogen Oxygen

17

o-xylene p-xylene

-

-

1.2

-

[43]

45.2

632.4

0.251

353.15-580.15

6.95

1.08

0.69

-

34.94

132.85

0.045

78.15-92.15

10.4

6.03

-

0.28

77.0

417.0

0.090

233.15-243.15

4.1

0.29

-

-

45.57

556.3

0.193

313.15-523.15

7.77

4.12

2.62

-

55.0

536.5

0.210

303.15-353.15

2.26

2.69

0.76

-

32.09

631.0

0.326

353.15-373.15

1.78

1.73

1.22

-

36.4

466.7

0.281

273.15-443.15

9.24

3.11

50.41

282.34

0.087

163.15-173.15

6.37

1.27

-

0.91

36.09

617.15

0.304

343.15-570.15

7.19

1.65

1.18

-

33.97

569.5

0.270

323.15-333.15

0.85

0.32

0.23

-

45.5

560.1

0.244

313.15-353.15

2.64

0.72

0.21

-

85.1

363.2

0.069

183.15-223.15

5.09

0.1

-

-

45.2

721.0

0.249

403.15-433.15

0.83

1.92

1.26

-

55.0

209.4

0.005

116.15-199.15

7.9

2.54

2.54

0.99

16,17

Toluene Xenon

16,17

16,17

1-Hexene 1-Octene

[16]

[18]

[43]

16

[2]

16

[12]

[4]

[39]

Propylbenzene

[3]

[48]

[33]

16,17

16

[13]

[7]

1-Chloro butane

16

2-Methyl pentane 3-Methyl pentane 2-Methyl hexane 3-Methyl hexane

-

1.95

[5]

16

1.8

0.21

6.94

[4]

[4]

-

4.46

14.55

[3]

16

1.35

2.18 1.46

[39]

16

4.56

404.45-457.15 293.15-323.15

[57]

16,17

83.15-143.15

373.15-423.15

Hydrogen bromide

16

ref. 19

0.119

Ethylcyclopentane

Iodo benzene

ref. 18

0.251

16,17

16

0.362

AGT

584.1

16,17

Fluro benzene

0.001

699.4

GT

670.0

[9]

[3]

150.86

42.2

AAD (%)

103

[30]

17

49.0

T (K)

45.2

[4]

16,17

ω

[6]

Carbon monoxide Chlorine

Tc (K)

[4]

343.15-373.15

1.11

1.53

0.87

-

71.15-120.15

8.76

3.06

-

0.42

50.4

154.6

0.025

87.15-147.15

7.41

3.9

-

1.2

37.32

630.3

0.312

353.15-373.15

2.08

2.34

0.62

-

35.11

616.2

0.322

343.15-373.15

1.19

1.58

0.7

-

32.0

638.35

0.345

353.15-373.15

2.7

2.6

1.04

-

41.08

591.75

0.264

330.15-560.15

7.12

2.63

2.2

1.45

58.4

289.74

0.008

163.15-274.15

7.84

2.23

0.88

0.88

31.7

504.0

0.285

283.15-333.15

3.86

0.88

0.71

-

28.2

585.0

0.323

283.15-373.15

1.86

3.37

0.25

-

542.0

0.228

303.15-333.15

6.28

3.65

3.23

-

0.270

278.15-333.15

4.26

0.56

0.84

-

31.2

504.5

0.272

298.15-333.15

3.37

0.19

0.26

-

[9]

27.3

530.1

0.331

293.15-363.15

3.43

0.66

0.42

-

[8]

28.1

535.2

0.323

298.15-363.15

3.32

0.24

0.31

-

[30]

25.3

549.8

0.339

303.15-373.15

3.94

0.88

0.83

-

[7]

25.6

553.5

0.344

313.15-373.15

3.42

0.27

0.71

-

[8]

24.9

550.0

0.357

303.15-373.15

3.81

1.07

0.82

-

[7]

25.5

563.6

0.371

313.15-373.15

2.48

0.48

0.18

-

[7]

25.4

561.7

0.371

313.15-373.15

2.03

0.76

0.17

-

2,2-Dimethyl hexane 2,5-Dimethyl hexane 3-Methyl heptane

0.327 0.037

497.5

2,4-Dimethyl hexane

4-Methyl heptane

617.0 126.2

36.8

[7]

16,17

16

35.41 33.98

30.2

16

16

0.49

16 16

[12]

16 16 16

20


Page 21 of 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Table 5: Calculated AAD (%) for pure liquids not listed in Lee and Kesler.

11

The superscript

indicates the data source, and the number of data points is in square brackets

Pc (P a)

Liquid Cyclohexene

16

Nitroethane Propene

16

16

1-Heptyne

16

GT

AGT

43.4

560.4

0.210

313.15-348.15

2.93

3.17

48.5

595.0

0.340

333.15-343.15

4.24

4.17

[2]

16

2-Methyl heptane 1,4-Dioxane

AAD (%)

T (K)

[2]

[7]

16

ω

[7]

[5]

1-Heptene

Tc (K)

16,17

17

2-Methyl propene 1,2-Propadiene

[7]

[4]

16

2-Methyl propane 2-Methyl butane

[6]

[4]

16

16

[49]

[5]

2,2-Dimethyl butane 2,3-Dimethyl butane

16 16

[5] [6]

2,2,3-Trimethyl butane 3-Ethyl pentane

16

[7]

2,2-Dimethyl pentane 2,3-Dimethyl pentane 2,4-Dimethyl pentane 3,3-Dimethyl pentane

16

16 16 16 16

[8]

[7] [6] [7] [7]

2,2,3-Trimethyl pentane 2,2,4-Trimethyl pentane 2,3,4-Trimethyl pentane 2,3,3-Trimethyl pentane

16 16 16 16

3-Ethyl 3-methyl pentane 3-Ethyl 2-methyl pentane

[7] [8]

3-Ethyl hexane

[7]

2,2-Dimethyl hexane 2,3-Dimethyl hexane 2,4-Dimethyl hexane 2,5-Dimethyl hexane 3,4-Dimethyl hexane 3,3-Dimethyl hexane

16 16 16 16 16 16

364.9

0.142

203.15-243.15

5.88

0.74

537.3

0.343

298.15-353.15

2.15

2.26

31.3

559.0

0.272

313.15-333.15

4.36

2.92

24.8

559.6

0.378

313.15-373.15

2.21

0.63

51.7

587.0

0.281

333.15-373.15

2.4

0.7

40.0

417.9

0.199

233.15-293.15

4.36

0.29

52.5

394.0

0.122

223.15-253.15

8.15

2.91

36.4

407.85

0.186

313.15-385.15

7.42

1.83

33.81

460.39

0.229

263.15-303.15

5.1

0.33

30.8

488.7

0.233

283.15-313.15

4.85

0.58

31.3

499.98

0.248

283.15-323.15

3.33

0.34

29.5

531.1

0.250

298.15-363.15

2.66

1.54

28.9

540.5

0.311

298.15-353.15

2.48

0.44

27.7

520.4

0.287

298.15-353.15

3.75

0.13

29.1

537.3

0.297

298.15-343.15

2.44

0.08

27.4

519.7

0.304

298.15-353.15

3.63

0.15

29.5

536.3

0.269

298.15-353.15

2.79

0.43

27.3

563.4

0.298

313.15-373.15

3.45

0.21

25.7

543.9

0.304

303.15-373.15

3.41

0.32

[7]

27.3

566.3

0.316

313.15-373.15

2.84

0.3

[6]

28.2

573.5

0.291

323.15-373.15

2.97

0.32

[6]

28.1

576.5

0.305

323.15-373.15

2.81

0.24

[7]

27.0

567.0

0.331

313.15-373.15

3.37

0.68

24.9

574.6

0.314

323.15-333.15

0.84

1.33

16 16

2,2,4,4-Tetramethyl pentane

16

46.0 29.2

16

[2]

26.1

565.4

0.362

313.15-373.15

2.5

0.52

[8]

25.3

549.8

0.339

303.15-373.15

3.94

0.88

[7]

26.3

563.4

0.347

313.15-373.15

2.92

0.16

[7]

25.6

553.5

0.344

313.15-373.15

3.42

0.27

[8]

24.9

550.0

0.357

303.15-373.15

3.81

1.07

[7]

26.9

568.8

0.338

313.15-373.15

2.75

0.24

[7]

26.5

562.0

0.320

313.15-373.15

3.43

0.41

21


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 32

Table 5 continued

Pc (P a)

Liquid Bromo chloro diuro methane Bromo triuro methane Butyl methyl ether

16

Chloromethane

16

Chloropropane

16

16

Dichloro methane

17

[7]

Dichloro uro methane

17

17

Diuro methane

Ethyl formate

16

16

16

17

[18]

17

[15]

[25]

16

[4]

[5]

[4]

[9]

16

Ethyl propionate

Ethyl propyl ether Methyl formate Methyl acetate

16

16

Methyl butyrate

16

16

340.15

0.174

190-320

10.27

3.04

33.7

512.8

0.316

288.15-313.15

1.45

1.97

66.8

416.2

0.151

283.15-303.15

6.47

0.99

45.8

503.0

0.235

283.15-313.15

1.66

2.74

49.86

369.28

0.221

210-350

8.34

2.89

38.73

301.84

0.175

170-280

9.36

2.38

61.0

510.0

0.199

293.15-313.15

6.19

3.52

51.87

451.52

0.207

250-420

7.05

1.51

41.3

385.1

0.179

220-360

9.38

2.35

36.4

466.7

0.281

288.15-303.15

6.02

1.44

58.05

351.26

0.278

268-333

17.33

5.27

28.8

500.3

0.331

288.15-333.15

3.41

0.11

30.3

530.6

0.369

298.15-333.15

2.39

3.86

47.1

508.4

0.282

283.15-313.15

2.5

3.19

38.5

523.3

0.363

293.15-373.15

4.39

1.24 0.11

60.0

487.2

0.254

283.15-353.15

2.94

2.16

46.9

506.8

0.326

283.15-333.15

4.71

1.85

34.8

554.5

0.378

313.15-373.15

2.37

0.75

[6]

40.3

530.6

0.349

293.15-343.15

2.35

0.84

[5]

44.0

437.8

0.236

288.15-323.15

7.13

1.54

40.3

538.0

0.310

303.15-343.15

2.43

1.13

17

17

[35]

[24]

17

16,17

1,2-Dicloro 1,1,2,2-tetrauro ethane 1-Chloro 1,1-diuro ethane

16

[33]

2,2-Dichloro 1,1,1-Triuro ethane [4]

549.73

0.390

303.15-373.15

2.37

1.25

471.1

0.195

260-440

8.04

1.92

48.36

298.97

0.267

210-280

13.36

3.24

40.59

374.26

0.262

210-350

9.49

3.56

33.78

487.4

0.249

270-460

8.84

2.99

45.16

386.41

0.276

220-343

5.54

4.89

42.5

477.35

0.225

275-343

4.55

0.82

54.0

561.0

0.278

313.15-358.15

3.92

1.82

[16]

32.37

418.9

0.244

240-390

8.77

1.98

[21]

31.29

353.1

0.251

200-333.15

9.63

3.09

40.48

410.3

0.231

230-390

6.69

2.35

36.74

456.9

0.282

253.15-423.15

8.3

2.82

36.8

520.6

0.267

293.15-333.15

6.94

3.91

45.1

489.0

0.196

283.15-313.15

3.7

2.65

17

[20]

[15]

[3]

[5]

33.7 44.72

[19]

1-Chloro 1,1,2,2,2-pentauro ethane

2-Chloro propane

39.7

1.38

1,1-Dichloro 1-uro ethane

16

3.29

2.23

1,1,2-Trichloro 1,2,2-triuoro ethane

16

240-350

4.61

1,1,1,2-Tetrauro ethane

2-Chloro butane

0.182

303.15-353.15

[8]

1,2-Dichloro ethane

426.9

288.15-333.15

[7]

1,1-Diuro ethane

42.6

0.391

16

16

7.73

AGT

0.333

Trichloro uro methane Triuro methane

GT

500.2

[5]

17

AAD (%)

546.0

[7]

16

T (K)

33.4

[6]

16

ω

33.7

[6]

[8]

Methyl ethyl ether Propyl formate

[6]

16

Methyl propionate

Propyl acetate

[12]

[3]

Diisopropyl ether

Ethyl actate

[35]

16

Dichloro diuro methane

Dipropyl ether

[12]

[3]

[4]

Chloro triuro methane

Diethyl ether

[12]

[4]

Chloro diuro methane

16

17

Tc (K)

17

17

17

[18]

22


Page 23 of 32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Table 5 continued

Pc (P a)

Liquid Acetone

16

[6]

Acetaldehyde Acrylonitrile

17

[5]

16,17

Acetonitrile Aniline

47.0

16

16

[5]

[11]

Butyraldehyde Butyronitrile

[8]

16

[5]

16,17

508.1

ω 0.307

T (K) 298.15-323.15

AAD (%) GT

AGT

2.24

1.3

55.7

461.0

0.303

283.15-323.15

1.85

4.53

48.3

545.5

0.327

303.15-333.15

1.87

0.52

45.6

536.0

0.350

298.15-333.15

2.58

1.49

53.1

699.0

0.384

403.15-453.15

2.79

4.48

43.2

537.2

0.278

303.15-343.15

4.97

2.18

37.90

582.2

0.373

323.15-363.15

3.14

3.53

73.74

304.12

0.225

217.15-289.15

20.16

7.84

Carbon disulde[6]

79.0

552.0

0.109

308.15-323.15

4.97

5.95

Cyclohexanone

40.0

653.0

0.299

363.15-373.15

2.88

2.96

46.0

624.5

0.288

353.15-373.15

2.1

0.39

Carbon dioxide

16,17

16

Cyclopentanone Diethyl sulde

16

Ethyne

16

16

[5]

Methenthiol Styrene

17

16

[2]

Propylamine Propionitrile

[22]

[2]

16

[3]

[3]

39.6

557.0

0.295

313.15-333.15

1.7

3.15

55.3

503.0

0.191

283.15-293.15

1.69

1.49

[3]

54.9

499.0

0.191

288.15-303.15

2.51

2.11

61.14

308.3

0.189

183.15-223.15

4.05

2.15

[4]

72.3

470.0

0.150

288.15-313.15

8.66

2.03

Dimethyl sulde Ethenethiol

[8]

Tc (K)

16

16

[3]

[4]

16,17

Trimethyl amine

[5]

16

[4]

2-Methyl pyridiene

16

[1]

2,4-Dimethyl pyridiene 2,6-Dimethyl pyridine

16

16

3-Methyl 1-butanethiol 3-Methyl pyridiene 4-Methyl pyridiene

16 16

17

[1] [1]

[1]

[1] [2]

38.2

636.0

0.295

353.15-363.15

3.79

3.69

48.0

497.0

0.283

288.15-313.15

2.25

5.4

41.8

564.4

0.313

313.15-353.15

2.78

1.46

40.75

433.3

0.205

288.15-313.15

6.54

0.52

46.0

621.0

0.299

358.15

1.02

2.27

38.7

647.0

0.351

358.15

5.4

3.66

39.8

623.8

0.373

358.15

5.65

5.83

35.0

604.0

0.191

339.45-362.55

5.34

2.6

44.8

645.0

0.279

358.15

0.82

1.57

46.6

645.7

0.305

358.15

3.01

1.87

For liquid mixtures also, the GT model performs worse than the AGT model (Figure 9). The AAD is in the range 0.6 - 20.6 % for the former and 0.4 - 6.5 % for the latter. The model of Escobedo and Manosoori

20

gives an AAD in the range 1.9 - 2.4 %. The AAD is

less than that obtained with the AGT model in some cases and more in other cases. For all mixtures, experimental data is from ref. 17.

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Figure 9:

∗ ∗ for liquid mixtures: GT model (◦); AGT model (∗). σexpt vs σcal

24


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Page 25 of 32

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Langmuir

Table 6: AAD for liquid mixtures. For mixtures with a superscript

∗ω

is calculated as per

ref. 11

Mixture Pentane - Cyclohexane ∗ Pentane - Benzene ∗ Hexane - Octane ∗ Hexane - Heptane

∗

Cyclopentane - Benzene ∗ Benzene - Cyclohexane

∗

Benzene - Methylcyclohexane ∗ Benzene - Hexane Benzene - Heptane

∗

T (K)

GT

AGT

ref. 20

288.15

2.92

1.53

-

288.15

8.75

6.48

-

313.15

1.84

0.44

-

303.15

1.19

1.15

-

298.15

2.50

1.28

2.42

313.15

1.17

0.93

-

303.15

7.75

3.24

-

298.15

5.36

3.46

4.04

2.88

3.18

AAD (%)

313.15

2.69

∗

313.15

5.29

4.72

-

∗

303.15

18.13

5.34

-

318.15

20.58

5.23

-

303.15

1.83

0.83

-

293.15

3.15

1.36

-

298.15

1.89

0.75

-

303.15

1.25

0.65

-

308.15

1.17

0.59

-

Methane - Propane

Cyclohexane - Heptane ∗ Cyclohexane - Hexane

∗

Toulene - Octane

303.15

7.64

1.39

-

Toluene - Heptane

308.15

3.66

2.35

-

318.15

3.36

2.63

-

323.15

5.78

4.51

-

333.15

3.42

4.29

-

323.15

2.99

1.89

-

333.15

2.09

1.56

-

Toulene - Cyclopentane

298.15

2.21

1.11

1.88

Nitroethane - Hexane

303.15

2.63

2.24

-

Butyronitrile - Hexane

323.15

5.87

4.90

-

333.15

4.85

4.48

-

343.15

4.49

3.84

-

288.15

5.88

4.99

-

Toluene - Cyclohexane

Pentane - m-xylene Pentane - p-xylene

288.15

7.45

6.21

-

Methyl acetate - Pentane

298.15

3.79

3.01

-

CCl4 - 1,4-dioxane

293.15

3.81

4.58

-

CS2 - CCl4

308.15

5.24

3.85

2.81

313.15

4.49

3.85

2.75

CCl4 - Iodomethane

308.15

0.63

1.19

2.37

Cyclohexane - Methylcyclohexane

303.15

1.30

3.41

-

Cyclohexane - Trimethylamine

303.15

1.96

2.76

-

Cyclohexane - Ethylbenzene

308.15

1.03

6.24

-

Cyclohexane - o-xylene

313.15

2.18

8.28

-

Acetonitrile - 1,4-Dioxane

298.15

5.25

1.43

-

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Page 26 of 32

Table 7: AAD for liquid mixtures

T (K)

GT

AGT

CCl4 - Cyclopentane

298.15

0.64

1.16

CCl4 - 2-Propanone

313.15

0.88

2.60

CCl4 - Methylcyclohexane

303.15

5.35

1.46

CCl4 - Benzene

298.15

0.97

4.42

CCl4 - Cyclohexane

293.15

3.54

2.50

Mixture

AAD (%)

CCl4 - Toluene

333.15

2.08

1.33

CH2 Cl2 - Pentane

298.15

5.13

2.66

CCl4 - CHCl3

298.15

5.03

2.32

CCl4 - Acetonitrle

308.15

1.58

1.64

318.15

1.09

0.91

298.15

4.55

3.08

303.15

2.91

2.80

313.15

0.43

1.86

308.15

8.47

3.82

303.15

8.56

2.60

2-Propanone - Benzene

1,2-Dibrormoehtane - Benzene

298.15

8.87

1.47

1,2 Dichloroehtane - Benzene

308.15

1.69

4.61

CS2 - Benzene

298.15

7.75

3.28

1,2-Dibromo ethane - Toluene

303.15

9.85

2.99

N,N-Dimethyl formamide - Toluene

318.15

7.70

1.54

Triethylmanine - Toulene

303.15

5.05

1.44

1,2-Dibrormoehtane - Cyclohexane

303.15

6.62

1.50

298.15

6.19

0.61

Acetaldehyde - 2-Propanone

293.15

4.10

4.31

Nitromethane - 1,4-Dioxane

303.15

8.40

0.75

Toluene - Methylcyclohexane

303.15

10.25

3.23

313.15

0.72

0.82

125.00

2.41

1.93

130.00

3.17

2.62

139.00

3.38

5.87

Argon - Nitrogen

83.82

1.75

3.80

Krypton - Methane

125.17

1.36

4.21

150.90

2.23

6.00

Argon - Krypton

160.30

1.12

1.89

CO - Nitrogen

83.80

6.28

6.64

CS2 - CH2 Cl2

293.20

1.24

0.77

308.20

2.66

4.75

Cyclohexane - p-xylene

313.15

0.78

5.82

323.15

1.00

3.71

Cyclohexane - m-xylene

313.15

0.82

5.86

323.15

1.02

3.77

293.15

9.35

3.87

Nitromethane - Cyclohexane 26


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Langmuir

Conclusions Overall, the AGT model for surface tension is easy to use and is shown to have exceptional correlating ability. Its only defect is that some of the parameters which should be positive turn out to be negative. The AGT3 model is particularly attractive if the limiting values of

σratio

and

Pratio

can be determined accurately at the critical point.

Acknowledgement One of us (MSA) is pleased to submit this article as a contribution to the special issue of Langmuir brought out in honour of his rst and only research mentor Prof. K.E. Gubbins. He is also pleased to acknowledge with gratitude the warm hospitality of the Departments of Chemical Engineering in the Indian Institute of Science, Bangalore and in the Indian Institute of Technology Bombay, where part of this work was done.

Notation

Alphabets

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a

Thickness of the transition layer in the vapour phase

b

Thickness of the transition layer in the liquid phase

c

Curvature

k

Boltzmann constant

m

Empirical constant

P

Normal pressure

R

Gas constant

T

Absolute temperature

x

Distance measured from location of surface of tension

y

Distance measured from the surface of discontinuity

yt

Distance between surface of tension and surface of discontinuity

Z

Compressibility factor

Greek symbols α

mPc 4κ2 λ

β, γ

Constants in equation (18)

κ

δP δρ

λ

Characteristic length of the transition layer

µ

Chemical potential

ρ

Density

σ

Surface tension

ω

Acentric factor

Subscripts

28


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Langmuir

c

Critical properties

cal

Calculated

expt

Experimental

max

Maximum

min

Minimum

sat

Saturation

v

Vapour phase

l

Liquid phase

Superscripts c

Limiting value at critical point

(0)

Simple uid

(r)

Reference uid

∗

Non-dimensionalised quantity

References (1) Gibbs, J. W. On the Equilibrium of Heterogeneous Substances.

Am. J. Sci.

1878,

441458.

(2) Tolman, R. C. Consideration of the Gibbs Theory of Surface Tension.

J. Chem. Phys.

1948, 16, 758774. (3) Tolman, R. C. The Supercial Density of Matter at a Liquid-Vapor Boundary.

J. Chem.

Phys. 1949, 17, 118127. (4) Mathias, P.

Thermodynamic Modelling of Surface Tension: Project Report ; 1974.

(5) Sundaresan, S.

(6) Rao, K. K.



(7) Hill, T. L. On Gibbs' Theory of Surface Tension.

J. Chem. Phys. 1951, 19, 12031203.

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Page 30 of 32

(8) Hill, T. L. Liquid-Vapor Transition Region and Physical Adsorption According to van der Waals' Equation.

J. Chem. Phys. 1951, 19, 261262.

(9) Rowlinson, J.; Widom, B.

Molecular Theory of Capillarity, International Series of

Monographs on Chemistry ; 1982. (10) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N.

Transport Phenomena ;

John Wiley &

Sons, 2007.

(11) Lee, B. I.; Kesler, M. G. A Generalized Thermodynamic Correlation Based on ThreeParameter Corresponding States.

AIChE J. 1975, 21, 510527.

(12) Tumram, S. The Gibbs-Tolman Model for Calculating Surface Tension from Bulk Equations of State: Thesis. 2014.

(13) Reid, R. C.; Prausnitz, J. M.; Poling, B. E.

The Properties of Gases and Liquids , 4th

ed.; McGraw-Hill New York, 1987.

(14) Poling, B. E.; Prausnitz, J. M.; John Paul, O.; Reid, R. C.

The Properties of Gases

and Liquids, 5th ed.; McGraw-Hill New York, 2001. (15) Sinnott, R.

Heat Transfer Equipment, Coulson & Richardson's Chemical Engineering ;

Butterworth-Heinemann, 2005.

(16) Jasper, J. J. The Surface Tension of Pure Liquid Compounds.

J. Phys. Chem. Ref.

Data 1972, 1, 8411010. (17) Lechner, M.; Wohlfarth, C.; Wohlfarth, B.

Surface Tension of Pure Liquids and Binary

Liquid Mixtures ; Springer, 1997. (18) Escobedo, J.; Mansoori, G. A. Surface Tension Prediction for Pure Fluids.

1996, 42, 14251433.

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Langmuir

(19) Mulero, A.; Cachadiña, I.; Parra, M. Recommended Correlations for the Surface Tension of Common Fluids.

J. Phys. Chem. Ref. Data 2012, 41, 043105.

(20) Escobedo, J.; Mansoori, G. A. Surface-Tension Prediction for Liquid Mixtures.

J. 1998, 44, 23242332.

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Langmuir

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