Perturbed hard sphere equations of state of real liquids. I. Examination of a simple equation of the second order

235

Fluid Phase Equilibria, 12 (1983) 235-251 Elsevier Science Publishers B.V., Amsterdam

PERTURBED

HARD

EXAMINATION K.

and

AIFl

of of

A

EQUATIONS

SIMPLE

OF

EOUATION

Process

Chemical

Sciences,

(Received

in The Netherlands

STATE

OF

OF

THE

REAL

SECOND

LIQUIDS.

I.

ORDER

NEZBEDA

I.

Institute Academy

SPHERE

OF

-Printed

January

165

Prague

1982;

8,

Fundamentals,

02

6,

accepted

Czechoslovak

Czechoslovakia

in

final

form

May

16,

1983)

ABSTRACT K.

and

state

of

Aim,

the

Utility

I.

Fluid

a

perturbed

of

the

hard

An

both

attempt

their

is

temperature

properties. perturbed

hard

sphere

made

to

some

sphere

and

a

sphere

equations

simple

equation

12: of

square is

in

general

state

well

based

parameters

terms

of

for

of

state

are

a

orthoof

of

the

independent on

on

fluid

examined

conclusions

of of

235-251.

supercritical,isotherms

interpret

equations

a

equation

for

contributions sub-

dependence

Finally,

hard of

Equilibria,

theory

three-body and

Perturbed Examination

Phase

Barker-Henderson

densities

fluids.

1983.

I.,

liquids.

order.

of

inclusive

and

real

second

simplified

baric

Nezbeda,

the

simple

equation molecular

family

of

drawn.

INTRODUCTION Theoretical

statistical

state

aim

fluid

rather

data.

Nonetheless,

to a

usually than

interpret, few

if

molecular

require

very

which

be is

understanding

at

obtaining

an not

accurate However,

advantage predict,

considered quite

equation

origin

this

the

beyond

the

the

approach

other

hand, of

is

of of

experimental

that

by

it

means

chemical its

equation

equations properties

with

properties

regardless

else

of

agreement

bulk

empirical

anything

useless

of the

On

any

calculations‘on the

accurate

parameters.

justification. hardly

mechanical

at

allows of

theoretical

of

state

than

an

interpolation

range

of

available

(EOS) scheme

experimental

data. In liquids

the

last

has

0378-3812/83/$03.00

made

two

decades

considerable

the

statistical progress

mechanics

and,

0 1983 Elsevier Science Publishers

B.V.

thanks

only

engineers

of to

simple

various

can

236

ingenious now

integral

available

simple

liquids

1980;

Smith

are

not

two

reasons:

about

within and

complexity

of

the

actual

intermolecular

of from

on

suitably

number defined

interactions an

directly

the

construct

sphere

of

density-dependent attractive

diameter) (F or

the

first

derived

version

permanent

make

use

attempts

well.

The

primary

can,

at

least

in

principle,

is

the

stems

repulsive gave systems

resulting term to

become

it.

EOS

with

e.g.

Based

in

is

either

popular

or

combined

EOS

being

on

to the

same

temperature-

terms

accounting

Leland,

1980

and

of

the

the

being

above

type

Contemporary known made

question

accurate to

is,

lead

to

is

hard

improve

whether

an

the

van

improvements sphere

the

the term

correction

such

accurate

der

of

an

and

approach

relatively

EOS.

This

paper

perturbed

hard

practical

calculations.

employing

assess and

of

1873.

are

as

by

has

see

fluids of

finding

semiempirical

review

EOS

in

term simple

the

temperature-

is a

to

1979).

Historically, equation

a

(with

for

allowed

intermolecular

related

it

is

reference

leading

somehow

result,

spheres

forces.

Robinson,

original

EOS or

for

originally

of

This

of

the

purposes

hard

for

and

with

are

role

with

Consequently,

series

of

scheme.

predominant

part

why

compensate

the

structure.

repulsive 1977).

practical

EOS

and

and

the

is

theory

theories

theories

mechanical

for

: the

Waals

by

hard

the

liquid

perturbation

expansion

statistical

Chao

the

of

(Boublik,

by

of

To

such

uncertainties

the

degenerate

mechanical

knowledge

as

(ii)

parameters

correlation

statistical

the

model.

may

another

of

al.,

because

That

potential

which

establishing

a

potential

et

theories

and

version

approach,

into

the

the

are

classical

(Boublik

interactions.

pair

of

calculations

theories

simplified

variables,

procedure

primarily

a

such

state

Success

way

a

with

of

rigorous

given

However,

(i)

methods

properties errors

practical

on

to

1978).

theories,

the

to

inaccuracies

this

Henderson,

combined

forces

perturbation

calculate

pseudoexperimental

calculations

depend

to

directly

the

usually

and

us

applicable

practical

rise

equations

allowing

the

limits.

sphere

a

The

of EOS

-Henderson

theory

parameters

adjustable,

part,

equations The

quite

family

first

for

general these employed

goal

a

of

state

and

their

the

paper

is

of

series

perturbed

is

examine

of

hard

equations

a square-well we

uf

by

based

on

a

of

devoted

to

application

in

two-fold.

sphere

finding

fluid. first

papers

its

EOS we

try

potentialities

simplified

Barker-

Leaving

all

all

utility

the

First,

the

EOS (i.e.

to

237 ability

to

correlate

orthobaric

densities we

Second, potential of

make

and

properties

conclusions to

the

of

the

entire

of

state.

equations

and

find

a link

independent

by

the

data)

sub-

only

paper

family

of

possess

of

between

the

equation

of

isotherms.

We

more

general

simple

This

is

molecular

it possible EOS.

for

obtained

quantities.

existence

an

the

super-critical

makes

means

of

physical

the

parameters

liquids

of

to

because

of

of

kinds

both

attempt

importance

interpretation

apply

and

an

parameters

high

the

particular

to

believe

predict

the

that

most

validity

and

perturbed

of

hard

sphere

pair

potential

THEORY Equation by

of

approximate,

an

effective oo

USW

because

Red

= -E

for

d(R
0

for

R -z R,

d

well,

and

is a hard

potential

with

correctly From that

the

their

on to

pressure.

the

Axilrod-Teller

was

estimated

and

Dividing perturbation of

the

total

parts,

Helmholtz

a pair

argon

of

real

for

et

+ upert,

to

this reproduce

1971)

it follows

negligible up to

50

and per

effects,

contribution

the

al.,

potential

potential

is able

three-body

whose from

the that

al., not

represent the

of

known

molecules.

et are

P, may

(Barker

energy,

It is

forces

account

u = uref

is a depth

(Barker

potential

pair

free

true

parameters

analytically

coworkers

the

chosen of

To

E

potential.

pressure,

pressure Barker

the

three-body

consider by

diameter,

liquid

non-additive

total

of

properties

study

contribution

the

sphere

appropriately

basic the

the

potential,

(1)

is a range

R.

of, simplicity,

square-well

for

where

of

state

We

perturbation

cent we

to theory

1968).

into the

F, assumes

reference

and

perturbation a form

(Boublik

expansion et

al.,

1980): F/(NkT)

= F ref/(NkT)

For

first-order

F(l)

the

= 2rp

7 0

+ F(')/(kT) perturbation

gref(r)

upert(r)

+ F(*)/(kT)* term r* dr

we )

+

...

(2)

have: (3)

238 where

p

is

function

the

number-density For

(rdf).

compressibility is,

the

only

the

the

than

therefore

distribution

local

used;

(Barker

has

radial

usually

better

approximation

approximation

is

term

is

marginally

compressibility

g

second-order

approximation

however,

latter

and

this

the

and been

approximation

macroscopic

Henderson,

chosen

1976).

because

The

of

simplicity: m F(2)

= -*PkT

where

subscript

For

the

system d c

(2) aP

pair

is R <

ghs(r)

hs

i

hs

denotes

potential

the of

straightforward, R,

and

tupert(r)12 hard

form =

otherwise.

the

the

s

(4)

system. choice

uhs(R;d)

For

dr

sphere

(1)

uref

zero

r2

of

and

u

pert

first-order

the

reference -E

= term

for

then

it

holds: F(‘)/(kT) where the

p

x

3

= Pd

integrals

method that

the

RO

, T” in

due

square

to

and

sphere

and Renon

rdf

and

x

(5)

analytically,

=

equals

Ro/d.

s

(19 76)

(5)

unity

order

In

we

based

on

beyond

the

the

to

express

follow

the

approximation range

of

the

-

11/(4”P)

we1 1 potential, dr

r2

Approximation general,

(6) it

The

E 7’ d

is

hard

r2

shown for

+

ap hs (,p)T,”

[kT

.

be

justified

R,

>s

is

accurately

(Carnahan

+ y2

-

and

for

R,/d

=

1.5

but,

-

in

d. described

Starling,

by

the

1969),

y3 ,

(1

(7)

Y)3

= 11p’/6.

Using

usual

thermodynamic

factor,

i!

=

z

+

*z(l)

= Zhs

3 R,/3

to

= y

=

any

reference EOS

1 + y

dr

(6)

was

sphere

(P/PkT)hs

ghs(r)

correct

Carnahan-Starling

where

(4)

,

1 r 2dr

ghs(r

= T/(E/k),

Eqns

Ponce

hard

ghs(r)

;

:,

= -2nPx(l/Tx)

where

A.z(~~)

After

evaluating

finally

get

P/pkT,

may +

dz(2)

stands the

+

for the

following

relationships

be

expressed . . .

the

the

LQ(3b)

to

(2), (8)

due

to

compressibility

expressions

Eqn

,

contribution

isothermal

compressibility

similarly

for

three-body from

AZ

(i):

Eqn

forces. (7)

we

239

T"

a~(')

= -4yX3

(4y

+

- 2y3).(1

+ 1oy*

- y)3 ,

DL AZ(~)

Tx2

= -6yCI

- 6y21

"c

a1 -

- 6y2C

ay

(9)

,

(10)

ay

where D = 1 + 4y

+ 4y2

(1 - Y)~/D

c =

I = 13/3

+

of

the

only

These

the

free

energy

was

quite

well = y2

YN where written

where

y

with

sphere

though

to

introduce

the

- 0.08477~

+ 0.05499y2

The

constant.

the

~~(1

and

factor

al.

found of

to

that

the

this

form

(12)

’

corresponding

compressibility

make et

contribution

Pade' approximant

0.07516

(7)

first

may

order

be conveniently

form, + 5y

of

to

given

EOS

of may

density any

+ 15y2

(11) state

seem

in

- 3y3)

,

Eqn

(13)

define

the

perturbed present

we

have

polynomials,

of

first

order

as

PHS2

retained the

order

EOS,

hard

paper.

preferring

Throughout

the

second

is denoted

the

in the

complicated,

appearing

equation

whereas

(lo),

(13)

simplifications.

sphere

PHSl

and

investigated rather

the

additional

hard as

by

Numerical

through

equation

perturbed

referred

in

not paper

l/TX

involving

is

the

equation.

procedures EOS

sphere

and

the

the

pair

fourth

by

we

to Barker

the

reference

- 0.31570y2

the

orders

hard

numerically sphere

forces

due

= ,/(,d').

(PHS)

The

hard

three-body result

+ 0.88445~

to

all

term

the

0.74240

Expressions Even

the

represented

= 20y2

x

of

analytical

calculated

in a simplified

4~(~~)

the

contribution

y is a force

contribution

.

available

authors

F(3b)dg

, (11)

(C - 1)/(24y)

the

(1968).

t y4

,

Concerning use

T"

- 4y3

derived

is essentially

diameter

reduced

range

potential

parameter,

(d), of

model the

the

depth

attractions and

were

temperature

a four-parameter of

the

equat ion.

potential

(A)

are

subject

to

independent

we1

directly adjustment. three-body

The

1 (E/k),

re lated The force

to

240 strength

has

(y),

al., To

study

were

the

the

best

for

argon

fit

of

(Aim,

It

violated

and

weighted

least

estimates. opinion

the

the

resulting

(Barker

et

(Pe

P is

the

been

t

all

taken

the

It

of

the

The

where

a

and

V

of

T;)2

2 ST i

in

this

data

point

(V”c

-

V8

are

the

in in

K,

expression

=

(14)

(14)

proved

have

a

and been

the

isothermal

be

reduced

one-step a

set

of

calculated

coexisting P-V

on

to

has

to

temperature

these

Pe, calculations

which

function

true

paper

0.0015 the

deleted,

of

and respective

this

sp

vo 1 umes

and

the

. During

values

(14)

molar

experimental

v;

for

3

sVG and

= 0.01

given

V;)2 2

so

model as

phases.

data

outside

the

was

(TP -

if1

ihood

is to

respectively,

: fugacities

used

1 i kel

experimentally

+

errors

therefore

For

are

function

orthobaric

reported sT

objective

function

region

index

VE,

equality

objective

N =

Pt,

the

two-phase

the

procedure,

parameters, satisfy

been

assumptions 1981)

maximum

the

V:)2

between

computations

has

optimization

minimization

-

phases,

= 0.00015 = 0.00015 VF, and sv sVl. G orthobaric data the first term in insignificant.

an

experimental

as

the

errors.

VI

distinguish

the

objective

criterion

2

vapor

invariably

parameters

represent

sVL

estimated

In

variables.

(V;

pressure, and

and

are

of (Aim,

the

all

from

+

saturation

e SK

than

obtain

was

Pt)2

2 sP

liquid

superscripts values;

-

some

objective

to

to

algorithm

parameters

that

these

as

iterative

optimization

minimized

so The

an

of

fact

subject

parameters,

symmetric

method

rather

parameters

+

EOS level

case

symmetric

the

are

Tt)2

2 ST

our

appropriate

function -

(Te

in

using

values

for

the

likelihood

likelihood

the

EOS

of

consideration.

by

estimates

the

equilibrium

have

that

maximum

variables

objective

where

clear

allows

evaluate

=

1980)

the

most

it

under

HanEil,

square

the as

determined

s2

literature

temperature

maximum

Nevertheless,

problem,

To

so

each

property

is

with

at

minimizing

and

1981).

our

the by

(Rod

associated

to

from

dependence

separately

estimated

function

of

taken

temperature

evaluated

were

s,

been

1971).

runs

(Pf +

over

-

P;)2

2 sP all

points

(V;

-

V:’

+

(15) considered

on

an

isotherm.

241 When

orthobaric

simultaneously function s3

had

= m.S,

where the

and

to the

the

isotherm

EOS

data

parameters,

were

the

used

objective

form

t S2

,

(1’3)

m is a factor orthobaric

liquid-phase

determine

from

data

the

point

range

1 to 0.67N

relatively

to the

used

to over-weight

points

on

liquid

isotherm. The

optimum

determined

parameters

at

by minimizing

the

the

critical

objective

point

have

been

function

(17)

f where

subscript

the

model

Ptc,

and

t V, were d2P

(Z)2

To

DISCUSSION additional

with our

krypton,

EOS

liquid

factor,

the

(iv)

coefficient. individual The

compilation were

also

basis

of

stemming

typical

methane.

for

critical

Given TE,

criterion

The

uncertainties

complex

liquids,

properties fit

- volume point

from

we

of

the

the

PVT

data,

and

(v)

to assess

the

relative

in

tried Eqn

source

the

(Vargaftik, from

of

triple

of

data,

(iii)

the

second

virial

relevance

of

(8).

range the

(i) data

compressibility

supercritical also

on were

orthobaric

(PVT)

critical

we

argon,

focused

both

- temperature

and

molecules

namely

We

taken

point.

coordinates

the

more

simple

simultaneous

pressure

from

The

the

models

critical

investigated

MPa.

the

(18)

to

(ii)

terms

temperatures 100

and

phase of

at

critical.point

problems

only

data,

location

on

interaction

xenon,

orthobaric

the

.

AND

applied

and

localized

avoid

connected

quantities

values,

-0

+(z)2

RESULTS

c denotes

parameter

variables to 450

PVT

1975). the

state point

NBS

krypton

from

the

work

of

methane

from

API

Res.

Project

data

employed

In addition, monograph

Theeuwes 44

covers,

K and

and

was the

(Gosman Bearman

(1968).

roughly,

pressures

up

to

Vargaftik's PVT

et

data

al.,

(1970),

for

1969), and

for

argon for

1

R,/d

vapor

89.83 90.11 100.22 118.81 132.11 140.72 152.43 177.89

1.825 1.828 1.771 1.679 1.624 1.592 1.552 1.479

1.781 1.795 1.787 1.774 1.719 1.682 1.682 1.562 1.542 1.492

R,/d

to PHS2

are

equation.

and

are

by PHSl

volumes

K

76.02 74.44 75.84 77.81 86.17 92.64 92.93 117.76 122.56 135.93

E/k,

data

uncertainties

correspond

experimental

and

3.7054 3.6515 3.5997 3.5459 3.5376 3.5398 3.5441 3.5502

110.00 130.00 150.00 170.00 175.00 180.00 183.00 186.00

liquid

3.3758 3.3549 3.3306 3.3140 3.2949 3.2666 3.2309 3.1968 3.1781 3.1531

87.28 94.39 102.85 108.58 114.99 124.19 134.83 142.69 145.99 149.00

'Both

d, 1

fit of orthobaric

T, K

Best

TABLE

0.08789 0.36710 1.0330 2.3380 2.7880 3.2880 3.6180 3.9800

therefore

not

shown.

%

The

parameters

d, E/k,

%

0.02 0.02 0.26 0.45 1.41 1.11 1.11 1.17

-0.07 -0.03 to.03 0.11 0.12 0.70 2.05 4.05 6.34 10.02

AP,

and

within

PHSE

well

0.08791 0.36717 1.0357 2.3485 2.8272 3.3246 3.6581 4.0268

0.10126 0.20258 0.40541 0.60859 0.91304 1.5275 2.5850 3.6899 4.3099 5.0168

P, MPa

equations

2.25 4.40 8.33 9.11 11.1

0.92

1.08 1.74 2.67 3.36 4.56 6.03 a.78 13.2 16.4

AP.

PHS2

PHSl

and

0.08870 0.37537 1.0785 2.5328 3.0421 3.6528

by PHSl

e

0.10242 0.20617 0.41612 0.62838 0.95348 1.6116 2.7556 4.0160 4.7183

P. MPa

equations'

t h a_n

0.10133 0.20265 0.40530 0.60795 0.91193 1.5199 2.5331 3.5464 4.0530 4.5596

argon

exDt1

reproduced

me

PHS2

P, MPa

and

243 Subcritical

region

Comparing performs EOS

the

much

of

the

term

improvement

et

of

this

as

strength

as

the

For

but

both

(y

of

argon

=

some

lead

the

PHS2

the

inclusion

three-body

data

trial

isotherms

erg.cm', of

the no

and

liquid

values to

into brings

data

and

73.2~10~~~

rather

equation

forces

orthobaric

orthobaric

value

well

for

equations,

respects, for

of

theoretical

1971)

PHS2

all

region.

treatment

the

al.,

force

and in

accounting

in

simultaneous use

PHSl

better

the

the

see

Barker

three

body

slight

deterioration

on

subcritical

of

results. Therefore

we

restrict

the

of

Eqn

to

forms

results

obtained

for

those

for

argon

argon

and

methane.

For

orthobaric

volumes is

are

superiority Table

very

of 1.

good

a

low

become

the

calculated

equation orthobaric

methane

displayed

on in

the

by

the as

selected

Table

mostly

PHSl

data

pronounced

results

largely

facts

of

the

EOS

significant is

PHS2

documented

equation

is

pressure

critical

region

for

with

about

equilibrium

equation the

isotherms

1 along

to

The

positive

the

the

analogous on

in

region

Since

inadequacy

pressure.

over

but

term.

deviations

whereas

of

more

The

are

discussion

small

high

the

AZ (3b)

the

xenon

typically

temperatures,

approached. are

our

PHS2

fit

of

and

base

or

too

the

The

at

departures

can

data

by

discussion

exclusive

krypton

negligible

reflected

in

we

the

(8)

the

is

argon

and

effective

potential

parameters. It

is

interesting

equation values

of

these

are

second

The

as

if

the

not

m

is

the

individual

related shown

terms

factors.

in

to

the

Table

2.

with

opposite

important

despite

the

PHS2

hard-sphere Since

signs its

in

Characteristic

the

the

first

inclusion

relatively

two of

the

small

=

was

of

unity

data

factor thus

to

PVT

by

the

from

reality; N in

and

m

in

be

and

PHS2

Eqn

confirming

0.67

included

found

liquid-phase

incapable

of

orthobaric equation

describing

data

simultaneously.

yields,

roughly,

the

follows:

distorted,

equation -

and

equation the

treatment

for

is

how

value.

results -

note

compressibility

are

large

term

PHSl

reasonably Such

factor

very

order

absolute

to

contributions

compressibility terms

to

contribute

the

VB

optimization,

(16)

the

term

in the

the

existing

the

fit

of

orthobaric

departure

expression

representation

of

for of

S,

data the

(14)

orthobaric

is

244 TABLE

2

Relative contributions the compressibility

of factor

individual

Z/Zhs

= Zhs/zhs

terms +

in

the

Z(')/Zhs

PHS2

equation

+ Z(2)/Zhs

near triple point

liquid vapor

0.0005 0.962

1 1

-1.0390 -0.027

0.0395 -0.011

near T_

liquid vapor

0.067 0.405

1 1

-0.972 -0.527

0.039 0.068

data

ranges

IAP] ~10

from

per

= 0.85;

T/Tc good

at

the

the

P (

PC,

pressure The

and

good

cent,

where

parameters data

increasing

temperature.

is of

the

about

of

yx

by

The

high

the

isotherms the

pressure

per

at

cent is

higher

cent

at of

more

at

very

pressures

50

MPa.

liquid-phase

evaluated

becoming

a

zc,

The

minimum

parameter

value et

is

viz.

from

pronounced

zc

(1971).

increases

equations

of

critical

that

by

namely

with

value

of

zc

at

x

the

the

the

yielded

and

the

proceed shifts

PHSP

family,

by

1.35.

three-body

marginal

we

PHSP

relative

=

improvement

approximately

As

again

this

given

0.3280

only

corresponding

al.

as

parameter,

surprisingly,

indicate

the

third x.

values

to

to to

yx

greater

X.

equation,

fails

to

that

higher as

well

as

describe

region.

region

expected,

equation

fourth

Barker

zc

Supercritical

of

deviations per

isotherms

those

AZ (3b) term)

the

brings,

minimum

other

adequately

of

the

= 0.02,

given the

fit

~20

treatment from

factor,

the

well,

(without term

values,

As

differ difference

of

potential

Introduction

all

function

equation

y

10

maximum

lAVg/ but

cent, to

simultaneous

compressibility

a

interaction at

per up

the

to

and

subcritical

increase on

up

cent,

of

somewhat

only,

critical

equation

for

per

point

The

PHS2

temperatures

IbPl 5 0.5

based data

orthobaric

width

low

representation

deviations

orthobaric

Critical

at

lbVLl ~0.7

to

the

without of

argon,

experimental critical

examination

AZ (3b)

the

krypton, data

point.

deviations

At

the

to

xenon,

is

exhibit

of

term least

T/T, the

ability

correlate and

of

methane

satisfactory

Q, 1.02, trend

typically, as

the

PHSE

supercritical

follows:

revealed in

close the

that

the

vicinity maximum

IAPI 2 0.5

per

245 cent 100

at

1 MPa,

MPa.

values

Temperatures equation Potential

% 1.3

the the

0.8,

and

are

both

well

\APl fit

above

4 per

PVT

range

cent

at

data

mentioned

cent,

maximum

respectively.

reproduced

supercritical

10 per

5 of

by the

PHSE

investigated.

parameters

Potential particles

parameters Dealing

usually The

better

variables extent,

a real

as

the

model

the

less

can

be expected.

to assess

potential

and

the

parameters

thus

many-body

and

an

less the

by looking

obtained

from

discussion

we

the

approximate

at

parameters

fitting

for model

calculations to

state

possible,

the

of

state

potential

parameters

It is therefore

EOS

pair

on

to compensate

inadequate

the of

isolated

depend

system,

in order

for

sensitivity

of an

cannot

adjustable

calculations

used,

properties

speaking,

with

treated

approximate

reflect

strictly

and,

variables.

used.

whole

10 MPa,

0.5,

volumes

the

at

isotherms

At T/T,

become

and

over

cent

higher

considerably.

deviation

are

5 per

5 for

However,

improves

are

(API

to a certain

behavior

of

the

the

EOS

to experimental

the

subcritical

data. the

In since The

following

similar

problem

conclusions is,

that

the

parameters

is not

one

extrapolate

cannot Effective

diameter, properties pair

as

interactions. are

in their

size

There

are

several

explicit

dependence

decrease

with

kinetic

energy,

obvious. who

The

d on

whereas

simplest purely

theory

of

core

of

the and

so

at

to be

the

a small

theories

increase

due

providing

In

general,

d should

of

increase

in

density

power-type

of

is known

in pressure.

because

of d is that

molecular part

densities

so that

on

several

other

EOS

liquid

variables.

dependence

sphere

for of

type

mechanical

state

repulsive

as well.

region

repulsive

increment

temperature its

any

hard

interest in terms

other

enormous

statistical of

hard

to each

an

increasing

investigated

any

of

critical

effective

diameter

d parameter:

close

produces

The

primary

collision

to the

quite

of

region,

region

region.

interpreted

Further,

hi'ghly sensitive molecules

quantity the

in the

diameter.

be easily

such

this

on

supercritical dependence

function

over

sphere

is the

It can

focus

in the

temperature

a simple

hard

d,

reasons.

hold

to

is much

less

Rowlinson

potentials

and

(1964), found

that d *

(l/T)l'n

,

(‘9)

246 where

n

defines

(1967a) was

the

define

d

parametrized

(1972).

for

Chandler,

sphere

diameter

and is

The

equation

These

seem

be

smooth

obtained

for

data

The

0.92

and

indicating

that

their

Henderson

integral,

by

Verlet

liquids

is

theory

integral

but

of

to d on

at

obtained on

functions

of

methane

nob1 e gases.

In

an

diameters

(Sticking

dependence

*

sphere

experimental

parametrization.

simple

and an

which and

that

the

d

Weis

of

hard

depends

now

on

density.

to

consider

by

by

potential of

(1971).

again

Barker

theory

theory

Andersen

hard

to

repulsions.

Lennard-Jones best

given

and

effective

PHSP

of perturbation

the the

temperature

T/ Tc

their

Presumably

Weeks,

both

steepness

in

a

PVT

simple

T/T,

>

0.90

then

slightly for

we

from

minimum

employed

do

not

that

for

at

temperature

model

simple

dependence

differs

with

the

1.

to

picture

shallow

increase methane

Table

temperature

somewhat

d exhibit

even

d

the

in

amenable

physical The

fitting

shown

temperature

density).

calculated

by

are

again, may

thus

be

oversimplified. Influenced models

we

d

+

=

do

For

all

by

substances

The

b parameter

for

methane

same.

diameters of

does

the not Even

the

varies

depth

properties

are

proved

different though

of

all

a

was

in

by

to

potential

1964)

in

range

at from

same

in

a

molecule

the

to

form

very by

our

calculations

yielding approaching

for as

krypton that

per

to

in

to

d,

may

hardly than

be the

cent.

10.4

observed

correspond

-

c/k

o/k

almost Tc:

depth a

15

E/k

the as

potentials It

to

this

the

the

size

interpreted. hard-core

we same tends

the

obtained

at

that from each

more

fit.

this

has

sampled temperature

wide

range,

However,

frequently

gases

macroscopic

and

relatively

overall

well,

effective noble

parameter starting

a

potential

for

indicates

well:

covering

the

of

quantity

K.

as of

of

8.5

range

and

2

can

The

sensitive

guesses

estimates

about

do

pair

10

very

being

larger

unique

accurate

not

Tc

model.

as

about

was

temperatures

Concerning

while

well. be

(20)

lower

1981).

somewhat

Kihara

Eqn

at

(Nezbeda,

parameters,

always

by

good

error

the

potential

initial final

temperature

hard-core

correlation very

the

nearly

other

seem

diameter.

level

simple (Rowlinson,

experiments

two

do

used

Depth

been

found

is

core

that

the was

T % T,,

was

the

hard

We found

E/k,

other

result

It

scattering of

the

at

which

high-energy meaning

and

examined

the

reasonable

of

Kihara

Rowlinson’s

d, (l/T)l’b

qualitatively still

the

applied

to

with attain

241

higher

values

shown

in

The same

depth

but

of

for

an

e/k

=

the of

for

independent

of

that

for

more

Tc

+

E/k

values

0.4

with

T,

the

[(T

of

e/k

- Tt)/(Tc

steepness

of

fluids

useful

is

Tc

to

may

Tt)lK

are

versus

the

to

the et

relate

al.,

c/k

to

accounting

possess

a

form

I

(21)

T.

subcritical

examined

related

parametrization

-

E/k

for

be

(Hirschfelder

approaching

calculated

four

more Simple

T

may

potential

purposes

E/k

values

the

potential

realistic

constants.

in

K defines

sets

square-well

a

material

0.5

Representative

range.

practical

increase

where

the

1.

parameter

1964),

to

within

Table

Based

on

regions

parameter

the

entire

we

have

be

set

K can

found equal

3. Range

of

attraction of

more

attraction. range

of

realistic

1964),

but

Values always

potential

otherwise of

with

level,

parameters.

There

Ro/d

= A1

gives

a

the and

sum

so

have

A2/(E/k)

+

A3(E/k)

parameters

side

that

or

R,/d

of

et

obvious.

equation even

R,/d

are at

the

the

two

and

E/k

interpretation,

the

al.,

same

(like

but

temperature relation (22)

only

correlation

holds

considered.

is

not

the

parameters

between

stronger

that

relating

other

depth,

physical

found

which

provides

is PHS2

correlation

temperature-independent

two

substances

right-hand

a

weaker

we

R,

the

combinations

offering

exhibit

of

to

Hirschfelder

potential a

rules

R,, e.g.

fitting

so

However,

strictly

between range

+

from

algebraic

11)

combinations

dependence.

(see

interpretation

indicating

-

semi-empirical

potential,

increasing

are

e/k[(Ro/d)2

are

models

obtained

decreasing

these

an

R,/d

temperature

e.g.

There

square-well

In a

minor

approximately

over

Eqn

the

(22)

the

contribution

proportional

equation

entire last to

directly

temperature term the

on

the

total

to

(c/k)-l.

Second

virial

Though rather

it

coefficient is

known

insensitive

provide

a

EOS.

Eqn

B(T)

=

useful (8)

(2/3)

the

source

gives w

that

to

d

3

B(T) {l

-

the

second

actual

form

of

information

in

a

virial

coefficient,

of

pair

the

about

the

potential, parameters

B(T),

is

it of

may the

form

[(Ro/d)3

-

1]

(1

+

0.5

T")/T")

(23)

248 We

have

Eqn

found

(23)

provides d

is

to

reasonable from

to

A

question properties and

(22).

The

results

for

used

We

have

temperatures,

d,

E/k,

values same

the

ability

and

of

manner of

R,/d

B(T).

If

as

that

Eqn

(23)

unchanged. to

parameters

predict

calculated evaluated

ranged

definite

low

the

potential

be

parameters

have No

argon.

in

(20)),

whether

versa.

very

experimental

(Eqn

may

the

of

practically

is,

vice

using

for

parameters

temperature

remains

liquid by

with

properties

B(T)

properties

fair

vary

typical

equation

exception

independent

correlation

liquid

correlate

from

an

temperature

allowed

found

with

that,

with

from

density

B(T)

from

from

excellent

conclusion

Eqns

for

can

evaluated

low

the

(20)

methane

therefore

PHS2

through to

be

only

drawn.

CONCLUSIONS We

have

investigated

simplified We

fluid.

have

equation

by

and

sub-

both

been and

the

case.

~z(~~) term

(Eqns with

to

be

to

bulk

paid

The a

some

to

be and

way

to

of

for

a

the

critical

and

on

square-well point,

an

main

conclusions

l/T*

has

not

least

the

second-order

in At

agreement good

is EOS

to

(13))

should

may

the

itself,

or

Anyhow,

the

been

be

been

of

PHS2

vicinity

of

any

simple

improved

by

this of

the

was

that

of

attention

non-additive

not

term

incompatibility

feel

contribution

The

the

However,

the

to term

but

inadequacy

we

able

required.

feature have

effects.

either

used.

is

everywhere

a common

three-body

estimating

results hard

are

extremely

sphere

parameters

have

quite

due

approach

of

we

order

the

the

function

priori

The

data.

which of

smooth

a

the

based

the

this

ought forces

properties.

effective is

may (12)

the

be

point,

account

This

EOS of

properties

isotherms

quantitative

to

performance

into

first PVT

if out

critical

taking

the

introduced

The

general

parameters.

two-phase

turned

the

EOS.

sphere

theory

densities,

super-critical

of

the

be

for

orthobaric

the

hard

perturbation

follows.

equation

equation of

of as

represent must

perturbed

searching

examining

interpretation summarized The

the

Barker-Henderson

the from

the

considered improve

the

sensitive

diameter, and of

structure the

it

d.

seems

the

that

function

and might

the

the d

d of

to

consider

of

the

dependence

itself

= d(T)

dependence be

value

temperature

properties

temperature

results

to

The

may

of

or

at

least

be

determined

molecules

alone.

only,

possible

one

density

As

dependence

d

249

This

as well.

would

partly

remove

perturbation

terms

but

the

d on

could

be

simply

to

the

density

Comparing significant actual R,/d

value. and

e/k,

two-parameter somewhat

For

to

the

found

the

EOS

equations The

properties to

the

that

point

calculations (simply

the

PHS2 and

consider along

the

reported LIST ; & N ; RO

the

OF

Ro/d

to

play to

the

to but

to the

is able

two-phase

still

find

and Eqns

(20)

equation

(as

given

by

the

ability

has

simple

of

been

the

data

of

above, the

two

the

PVT

theoretical

outlined course.

We

directions

at

with

two

the

basis

temperature,

T'

= T/(E/k)

to of

describe

the

fact

clearly

investigations

results

in variable

function temperature

and

Daubert

encourages

the

SYMBOLS

error

and

assumptions the

PHSE

Hence,

hard sohere diameter Helmhoitz free energy radial distribution function Eoltzmann constant number of molecules pressure center-to-center distance attraction range of potential experimental

the

equations

same.

continue and

temperatures

(22))

Graboski

on

accurately

Detailed

through

the

derived

promising.

above

reduced

1.5.

PVT

describe

methane

is practically

in due

TX

of

reasonably

important.

by

approach

objective absolute

al.

value

of

semiquantitatively

we

argon

that

to

region

on

surface

rather

estimated

two-

reported

performance

a

exhibits et

liquid

those

a

its

parameters

strictly

and

parametrized

equation

sX S T

such

be essentially equation

set

to

the

equation

the

and

critical

revealed

defined

PHSZ of

the

PVT

of

same.

Redlich-Kwong-Soave (1978))

dependence

Dzialoszynski

orthobaric

region

temperatures

subcritical

with

seem

PHS2

(1976).

inferior

supercritical

that

comparative equation

the

insensitive

out

a similar

Renon of

not

the

fact,

than

and

results

inclusive

lower

close

the

between

turns

this

correlation

the

from

how

rather

coupling

PR equation

obtained

is

fact

Ponce

the

simultaneous in the

Despite

does

are

investigated

flexibility

examined

whereas

at

Due

of

remains

parameter

results

EOS

they

E/k

complexity

parametrized.

the

EOS.

methane

question

and

greater

-parameter (1980)

d,

role

the

X

will

be

us

to

260

packing fraction, compressibility Greek Y x Y Ax i P x P

letters three-body

y = nd3P/6 factor

interaction

force

constant

reduced force constant, yx = y/(cd') or deviation in contribution, change, depth of the potential well reduced range of attraction, X = R,/d number density, p = N/V 3 reduced density, px = pd

quantity

X

Subscripts C critical point hs hard spheres t triple point Superscripts experimental value e t true value three-body interactions (3b) Abbreviations EOS equation of state PHSl perturbed hard sphere (equation), PHS2 perturbed hard sphere (equation), PR Ponce-Renon (equation) PVT - volume - temperature pressure rdf radial distribution function

first second

order order

REFERENCES Aim,

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