Behaviour of the equation of state of liquid methane at high pressures

Materials

Chc,mistrJ, and Physics.

BEHAVIOUR ~ PRESSURES

OF

J. AMOROS,

THE

J.R.

18 (1987)

EQUATION

SOLANA

401 -408

OF

STATE

May

OF

LIQUID

METHANE

AT

HIGH

and E. VILLAR

Fundamental Physics Department. of Cantabria, Santander (Spain) Received

401

12, 1987;

Faculty

accepted

June

of

Sciences,

University

15, 1987

ABSTRACT The equation of state of liquid methane is analyzed at high pressures along a series of isotherms which cover a substantial part of the range of existence of the liquid phase. It is found that experimental data are well reproduced by means of an equation of state which was initially proposed for the high density hard-sphere system, provided that the effective excluded volume is properly determined for each isotherm. These effective excluded volumes decrease slightly as the temperature increases.

INTRODUCTION The

perturbation

ries of liquids. the as

repulsive a

can ries

be

considered

are

are based are

of

known

as

among

the

on the fact

dominant

them.

of the well

and

These van

attractive

are

der Waals'

a particular

case

not

theory

of

successful

that at high

the

ideas

more

theo-

densities forces

new:

they

which,

act form

in fact,

the perturbation

theo-

flj. In order

ve forces, system (and

They

forces

perturbation

the basis

theories

to take

into account

the hard-sphere

[21

for

sometimes

real

fluid

liquids,

density

the contribution has proven

as

dependent)

long hard

as

a

of the repulsi-

to be a good

reference

temperature

dependent

core

diameter

is properly

determined. With

these

facts

we havesuccessfullyused for

the

high

density

in

mind,

in

an equation hard-sphere

several

previous

papers

of state,

initially

to

correlate

system,

r3,41

developed the

high

0 Elsevier S~quoia/Printed in The Netherlands

402

pressure

experimental

Kr and

Ar, part

of

All

these

spherical to

liquids

when

les

are

liquids

frequently

THEORETICAL

FOUNDATIONS

starting [7], from

retically

by

In

way

point

the

as

al.

monoatomic

work

phase.

and

molecules we

[5,6]

thus

moredifficult with

initiate

of methane,

spherical

whose

and

a

the

molecu-

thus

it can

liquids.

of state

obtained

of the free energy for

liquid

to become

present

liquids

a substantial

the

of

to simple

[8]

of

composed

is the equation

et

covering

are

the case

the expression

Rudd

which

liquids

with

of the simple

existence

is expected

considered

in a similar

et al.

to

data

isotherms

of

molecules

structure.

be treated

Our

of

The model

applied

of molecular

of state

range

have

complicated

study

a number

temperature

in shape.

handle

more

over

Xe,

the

equation

the

by Alder

developed

hard-sphere

solid,

theoin

the

form:

PV/NkT

= 3/a + Co + Cl'y + C2s2

where

and

density

cO

It

to

the

have

other the

being

diameter. density

obtained

the regular

of

results,

packing

is the high

expansion

coefficients

dynamics

close

(1)

Expansion

virial

the

the molecular

= 3/a + 2,566

the

low

+...

Young

r91. eqn.

(1)

3 + 0.5501 - 1.19ar2 + 5.95ru + *..

hand,

equation

the of

self-consistent

stste

of

the

from

obtaining:

free

(2)

volume

hard-sphere

solid

theory in

the

1111:

PV/NkT

the

hard-sphere

of

[lo]

expresses form

the

by fitting

c3

On

o

analog

Alder to

PV/NkT

vO

3 3

c1 = (V - Vo)/Vo, V. = No3/fi

volume

and

+ca

is

= [l - (Vo/V)1'3]-1

has

been

replaced

hard-sphere reproduce

the amorphous

shown by

the

[12]

solid

state,

accurately hard

if

that,

corresponding

amorphous

very

(3)

the

spheres

Vz

the

regular

maximun

= 1.129Vo,

molecular equation

close

packing eqn.

dynamics of state,

packing

volume

for

(3) is able results

obtained

for when

403

the metastable With

ched.

this

[3]

value

that

Woodcock's 3

>

Moreover,

(2)

and

supplies

results

this

only

PV/NkT

= 3/a

in

neighbourhood

densities,

the

+ 2.566,

agreement

the

first

one of

addition

of more

with

amorphous even

when

-< ps3 < 1.085). eqn. (2), i. e.,

(0.943

terms

of

the same

point

terms

agreement

is satisfactory

nearly

melting

of

in a previous

hard-sphere

region

two

obtains

the

the

quen-

- 1, instead

shown

excellent

for

fluid

is rapidly

ti = V/V:

we have

solid,

to the metastable taking

no3 = 1.085

of

and putting

also

simulation

1.085)

extrapolated

the

of Vz

eqn.

[12]

(00

solid

at a density

- 1 as in the regular

ci = VjVo work

fluid

of eqn.

results

= 0.943.

po3

except

At

(2) will

lower

obviously

be necessary. In real

dense

liquids

forces.

The

of state of

the attractive

[13], which

on the structure

contribution

of

the

can be approximated

state,

as

long

as

we

by means

determine

previously. to attenuate

the

through

contribution

and

density thus

tion

it seems

in l/al may

the

system.

liquids

Ar, Kr and Xe

PV/NkT

for

the

a

to lower

in fact,

effective

of

of

the

internal

the

than

in the case

equation diameter

of

state

pressure,

linear

densities

equation

attractive

the equation

that

true

[4] for which

approxima-

in the hardof the

simple

the relationship:

liquid

phase,

due

to

pointed

out

[41

term

(4)

substantial

pendent

the

was,

to the

presence

of

effect

= 3/01 t Co

holds in

This

the

variation

to expect

be extended

-sphere

forces

suitable

In addition,

reasonable

little

by the repulsive

of the hard-spheres

a

forces with

have

mainly

repulsive

as mentioned tends

forces

is determined

Co,

the

and

of

range

Co was

although influence

of

formal

analogy

the the

densities found

attractive

along to be

each

temperature

forces.

between

Batschinski-Hildebrand

isotherm

eqn.

We

(4),

[14,151

have

dealso

except

for

equation

for

the viscosity:

n = AVB/(V

This

seems

transport of

- VB)

to

lend

properties

state,

[161,

(5)

within

in

the

support follow

same

experimental

way

to

the

idea

an expansion as

they

uncertainty

that

at

similar

follow

the

high

densities

to the equation virial

expansion

[17], at low densities.

404

RESULTS

AND DISCUSSION

Taking

into

be

considered

in

the

methane.

as

squares the of

fitting

for

than

different

The

99.9%.

The

studied

The effective

their

+ bTi)/(l

for

ordinate

were

data

as

in

origin

for

all way

the

can be fitted

by an equation

the

selected

in this

slightly

from

function

the approxima-

was,

obtained

decrease

a least

a

at

and

isotherm

obtained

[181, as

us to determine

of Vz

of Vz

through

values,

the data

pressure

For each

and error,

can

as mentioned high

the values

coefficient

volume

uncertainty,

shape,

pressure

which

values

in

follows.

the

molecule

hold

literature

correlation

isotherms

increases.

the

obtained:

from

methane

obtaining

test enabled

linearity

fit.

also

is as

high

from

3 was

the

by trial

the PV/NkT taken

slope

of

definitive

vz = (a

will

temperature

A previous

0’

te region

of

that

spherical

(4)

procedure

of

data

l/o! until

is then C

within

fact

of Vz was determined

smoothed

better

eqn.

The

a function

the value

the

to be essentially

introduction,

liquid cO

account

the

cases, for

the

temperature

to temperature,

of the form

[19]:

(6)

+ Ti)

where

T = T/Tc, with Tc being the critical temperature for methaR This equation has the convenient feature of having limiting

ne.

values

of

Vz

for

R in eqn.

the constants 3 -1 cm .mol . Figure methane

1 shows

= 0 and

which

noted

that

drawn

in

starting

cover

the

1, holds the

be

in

effective

not

liquids

due

the

Kr

change

excluded for.

and

This [4]

probably

and

mainly

forces

in

at

pressures,

low

equilibrium

these

line,

of

behaviour

Xe

repulsive

liquids in there

a

of

It

is

due

not

range

discrepanand

thus

[201, which observed

partially

be

discrepancy

the potential,

to

in

the

to the greater hardness as compared

to

(4). also

This

pressure was

liquid consi-

an extensive slight

for

b = 21.86

state

range.

over for

with

with

we

-liquid

in shape

obtained

isotherms

high pressures.

volume,

pressures

Moreover.

dealt

except

we

by expression

isotherm

at very

of

liquid

predicted

line,

values

cm3 .mol -1 and

the different

whole

for each

isotherms

accounted Ar,

the

melting

to

The

equation

along

behaviour

Fig.

cy may the

nearly

from

= m.

a = 27.02

experimental

of l/a

linear

in the two upper

TR

(6) are

the

as a function

dered,

have

T

to liquid the

we the

lower of

the

methane.

the

neighbourhood

of

is

a progressive

departure

vapourfrom

.

_

l -

406 linearity the

fact

thus of lines

of

the

that

as

isotherm

temperature

cy, covered

increases,

when

c1 <<

sure

region perhaps

may

depend

it would

not

only of

and density

the context The

on

the

range

the

is actually considered,

range

(4)

of

is

fact

to add that

temperature

strictly

hard

core

considered

more

also

and

and vapour valid

terms

of

only

volume

eqn.

excluded

on density.

by other

to

in the low pres-

effective

but

due

densities,

the melting

the predictions

neccesary

of perturbation

independent,

tractive

be

the effective

at

the

temperature,

between

approximation

has been

ordinate

rature

rises

to improve

consider

the dependency ture

and

to

increasing

by the isotherm

1. In order

and

with

In fact,

on both

authors

(1)

volume

tempera-

[21] within

theories.

origin

Co,

temperature is

which

in eqn.

dependent,

negative

due

to

(4)

and, the

is density

in the tempe-

presence

of

at-

forces.

-2

-4

co -6

-8‘

-1.0

Fig.2. reduced

1.2

Variation

of parameter

temperature

In Fig.2

it can be seen

ge considered,the

Co in eqn.

for liquid

values

of

(4) as a function

of the

methane.

that, Co

as

in the limited a

function

of

temperature the

inverse

ranof

407

the

reduced

temperature,

fall

practically

on

a

straight

line

of

equation:

=

cO

2.342

- 5.060/TR

If we assume

that

(7)

this behaviour

dered,

the

limit,

is reasonably

value

Co

to a hard-spheres

=

2.342

close

system

continues

at

to

TR

the

=

the range

i.e.,

the

m,

value

in agreement

beyond

Co

with

consi-

hard-spheres

= 2.566

corresponding

expression

(2).

CONCLUSIONS The eqn. of

results

(4) which

the

is

which

-sphere It hold Work

value is

for

the

work

allow

to be useful

in the case in

As

volume

in

limit

conclude

the behaviour

methane

the

case

whereas

(T

whose of

[4],

mole-

the

above

is a slightly

Co is a increasing

approaches

+ m)

that

Ar, Kr and Xe

Vz of methane

of the temperature,

hard-sphere

to

liquids

of liquid

shape.

the excluded

us

in predicting

of the simple

spherical

function in

present

state

satisfactory

liquids,

decreasing one

of

nearly

mentioned

the

was proven

equation

is equally cule

of

the

hard-

behaviour

will

Co = 2.566.

to

be

expected

that

liquids

composed

of

on this direction

the

same

molecules

is currently

general

non-spherical

in

shape.

in progress.

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