# An on the material and is only indirectly

An equation of state that has been
derived for gases is the nonlinear isotherm regularity (NLIR EOS). The NLIR EOS may be written
as:

where
Z and v are compressibility factor and molar volume,
respectively.

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,

and

are
temperature dependent parameters, which are given as:

The

,

and

parameters are determined by correlating
experimental data in the form of Eq. (1) and therefore the EOS is correlative type.

The
potential energy used in EOS is:

Where U is the total potential energy in N
molecules and I (?) is the coordination number.

In
this work, we used a modified Lennard–Jones potential (6, 3) for potential
energy with temperature dependent coefficients which is:

Where
N is the number of molecules and Ci are coefficients of potential
energy which are assumed to be temperature dependent. I (?) is the
coordination number which is supposed independent of density and r is
distance among molecules.

Physically, term proportional to

can
be justified in potential energy because a repulsive interactions, is expected
to be present for molecules of all types. Distance among molecules in gases is
high and therefore long range interactions are expected to be dominant and the

term would help to account for these
interactions. The same reasoning is also applicable to molecules with permanent
dipoles. The importance of this term might mainly depend on the balance between
repulsive and attractive forces, which varies depending on the material and is
only indirectly related to the interaction range in a particular substance.

Following Parsafar and Mason we assume that

Substituting

into Eq. (6)

by rearranging the Eq.
(6) we have

We
have assumed that

and

have linear relation with temperature:

where

,

,

, and

are constants. By inserting these relations
to Eq. (7) we obtain

For obtaining an EOS we begin with the following exact
thermodynamic relation like the method used for NLIR EOS derivation in 9:

In
which

is usually called the thermal pressure and

is called internal pressures (Pi). So

and

by differentiating from Eq. (10) relative to v,
we obtain

And
divide both sides of Eq. (12) by T2

at constant ? we have

using

the
result is,

by
substituting Eq. (14) in Eq. (18) and integrating the following relation
is obtained:

We
have chosen f (?) as the following function by considering the
ideal gas limit and the density dependence of Eq. (18):

on purpose.
Experimental data (by fitting) shows this choice works well and the parameters
?, ?, ? are m’o R, mo R and R respectively
(m’o and mo are constants).

This choice was used for simplicity and
it gives a final equation that works well with experimental data.
Inserting Eq. (20) into Eq. (19) and rearranging
gives:

The
compressibility factor Z is obtained by dividing Eq. (21) to ?R:

Therefore,

Considering
M´ (T) and M (T) as:

and
dividing Eq. (25) to ?2 the EOS is obtained

To use the equation of state for a gas, the

and

parameters must be known. To
find these parameters, we may plot

against 1/? for different isotherms.
The slope and intercept of the straight lines can be fitted
to Eqs. (24) and (25) from which

to

and

to

can be found, respectively.