An equation of state that has been
derived for gases is the nonlinear isotherm regularity (NLIR EOS). The NLIR EOS may be written
Z and v are compressibility factor and molar volume,
temperature dependent parameters, which are given as:
parameters are determined by correlating
experimental data in the form of Eq. (1) and therefore the EOS is correlative type.
potential energy used in EOS is:
Where U is the total potential energy in N
molecules and I (?) is the coordination number.
this work, we used a modified Lennard–Jones potential (6, 3) for potential
energy with temperature dependent coefficients which is:
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
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
into Eq. (6)
by rearranging the Eq.
(6) we have
have assumed that
have linear relation with temperature:
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:
is usually called the thermal pressure and
is called internal pressures (Pi). So
by differentiating from Eq. (10) relative to v,
divide both sides of Eq. (12) by T2
at constant ? we have
substituting Eq. (14) in Eq. (18) and integrating the following relation
have chosen f (?) as the following function by considering the
ideal gas limit and the density dependence of Eq. (18):
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
compressibility factor Z is obtained by dividing Eq. (21) to ?R:
M´ (T) and M (T) as:
dividing Eq. (25) to ?2 the EOS is obtained
To use the equation of state for a gas, the
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
can be found, respectively.