CHAPTER THREE: THEORETICAL CONSIDERATIONS
Chapter overview
The theory and equations used to compute infinite dilution
activity coefficients by two selected methods are presented. Where necessary,
enough details on derivation procedures are provided. Assumptions leading to
the derivation of these mathematical relationships are highlighted. They
determine the reliability of the experimentally generated data. The modeling of
mass transfer in the dilutor cell, as presented in the literature, is reviewed.
It represents an important basis on which the dilutor cell constructed for this
work was designed.
3.1. Gas liquid chromatography
Gas liquid chromatography is a well established method for
determining infinite dilution activity coefficient of volatile solutes in
nonvolatile solvents (Letcher 1980). It is logically used
C L
in this study of ionic liquids which are generally known as high
boiling solvents. Equations
K C M
involved in IDAC`s computations, discussed by Letcher (1980)
as well as Condor and Young (1979) and used in this work are given below. As
the solute (1) moves through the column, it is assumed that:
· The mobile or gas phase is an ideal mixture composed of
the carrier gas (2) and the solute (1);
· The stationary gas is a real liquid mixture, containing
the solute and the solvent (3);
· The gas phase (mobile phase denoted as M) and the liquid
phase (stationary phase, denoted as L, i.e. ionic liquid) are in thermodynamic
equilibrium, well described by the modified Raoult`s law.
The distribution coefficient, of the solute between the two
phases, according to Condor and
n
Young (1979), is:
(3-1)
Concentration of the solute in the liquid and gas phases are
given by
(3-2)
and, (3-3)
where and are the mole fractions of the solute in the liquid
and gas phases respectively,
is the number of moles of the carrier gas in the mobile phase,
is the number of moles of the
solvent in the stationary phase, is the volume of the gas phase
and
stationary phase. From equations (3-1) through (3-3), it can be
found that:
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is the volume of the
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(3-4)
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(3-5)
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and using modified Raoult`s law, i.e.
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(3-6)
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Equation (3-4) can be written in a simplified form as:
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(3-7)
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At mean column pressure (Laub and Peacsok 1978), the net
retention volume, , is related to the distribution coefficient and the volume
of the stationary phase by:
(3-8)
When this expression is used in equation (3-7), the following
equation determined by Porter et al. (1956) is obtained:
(3-9)
When the solute-solute and solute-carrier gas imperfections,
as well as the pressure drop along the column are considered, a more accurate
equation (Everett 1965, Cruickshank et al. 1966a and 1969) is derived:
(3-10)
where is the infinite dilution activity coefficient of a solute
(1) in a solvent (3), T is the
column temperature, is the column outlet pressure, the same as
the atmospheric pressure,
is the column inlet pressure, is the mean column pressure, , the
saturated pressure
vapour of the solute at temperature T, is the molar
volume of the solute, is the partial
molar volume at infinite dilution, equal to, in this particular
case. is the pressure
correction term given by
(3-11)
is the net retention volume given by
(3-12)
, the column outlet flow-rate, the retention time for the solute
and is the inert gas retention time.
The second virial coefficients of pure solutes were calculated
using the McGlashan and
Potter equation (1962).
(3-13)
The same equation allowed the calculation of mixed virial
coefficients
? ? ? ? ? ? . Other mixed
properties are determined by means of Hudson and McCourbey`s
(1960) mixing rules as follows:
(3-14)
(3-15)
(3-16)
(3-17) In the last four expressions, subscripts =1`, =2` and
=12` refer to the solute, the carrier gas and the mixture of these two
components, n is the number of carbon atoms and is equated to one
for
x f I y P
1 ? 1 = ?
1 1
compounds with no carbon atom. Critical properties and as well
as the ionization
energies, are found in the literature (CRC Handbook 2005).
Modified Rackett equation (Poling et al. 2001) gives the values of molar
volumes as follows:
(3-18) Saturated vapour pressures involved in equation (3-10)
were calculated using the Antoine, modified Antoine or Wagner equations in
compliance with the applicability range of each
x f oq I y P
3 ? 3 3 = ?
3 3 3
correlation. Constants for vapour pressure correlations which
are required in equation (3-10) as well as acentric factors were taken from the
literature (Poling et al. 2001).
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