THEORETICAL TREATMENT OF VAPOUR PRESSURE AND VAPOUR-LIQUID EQUILIBRIUM DATA
3.2 Correlation of Vapour Pressure data
3.2.2 Vapour Pressure correlations
The first family of equations for the correlation of vapour pressures, apart from the empirical relations, is based on an expression relating the equilibrium condition between the vapour and liquid phases of a pure component, known as the Clausius-Clapeyron equation:
(3.1)
where Pyap is the compound saturated vapour pressure, T is the temperature, Ris the universal gas constant and ~Hvand~Zvare the differences in the enthalpies and compressibility factors of the saturated vapour and liquid phases, respectively. A large number of vapour pressure correlation equations are based on an integration of Equation (3.1). The simplest exact solution to Equation (3.1) requires the assumption that the
~Hv
term is a constant and not dependentMZv
on temperature. Both ~Hv and~Zv decrease with increasing temperature and have weak temperature dependencies except near the critical point.
With a constant of integration i.e.A , Equation(3.1) becomes the following:
InPvap=A B
T (3.2)
h B ~Hv h' " .
were = - - - . T ISform of the equatIon IS sometImes known as the Clapeyron equation MZv
and is suitable for the representation of saturation pressures over small temperature intervals not exceeding 20 K (Malanowski and Anderko, 1992), even when ~Hvis temperature-dependent.
Over large temperature ranges, the equation represents vapour pressures quite poorly (Reid et al., 1987).
Antoine (1888a, 1888b) proposed a simple empirical modification of the Clapeyron equation in the form of a three-parameter equation, shown below in Equation (3.3).
InPvap
=
A - - -B T+C(3.3)
where A, Band C are constants and for C
=
0 , Equation (3.3) reverts to the Clapeyron equation.The Antoine equation is perhaps the most popular vapour pressure correlating equation for low- pressure VLE studies and Antoine constants have been tabulated for more than 8000 organic compounds (Malanowski and Anderko, 1992). The Antoine equation can be used to accurately correlate vapour pressures for non-associating organic compounds over the pressure range of 1- 200 kPa. When used for the representation of associating compound vapour pressures, the range of applicability i.e. pressure is reduced (for alkanols it is limited to 5 - 80 kPa).
The adjustable parameters or constants of the Antoine equation are empirical and the only reliable means of determining them is through the regression of experimental data (Reid et al., 1987) as advocated by Malanowski and Anderko (1992), who are guarded against the use of Antoine constants that are obtained by mathematical treatments. Each set of parameter values should be associated with temperature or pressure limits and the constants should ideally never be used outside these limits. Extrapolation outside the stated limits can sometimes produce erroneous results as a result of the Antoine equation often being incapable of reproducing the correct shape of the vapour pressure curve over the entire temperature range.
Other popular three-parameter empirical equations include those of Miller (4.4) and Cox (4.5).
The modified Miller equation was found by some researchers to provide a more accurate fit to experimental data than the Antoine equation (Reid et al., 1987).
(3.4)
I T Pvap . .
w 1ere Tr
= T
and P vapr= p
and PcandTcare the cntlcal pressures and temperatures,c c
respectively. The adjustable parameters in the Miller equation, shown above, areA, B and Pc'
Chapter 3. Theoretical Treatment of Vapour Pressure and Vapour-Liquid Equilibrium Data
The parameter Pc is an empirical parameter that exhibits large deviations from the real critical pressure and hence cannot be used for its prediction.
The Cox equation was found by Osborn and Scott (1978, 1979) to be capable of "adequately"
representing vapour pressures from the triple point pressure to 0.3 MPa and for extrapolations with "reasonable precision" over a 50 K range. This was in comparison with the Antoine equation that could not successfully be used by the authors for extrapolation outside the experimental range. The Cox equation was used in the following form by Chiricoet al. (1989):
(3.5)
where Prefand Trefrefer to a reference pressure and a reference temperature, respectively. For a reference substance, Pref and Tref represent an accurately known point on the vapour pressure curve of the substancee.g. the normal boiling point of the substance. The adjustable parameters, as for the Antoine equation, are A, B and C .
The three-parameter equations, discussed above, are generally not suitable for the representation of vapour pressures over a wide range of temperatures from the triple point to the critical point.
In an attempt to improve upon the use of empirical equations to represent vapour pressures, the number of adjustable parameters or constants in the equations was increased.
One of the most popular of these equations was the Frost-Kalkwarf equation (Malanowski and Anderko, 1992) or the Harlecher-Braun equation (Sandler, 1999). This relation, shown as Equation (3.6), has 4 adjustable parameters in the form of A, B, C and D.
In B Pvap
P
=
A + -+ClnT+D-yap T T2 (3.6)
The Frost-Kalkwarf equation was derived on the basis that the enthalpy of vapourization,~Hv, is a linear function of temperature and that the change in volume term between the vapour and liquid phases, ~Vv' can be estimated from the van der Waals equation of state. Due to the form of the equation, the vapour pressure must be solved for iteratively.
A modification of the Frost-Kalkwarf correlation was the form used by the DIPPR Compilation Project (Daubert and lones, 1990) as follows:
InPyap =A + -+ClnT+DTBT E (3.7)
where A, B, C and D. are adjustable parameters and E is assigned a value of either 2 or 6, depending on which value gives a better fit of the experimental data. Equation (3.7) with E = 6 is also known as the Riedel equation (Sandler, 1999). The fitting procedure for the equation involves either a constrained (at the critical point) or unconstrained (with a fit only over the experimental data range) fit. The constrained fit is necessitated due to requirement that the thermodynamically correct "slight S-shape" should be obtained for the vapour pressure curve between the critical and normal boiling points. This is achieved with positive values forAand D together with negative values for Band C . In an unconstrained fit, there is no assurance that the sign rule will hold, even with accurate experimental data. The above equation was used extensively by the DIPPR Compilation Project for the correlation of the vapour pressures of many industrially relevant chemical substances (Daubert andlones, 1990).
It has also been found to be advantageous to generalize the vapour pressure relations through the law of corresponding states i.e. through the use of reduced variables. An excellent example ofthis is one of the most highly recommended vapour pressure correlation equations in the form of the Wagner equation shown below:
Ax + Bx1.5+ Cx3+ Dx6
InPvapr
= - - - -
T
r (3.8)From Equation (3.8), A, B, C and D are the adjustable parameters and the independent variable i. e.x =
(1 -
Tr ) is a measure of the distance from the critical point. The above form is known as the "3 - 6" form of the Wagner equation since the base x associated with the coefficients C and D, is raised to the exponential powers of 3 and 6, respectively. An alternative form i.e.the "2.5 - 5" Wagner equation was employed by researchers such as Morgan and Kobayashi (1994) in fitting vapour pressure data. The Wagner equation is considered as being highly efficient (Malanowski and Anderko, 1992) and as being the only empirical equation capable of accurately representing experimental data with a few constants from Tr = 0 to the critical point.
However, one should be guarded against possible inconsistent thermodynamic behaviour with
Chapter 3. Theoretical Treatment of Vapour Pressure and Vapour-Liquid Equilibrium Data
regards to the shape of the vapour pressure curve, whilst giving an accurate fit to the experimental data.
Unlike the Antoine equation, the Wagner equation may be used to safely extrapolate outside the experimental data range due to the manner in which the constants have been determined.
However, the Wagner equation may not extrapolate well below reduced temperatures of 0.5.
Some researchers (Vetere, 1991) noted that the D parameter is important for obtaining excellent correlations with highly accurate experimental data. However, for data that is not reliable, the
D - term can in most cases be ignored, without greatly affecting the fitting of the data.
The equations presented above i.e. Equations (3.2) - (3.8) have in common that their adjustable parameters are all purely empirical, as the latter have no theoretical significance in terms of molecular parameters or interactions. One attempt to address this lack of physical significance of the parameters was by Abrams et al. (1974). They employed the use of the kinetic theory of polyatomic fluids developed by Moelwyn-Hughes (1961) to formulate a vapour pressure equation (AMP) with the same algebraic form as the empirical equation suggested by Miller (1964). The derivation of the AMP equation will not be presented here as it is beyond the scope ofthe author's work and the pertinent equations will merely be presented below as follows:
where
In P
=
A+ -B +C InT+DT+ET2 TA
=In[:" ]+-±H;] -In[{s-I)!J +Ina
B
=-~
R
C
=
--s3 2s-I D = [ ; ]
(3.9)
(3.1 0)
(3.11 )
(3.12)
(3.13 )
(3.14)
The two adjustable parameters of the AMP equation as shown in Equations (3.10) - (3.14) are s and Eo. The hard-core van der Waals volume
(Vw)
is obtained from the group-contribution correlation of Bondi (1968) and a. is a proportionality constant (independent of the nature of the liquid), and determined empirically to be equal to 0.0966. R is the universal gas constant (82.06 cm3.atm.morJ.KJ) such that ( ; ) is given in K, P is in atm and Tin K. Therefore, as opposed to the original form of the Miller equation (1964) with five adjustable parameters in the form of A, B, C, D and E, the AMP equation has only two adjustable parameters i.e. s and Eo. The parameter s is a function of the size, shape and flexibility of the molecule and was shown to increase with chain length and to decrease as the shape of the molecule became more globular or branched. The parameter Eo is an indication of the strength of the intermolecular forces;hence it was observed that Eo for a normal alcohol was larger than that for a normal alkane of the same chain length. When the Eo parameter was differentiated with respect to temperature, it was shown that the parameter corresponded with the enthalpy of vapourization of the hypothetical or model fluid at T=O.
A comparative study conducted by the authors with experimental values and those obtained from the Antoine correlation showed that excellent representation of the vapour pressures of pure liquids from 10 - 1500 mmHg could be achieved with the AMP equation. The authors also concluded that since molecular size and flexibility are taken into consideration in the kinetic theory, the equation should perform well for liquids with large molecules.
3.2.3 Modelling experimental Vapour Pressure data
There are essentially two correlation strategies for the fitting of the experimental vapour pressure data to vapour pressure equations i.e. a constrained and an unconstrained fitting approach. Malanowski and Anderko (1992) advocate that the constrained correlation approach should in theory produce results that are more accurate than those obtained by an unconstrained approach, since for the former relation the known or reliably predictable physical behaviour of the system is incorporated into the fitting procedure as a constraint. This would be particularly effective where the experimental data set is characterised by a great deal of uncertainty e.g.
when the investigated compound is prone to undergoing degradation at higher temperatures. It
Chapter 3. Theoretical Treatment of Vapour Pressure and Vapour-Liquid Equilibrium Data
IS imperative that the correlations obtained are both accurate (small residuals) and thermodynamically consistent in terms of the observed behaviour of the saturation curvei.e. the slight S- shape of the curve between the triple and critical points. As mentioned above, the
"constrained fit" approach was utilized by the DIPPR Compilation Project (Daubert and Jones, 1990) to ensure that the slight S-shape of the saturation was obtained for the DIPPR modified Frost- Kalkwarf correlating equation shown in Equation (3.7). The use of an unconstrained fit by the author was justified in this work as the conditions for the compounds investigated were always far from the critical point; hence no consideration of the proper shape of the saturation curve near critical conditions was necessitated.
With regards to the use of the correlating equations, the comparative studies illustrated in the works of Malanowski and Anderko (1992) and Reid et al. (1987) have shown that the Antoine equation is suitable for pressure measurements in the 5 - 200 kPa range for non-associating or weakly associating chemicals (for strongly polar compounds this range is reduced). The Wagner equation was considered as much more suitable for higher and wider pressure ranges i.e. for conditions nearing the critical (as Tr --+ 1).
The Antoine, Cox, Wagner, Frost-Kalkwarf and modified Frost-Kalkwarf vapour pressure correlations were chosen for the modelling of the experimental vapour pressures obtained in this work. Since the vapour pressures in this study were measured at low pressures, far away from the critical region, the use of the equation of state method for vapour pressure correlation was not justified. Also the use of the equation of state method, as discussed in Section (3.5.3.2), does not allow for an explicit solution of pressure i.e. it involves iterative calculations. The mathematical computational software, MATLAB® (version 7.0.1), was used to perform the fitting of the experimental data to a variety of correlation equations using the Nelder-Mead Simplex algorithm. A discussion of the functions and the features of the Optimization Toolbox are available from the on line Mathworks® resources.
The objective function used was simply the difference between the experimentally measured vapour pressure(pexp ) and that predicted(pealc) from the various vapour correlations presented above, as shown below:
n 2
OF
= I
(pexp _peale )i=1
Since it was a direct fit, no iterationalloops were involved.
(3.15)