Parameter Margules Model: 1484154

Summary of methods :
1 – parameter Margules model :

The Margules model for activity is a type of thermodynamic model which is used for the excess Gibbs free energy in case of a liquid mixture. It is also called the Margules activity or the activity coefficient model. The model is used for deriving the expression for the activity coefficients for a compound ‘i’ in a liquid using the concept of the activity coefficient. It gives a measurement of the deviation from the ideal solubility and referred to as the Raoult’s law. It is very old model but it can provide a description of the extreme point for the activity coefficient using a characteristic feature. The latest models such as Wilson and NRTL are not capable of describing these.
GE/RT plots :
The Table 1 shows the variation of the value of GE/RT w.r.t. ‘x1’.
Table 1
x1 GE/RT
0 0
0.1 0.01
0.2 0.04
0.3 0.09
0.4 0.16
0.5 0.25
0.6 0.36
0.7 0.49
0.8 0.64
0.9 0.81
1 1

Figure 1
The Figure 1 shows that the value of GE/RT rises w.r.t. x1 in a quadratic manner.

2 – parameter Margules model :
The generation of the P-x-y and the T-x-y diagrams is done for 1 mole of the binary mixture in the vapor – liquid equilibrium or VLE. The P-x-y diagram is given at a particular temperature. The T-x-y is given at a particular pressure. The liquid phase boundary or the bubble point and the vapor-phase boundary or the due point can be determined from the P-x-y and the T-x-y diagrams. The amount in moles for the liquid and the vapor in equilibrium can eb displayed on the bar chart. The lever rule can be used for the calculation of the mole fraction for a component in each phase for the liquid (denoted by x1) and for the vapor (denoted by y1) and the relative amounts. The 2 – parameter Margules model is used for modelling the non – ideal liquid mixture. In case of an attractive interaction between the 2 components, the attractive interaction is strong as compared to the average value of the interactions for the pure components. This shows a negative deviation from the Raoult’s law. In case of a repulsive interaction between the 2 components, the attractive interaction is weak as compared to the average value of the interactions for the pure components. This shows a positive deviation from the Raoult’s law. The degree of interaction can be changed by the help of the sliders by altering the value of the Margules parameters – A12 and A21 used for the calculation of the activity coefficients. The liquid is ideal as per the Raoult’s law if the Margules parameters are equal to zero and the activity coefficients are equal to 1. The system consists of an azeotrope if the value of the activity coefficients shows significant deviation from the value of ‘1’.

The Table 2 shows the variation of the value of GE/RT w.r.t. ‘x1’.

Table 2
x1 GE/RT
0 0
0.1 0.11
0.2 0.24
0.3 0.39
0.4 0.56
0.5 0.75
0.6 0.96
0.7 1.19
0.8 1.44
0.9 1.71
1 2

Figure 2
The Figure 2 shows that the value of GE/RT rises w.r.t. x1.

Van Laar model :
The Van Laar model is a thermodynamic activity model. It is used for describing the phase equilibria of the liquid mixtures. The Van der Waals equation is used for deriving the equation. The van der Waals parameters obtained originally were not enough to describe the vapor – liquid equilibria for the phases. Hence, it was found appropriate to fit the parameters to the results of the experiment. Due to this reason, the model does not have a connection with the molecular properties. Hence, for obtaining a correlation with the experimental results, it must be considered as an empirical model.

The Table 3 shows the variation of the value of GE/RT w.r.t. ‘x1’ and ‘y1’.

Table 3
x1 .y1 GE/RT
0 1 0
0.1 0.9 0.09
0.2 0.8 0.16
0.3 0.7 0.21
0.4 0.6 0.24
0.5 0.5 0.25
0.6 0.4 0.24
0.7 0.3 0.21
0.8 0.2 0.16
0.9 0.1 0.09
1 0 0

Figure 3
The Figure 3 shows that the value of GE/RT rises first w.r.t. x1 and then after reaching the peak point of 0.25, it falls again to a value of ‘0’.

NRTL model :
The NRTL (non random two liquid) model is the activity coefficient model. It is used for correlating the activity coefficients for the compound having the mole fractions in the liquid phase under consideration. It is used in the chemical engineering field for the calculation of the phase equilibria. The hypothesis on which the NRTL model is based is that of Wilson. It states that the local concentration around a molecule is not the same as that of the bulk concentration. There is a difference because of the difference in the central molecule’s interaction energy with the similar type of molecules and different type of molecules. For the local molecular level, a non – random behavior is introduced due to the difference in energy. 1
The NRTL model is a type of local composition model is a type of local composition model. The Wilson model and the UNIQUAC model are also of similar type. Such local composition model does not show thermodynamic consistency for single fluid model in case of a real mixture. This is because it is assumed that the local composition around a molecule ‘i’ is not dependent on the local composition around a molecule ‘j’. But Flemr showed that this assumption is false. There is a consistency shown in case of a 2 liquid hypothetical model.

The Table 4 shows the variation of the value of GE/RT w.r.t. ‘x1’ and ‘y1’.

Table 4
x1 .y1 GE/RT
0 1 0
0.1 0.9 0.18
0.2 0.8 0.32
0.3 0.7 0.42
0.4 0.6 0.48
0.5 0.5 0.5
0.6 0.4 0.48
0.7 0.3 0.42
0.8 0.2 0.32
0.9 0.1 0.18
1 0 0

Figure 4
The Figure 4 shows that the value of GE/RT rises first w.r.t. x1 and then after reaching the peak point of 0.5, it falls again to a value of ‘0’.
UNIQUAC model :
It is a model which is used for describing the phase equilibria and based on activity coefficient. It is a lattice model and is derived in the statistical thermodynamics using the 1st order approximation for the molecule surfaces which interact.
This model helps the chemical engineers in the prediction of the phase behavior for the chemical mixture having multiple components. It is used in the programs for process simulation for the calculation of the mass balance for the separation units. It employs the functional groups which are there on the molecules which constitute he liquid mixture for the calculation of the activity coefficients. It is a semi empirical system which is used to predict the non – electrolyte activity for the non – ideal mixture.

Discussion :

Data set :
The Table 5 shows the variation of the value of P w.r.t. ‘x1’ and ‘y1’.

Table 5
P[mmHg] x1 y1
780.52 0.0022 0.0192
827.64 0.011 0.086
931 0.035 0.191
1003.2 0.053 0.245
1235.76 0.121 0.434
1535.96 0.281 0.619
1624.12 0.352 0.662
1882.52 0.522 0.75
2115.08 0.667 0.824
2337.76 0.826 0.911
2508 0.932 0.969
2528.52 0.958 0.981

The above data is the experimental data for the binary mixture containing methanol (1) and water (2). This data is measured at 100 degree Celsius.
The Antoine constants are :
Table 6
A B C [ degree Celsius ]
Methanol 7.97007 1521.23 233.97
water 8.01767 1715.7 234.268

Antoine equation :

Here, p = vapor pressure, T = temperature and A, B and C are constants which are component-specific.
For methanol, log p = 7.97007 – (1521.23 / (233.97+T))
For water, log p = 8.01767 – (1715.7 / (234.268+T))
Interpretation of data :

P-x-y plots :

Figure 5

Figure 6
The Figure 5 shows that there is an increase in the value of P w.r.t. ‘x1’. The Figure 6 shows that there is an increase in the value of P w.r.t. ‘y1’. The rise in the value of P w.r.t. ‘y1’ is very steep as compared to ‘x1’.