This task is supposed to make you familiar with the generation of $$xy$$-diagrams that represent VLEs of binary mixtures under isobaric conditions.
We here consider a binary mixture of benzene (1) and p-Xylene (2). I chose benzene and p-xylene, as this binary mixture can be taken as an ideal mixture. This simplifies the equations that follow, as we can set all the parameters accounting for non-idealities to 1.
The table below shows vapour pressures of benzene and p-xylene as a function of $$T$$. I generated these vapour pressures using the Antoine equation with the Antoine parameters for benzene and p-xylene, which are provided by the NIST services (just in case you want to reproduce them). We are going to complete the table task by task.
TASK 1: x-axis of the $$xy$$-diagram
We know from the lecture that a non-hetero-azeotropic, liquid, binary and ideal mixture boils when
$$x_1 p_1^{sat}+x_2 p_2^{sat}=p_t$$
where we choose as total pressure $$p_t=0.10130MPa$$.
Use this equation for the computation of the compositions of the liquid mixtures that boil at a total pressure $$p_t=0.10130MPa$$ at the temperatures specified in the table. Write the values into the column $$x_1$$. You will find out that pure benzene at $$p_t=0.10130MPa$$ boils at $$T=353K$$ and that pure p-xylene at the same pressure boils at $$T=412K$$. Per definition, the high-boiler is compound 2 and the low boiler is compound 1. The mixtures of both pure compounds feature at the same pressure boiling temperatures that are in between the pure compound boilg temperatures.
TASK 2: y-axis of the $$xy$$-diagram
Alternative A
We know for a non-hetero-azeotropic, liquid, binary and ideal mixture the relation
$$y_i=x_i p_i^{sat}/p_t$$
Use this relation for the computation of the molar fraction of compound 1 in the vapour phase $$y_1$$ and add the missing values to the table above.
Alternative B
We know from the lecture that the vapour and liquid phase compositions can be correlated using the separation factor (relative volatility) $$α_{ik}$$ with this equation
$$y_i=α_{ik} x_i/(1+∑_{j=1}^{j=k-1}(α_{jk}-1) x_j)$$. For the binary mixture this equation becomes $$y_1=α_{12} x_1/(1+(α_{12}-1) x_1)$$ and we already know that for non-hetero-azeotropic, liquid, binary and ideal mixtures $$α_{12}=p_1^{sat}/p_2^{sat}$$ .
Compute first $$α_{ik}$$ and then $$y_i$$ (with the equation just above) and complete the respective columns in the table. Do not be surprised that the values für $$y_i$$ are identical for both Alternatives A & B.
TASK 3: Correlation of the vapour and liquid phase compositions
$$K_i$$-correlation
We know from the lecture that the vapour and liquid phase compositions are correlated via the partition coefficient
$$K_i=y_i/x_i=$$ and we know for non-hetero-azeotropic, liquid, binary and ideal mixtures that $$y_i=x_i p_i^{sat}/p_t$$. From this we deduce that for non-hetero-azeotropic, liquid, binary and ideal mixtures
$$K_i=p_i^{sat}/p_t$$. Compute $$K_i$$ and insert the missing values in the table above. You will see that $$K_i$$ strongly depends on temperature and on the mixture composition (min valiue 1 and max value 4.36->Factor of 4.36 in between min and max value).
$$α_{12}$$ correlation
In Alternative B we aready made use of a correlation equation $$y_1=α_{12} x_1/(1+(α_{12}-1) x_1)$$ with the relative volatility $$α_{12}=p_1^{sat}/p_2^{sat}$$. The values of $$α_{12}$$ varied only between a max value of 6.52 and a min value of 4.55. Thus, the largest value is only 1.43 the smallest value. Remember that the maximum partition coefficient $$K_i$$ was 4.36 times the smallest value.
TASK 4: Quick and dirty estimation of the $$yx$$-diagram
In Alternative B we computed $$y_1$$ using the relative volatility $$α_{12}$$ and realized that it did not change too much from pure p-xylol $$x_1=0$$ to pure benzene $$x_1=1$$. The very right columns in the table above show the $$y_1$$-values one would estimate when using the mean relative volatility value $$α_{12}=5.07$$ as constant. The figure below shows the similarity of the accurately computed $$yx$$ data and the ones estimated via the quick and dirty method. The similarity is not too bad. This means that whenever estimated $$xy$$ data are sufficient, you can simply choose an $$α_{12}$$ that is somewhare in between $$x_1=0$$ and $$x_1=1$$. Using this $$α_{12}$$ as a constant, you then can compute $$y_1$$-values for arbitrarily chosen $$x_1$$-values. Recall that in task 1 we had to compute the $$x_1$$-values according to the boiling point criterium first. These computations are skipped when using the quick and dirty estimation.
