Thermodynamically Proven: In Binary Azeotropes, Where the Light “Becomes” the Heavy
Keywords: #Thermodynamics, #Azeotropes, #Distillation, #DesignIssues
During distillation training sessions, process engineers are often presented with this simple conclusion: for all binary homogeneous maximum-boiling azeotropes, the light component (LK) always acts like a heavy key (HK) in the dilute region until the azeotropic point is reached; similarly, for all binary homogeneous minimum-boiling azeotropes, the light component always acts like a heavy key from the azeotropic composition to purity. For example, in the HF-water system, HF and water form a maximum-boiling azeotrope at 112.5 °C under 1 atm (Figure 1). Pure HF boils at 19.5 °C while water boils at 100 °C. Therefore, HF is the light key (LK) and water is the heavy key (HK). However, in the region from dilute HF to the azeotropic composition, HF acts like a heavy key because it is richer in the liquid phase than in the vapor phase (on the x-y diagram, the equilibrium curve lies below the y = x diagonal). The author has used the following pair of visual statements to memorize the conclusions:
- Maximum-boiling (Tmax) azeotrope: the x-y equilibrium curve starting from x = 0 always first lies below the y = x diagonal (LK acts heavy in the dilute region up to the azeotrope).
- Minimum-boiling (Tmin) azeotrope: the x-y equilibrium curve starting from x = 0 always first lies above the y = x diagonal (LK stays light); consequently, from the azeotrope to pure LK the curve lies below the y = x diagonal and the LK acts heavy.
Since an engineering training session rarely dwells on the fundamentals, few would have the chance to reflect on the premise: does this order of relative volatility (commonly denoted as α) between LK and HK hold true for all azeotropes, or is it merely another rule of thumb? The answer is that the rule is thermodynamically rigorous. While an algebraic proof appears in advanced thermodynamics texts, this essay presents a graphical proof that is more intuitive to engineers.
Figure 1. The x-y and the T-x-y diagrams of the HF-water system (maximum-boiling azeotrope). Pure HF boils at 19.5 °C and is therefore the light key (LK); pure water boils at 100 °C and is the heavy key (HK).
For the graphical proof we first define the points that shape the diagram. In distillation, a light key (LK) is the component of concern that possesses the lower pure-component boiling point (Tb, LK), while the heavy key (HK) has a higher pure-component boiling point (Tb,HK). For a binary homogeneous system that contains only 1 maximum-boiling azeotrope, the mixture exhibits a single composition (xa, LK = ya, LK), at which the bubble point (Ta) is higher than both the pure-component boiling points. As the convention, x denotes the liquid mole fraction and y denotes the vapor mole fraction. A generic T-x-y diagram of this type can be constructed as in Figure 2:
Figure 2. Representation of any binary homogeneous system that possesses only 1 maximum-boiling azeotrope. The azeotropic composition, (xa, LK, ya, LK), may lie anywhere between 0 and 1, and the azeotropic temperature Ta must be higher than both the Tb, LK and Tb, HK. The bubble-point curve and the dew-point curve are continuous and may adopt any contour, but they may neither cross nor exhibit additional extrema as prevented by the Gibbs-Konovalov theorem.
Figure 2 shows the bubble-point and dew-point curves of any binary homogeneous maximum-boiling azeotropic system that contains only a single azeotropic point. The azeotropic temperature Ta may take any value, provided it exceeds both pure-component boiling points, Tb, LK and Tb, HK, which is the intrinsic definition of a maximum-boiling azeotrope. The curves can be of any shapes, yet two thermodynamic constraints apply: (1) they cannot cross except at the azeotrope, and (2) they cannot possess any other local extrema. The Gibbs-Konovalov theorem rigorously establishes that all local maxima or minima of the bubble-point and dew-point curves occur only at azeotropes, where they cross (Castellan, 1983). In addition, classical thermodynamics requires the curves to be continuous over the entire temperature and composition range.
Figure 3. 1) In the region between pure HK (xLK, yLK =0) and the azeotrope, the liquid mole fraction of the LK is greater than its vapor mole fraction (xi, LK > yi, LK). Consequently, for a binary system, the HK is enriched in the vapor and is the more volatile component. 2) In the region between the azeotrope and pure LK, the vapor mole fraction of the LK exceeds its liquid mole fraction (yj, LK > xj, LK.), restoring the normal light-key behavior.
At any temperature Ti that lies between Tb, HK and Ta, the equilibrium composition of the LK must fall between 0 and the azeotropic composition xa, LK (= ya, LK). A horizontal tie-line drawn at T=Ti intersects the bubble-point curve at xi, LK and the dew-point curve at yi, LK (Figure 3). Because the bubble-point curve lies to the right of the dew-point curve in this region, LK’s liquid mole fraction xi, LK is always greater than its vapor mole fraction yi, LK. Or, mathematically:
– Ineq.1
Where,
xi,LK is the liquid mole fraction on the bubble-point curve at Ti;
yi,LK is the vapor mole fraction on the dew-point curve at Ti.
Similarly, at any temperature Tj that lies between Tb, LK and Ta, the equilibrium compositions of the LK must fall between the azeotropic composition xa, LK (=ya, LK) and pure LK. A tie-line at T=Tj now intersects the curves at xj, LK and yj, LK. Because the bubble-point curve now lies to the left of the dew-point curve, LK’s vapor mole fraction yj, LK is always greater than its liquid mole fraction xj, LK. Or, mathematically:
– Ineq.2
Where,
xj,LK is the liquid mole fraction on the bubble-point curve at Tj;
yj,LK is the vapor mole fraction on the dew-point curve at Tj.
Now given the established relationship between the LK’s vapor and liquid mole fractions across the azeotrope, relative volatility can be used to determine whether the LK “acts” light or heavy in the respective “i” and “j” regions. Relative volatility is defined as (Kister, 1992)
In the dilute LK region before the azeotrope, from Ineq.1, we have xi, LK > yi, LK and therefore (1-xi, LK ) < (1-yi, LK), which means xi, HK < yi, HK (in a binary mixture). Consequently KLK= yi, LK / xi, LK < 1 while KHK= yi, HK/ xi, HK > 1, so that α=KLK / KHK < 1. The light key is therefore less volatile than the heavy key and effectively “acts” as the heavy key. Thus, the statement “for all binary homogeneous maximum-boiling azeotropes, the light component (LK) always acts like a heavy key (HK) in the dilute region until the azeotropic point is reached” has been proven.
In the region between the azeotrope and pure LK, the opposite inequality holds, α > 1, restoring the normal behavior of a light component.
When the corresponding x-y diagram is drawn for a Tmax-azeotrope, the inequality xLK > yLK in the dilute LK region forces the equilibrium curve of the LK to lie below the y = x diagonal (Figure 4). Consequently, the curve that starts at x = 0 first descends below the y = x diagonal, later crosses the diagonal line at the azeotrope, and finally remains above the diagonal as a conventional light component would. Thus, the maximum-boiling case for the statement “the x-y equilibrium curve starting from x = 0 always first lies below the y = x diagonal” has been proven.
Figure 4. Complementary schematic illustrating the relative positions of the x-y equilibrium curve and the 45° x =y diagonal.
A few notes:
- The x-y and the T-x-y diagrams are almost always constructed with the light key as the plotted component.
- The proof for minimum-boiling azeotropes is analogous (the temperature ordering is simply inverted). The same geometric reasoning, applied symmetrically, shows that the x–y curve starting from x = 0 first lies above the y = x diagonal; therefore, from the azeotrope to pure LK the curve lies below the diagonal, and the light key acts as a heavy key.
Binary systems that possess more than one azeotrope are uncommon in industrial distillation. Nevertheless, the graphical method presented here can be extended, region by region, to those more complex cases.
References
[1] Castellan, G. W. (1983). Physical chemistry (3rd ed.). Addison-Wesley.
[2] Kister, H. Z. (1992). Distillation design. McGraw-Hill.