VLE diagram and relative volatility [Tutorial]

Every time I read distillation, I thank Raoult and Dalton for inventing the wonderful subject of partial pressure. Understanding Vapor-Liquid equilibrium, which is fundamental to many complex unit operations, may have been impossible.

What is a VLE?

The boiling point properties of the components in the mixtures being separated are used to design distillation columns. As a result, the sizes, particularly the heights, of distillation columns are determined by the mixtures' vapor liquid equilibrium (VLE) data.

How does vapor-liquid equilibrium occur?

The term phase equilibria refer to when two or more phases coexist (in equilibrium). When two phases coexist, they are in a state of saturation. ?The chemical potential can predict phase stability because the most stable form of the substance has the lowest chemical potential at the given temperature and pressure. The chemical potential of a substance introduced by Gibbs is simply the Gibbs free energy or chemical energy per mole of that substance

The vapor-liquid equilibrium (VLE) describes the distribution of a chemical species between the vapor and liquid phases. A component distributes itself between vapor and liquid phases in the ratio of mole fractions y/x. This is called the K value of the component. The ratio of K of the more volatile phase to the less volatile phase is relative volatility.

?The pressure in a distillation column is generally constant. The temperature varies from bottom to top. At equilibrium, both phases have near equal Gibbs free energy at a minimum while the components of each phase can have a different composition.

Consider a tray in a distillation column with a liquid phase that has its own vapor pressure derived from the sum of two unequal partial pressures of two components [binary system]. The tray also has a vapor phase made up of two vapor components with unequal partial pressures. The total pressure is the sum total. As a result, there are four variables. Because such an unstable system cannot exist, they all work towards the same goal of minimizing chemical potential and achieving equilibrium.

The most convenient thing that happens is the liquid phase gives up more volatiles and thus becomes more concentrated with high boilers on each plate, whereas the vapor phase gives up less volatiles and thus becomes more concentrated with low boilers on each plate. This continues until the two phases have separated. Finally, both phases have reduced their chemical potential and achieved equilibrium.

To summarize, distillation takes advantage of differences in concentrations of components in the liquid and vapor phases on each tray.

Draw a vapor-liquid equilibrium curve with the following Benzene – Toluene vapor pressure data.?

Step by step

Write Raoult's Law for a binary mixture of A and B;

pA = Pvp,A xA ; pB = Pvp,B xB = Pvp,B (1 - xA)

pA and pB are the partial pressures of A and B. Pvp,A and Pvp,B are the vapor pressure of A and B

The total pressure, PT = pA + pB

Replacing for the partial pressures and re-arrange, we have:

P tot = P vpA Xa + [PvpB – PvpB Xa]??????????????????????????

P tot = P vpA Xa + PvpB - PvpB Xa

P tot = Xa [P vp A – PvpB] + PvpB ??

Re-arrange, we have the expression for xA:

Fundamental equations

Xa = [P tot – pvpB] / [pvpA- pvpB]

Since Pvp,A Xa = yA P tot; we have the expression for yA: [combined Daltons + Raoult’s law]

Ya = [PvpA/Ptot] Xa

The VLE data can be calculated at each temperature by substituting for total pressure (101.32 kPa) and the appropriate vapor pressures. For example, at 85.0 degc,

Let A is benzene and B is toluene

Xa = [101.32 – 46]/ 116.9-46] = 0.7803

Ya = [116.9/101.32] 0.7803 = 0.9002

K1= 0.9002/0.7803 = 1.15 [y/x]

We have the following final results:

Similarly for toluene

Xb = [P tot – pvpA] / [pvpB- pvpA]

Yb = [PvpB/Ptot] Yb

Yb = [101.32 -116.9]/ [46-116.9] = 15.58/70.9 = 0.2197

Yb = [46 /101.32] 0.2197= 0.0997

K2 = 0.0997/0.2197 = 0.45 [y/x]

Relative volatility = 1.15/0.45 = 2.55

Definition of relative volatility

Alpha = the relative volatility of the more volatile component to the less volatile component

Alpha = K1/K2 = [ y/x] in benzene / [ y/x] in toluene

?VLE diagram

Credit: Google