Vapor-liquid equilibrium is a state in which a pure component or mixture exists in liquid and vapor phases, with mechanical and thermal equilibrium and no net mass transfer between the two phases. Vapor and liquid are separated by gravity and heat (Figure 1). The liquid mixture is inserted into the system, which is put into a vacuum state with a vacuum pump. The vapor is condensed and returned to mix with the liquid, which is then passed back to the boiling chamber. Differences in the boiling point results in some separation of the mixture. The boiling point of water is higher than that of the added components, so the volatile components begin to evaporate.

Figure 1: A depiction of the apparatus

An activity coefficient is defined as the ratio of a component's fugacity in an actual mixture to the fugacity of an ideal solution of the same composition. Fugacity is a property used to show differences between chemical potentials at standard states. Vapor phase fugacities can be expressed in terms of a fugacity coefficient [φ: f_i^V = φ_i y_i f_i^0V ], with y_i = mol fraction of i in the vapor phase, and f_i^0V = the vapor standard state fugacity (the fugacity of pure component vapor at T and P). For low pressures, as in this experiment, φ_i = 1 and f_i^0V = P. Liquid phase fugacities can be expressed in terms of an activity coefficient γ_i: f_i^L = γ_i x_i f_i^0L , with x_i = mol fraction of i in the liquid phase, and f_i^0L = the liquid standard state fugacity.

At the saturation pressure (P_i^s) of this T, the pure component liquid fugacity would be P_i^s, because the pure vapor and liquid are in equilibrium. Since liquid fugacities are only weak functions of pressure, we can approximate the pure component liquid fugacity at T and P (f_i^0L) as P_i^s, as long as the difference between P_i^s and P is not large. This approximation is usually called "neglecting the Poynting correction". If experimenters use a VLE apparatus to measure the compositions of the vapor and liquid which are in equilibrium, experimenters can directly calculate the activity coefficients provided to also measure P and T. T must be measured to determine P_i^S for all i.

The heart of the VLE device, used in this experiment to determine compositions of mixtures, is a Cottrell pump which "spits" boiling liquid into a well-insulated, equilibrium chamber. Two magnetically operated sampling valves allow for withdrawal of liquid and condensed vapor samples. A large reservoir helps to dampen pressure pulses in the system as the on-off control valve switches, and from fluctuations caused by the Cottrell pump. A slow leak can be used to create a balance between the rate of withdrawal of air and the rate of input of air to maintain a constant pressure, if necessary.

A comparable way to solve for vapor-liquid equilibrium is to use a variety of models. Raoult's law, Dalton's law, and Henry's law are all theoretical models that can find the vapor-liquid equilibrium concentration data. All three models are related to the proportionality of partial pressures, total pressure, and mole fractions of substances. Wilson's equation has been proven to be accurate for miscible liquids, while not being overly complex. Additionally, Wilson's model incorporates activity coefficients to account for deviation from ideal values.

The activity coefficients of the data do not show significant deviations from a mean value for each component (Table 1). This is as expected because for intermediate component compositions there are not large variations. However, components near 1 have γ's near 1. Low composition components have high γ's. Components highest in concentration in a mixture which will have a reduced deviation, therefore it will be closer to ideal (γ = 1). Components with lower concentrations in a mixture will have higher deviations, so their γ's will be greater than 1.

Table 1: Results of each sampling of the experimental data.

The data were fit to Wilson model parameters and the coefficients were calculated (Table 2). A simple reduction in the sum of squared residuals between experimental and Wilson equation (1) activity coefficients was used. This was achieved using Excel's solver function. The parity plot shown relates the Wilson's Equation model activity coefficients to the experimentally found activity coefficients. The experimental activity coefficients were calculated and graphically compared to the calculated model coefficients.

Table 2: Results of fitting the data to the Wilson model parameters.

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The parameter values found were the best fit (Table 3). Ideally the correlation is along the y=x line; however, a significant correlation resembling the ideal scenario was found (Figure 2). The activity coefficients of the data did not show significant deviations from a mean value for each component, as expected. A reduction in the sum of squared residuals between experimental and Wilson equation activity coefficients was used with Excel's solver function. The parity plot relates the Wilson's Equation model activity coefficients to the experimentally found activity coefficients.

Table 3: Model parameters with water (a), MeOH (b), and IPA (c). The experimental values are compared to expected values.

Figure 2: Depiction of the correlation between the experimental activity coefficients and the model activity coefficients.

This experiment demonstrated the equilibration of methanol - isopropanol - water vapor-liquid mixtures at a constant P = 700 mm Hg and how to measure temperature and composition and calculate activity coefficients. The activity coefficients of the data did not significantly deviate from a mean value for each component, as expected. A reduction in the sum of squared residuals between experimental and Wilson equation activity coefficients was used with Excel's solver function. The parity plot relates the Wilson's Equation model activity coefficients to the experimentally found activity coefficients.

In the petroleum industry, distillation is the primary process for separation of petroleum products. Many oil refineries use distillation for crude oil¹. Light hydrocarbons are separated from heavier particles, separating based on boiling points¹. Heavy materials like gas oils collect in the lower plates, while light materials like propane and butane rise up¹. Hydrocarbons, such as gasoline, jet, and diesel fuels, are separated¹. This process is often repeated many times to fully separate and refine the products¹. Refineries run these processes at steady state, constantly creating new products at maximum capacity, so efficiency is key¹. Chemical engineers working on these processes focus on optimizing the efficiency of the production¹.

Tray distillation columns are also used to separate a variety of chemical products. Ethanol is one such product. Through closely related processes, a variety of products such as fuel-grade ethanol, beer, and liquor can all be distilled². Specific amounts of alcohol can be separated from water in order to create a specific proof². This process is limited to reducing the percentage of water in the product, but cannot completely eliminate it². In order to remove water completely, azeotropic distillation is required, which uses extractor chemicals to separate water from ethanol².