A Simple and Practical Theoretical Model for Interpreting Binary Azeotropes

We put forth a simple and yet practical theoretical model generalized from Raoult’s law and Henry’s law and show that it can be reduced to these two laws under limiting conditions. The model entertains a hybrid parameter h_B with activity coefficient bundled into it, which smoothly bridges the p_B^* and K_B coefficients in Raoult’s law and Henry’s law. The value of h_B falls in the interval of [K_B, p_B^*] or [p_B^*, K_B] in the case of negative or positive deviation from Raoult’s law, respectively. We uncover an overlapping rule for the ranges of h_A and h_B, which binary mixtures must obey to form azeotropes. We also provide straightforward ways to analyze the characteristic mole fraction and pressure for azeotropes and to understand the relative positions of vapor composition curves with respect to the liquid counterparts. We rely heavily on experimental data available in the literature for representative binary mixtures with both negative and positive deviations from Raoult’s law to illustrate the algebraic derivations. The knowledge gained is useful in the analysis of experimental data from vapor-liquid equilibrium measurements and possess pedagogical merit in various relevant fields.

A Simple and Practical Theoretical Model for Interpreting Binary Azeotropes

We put forth a simple and yet practical theoretical model generalized from Raoult’s law and Henry’s law and show that it can be reduced to these two laws under limiting conditions. The model entertains a hybrid parameter h_B with activity coefficient bundled into it, which smoothly bridges the p_B^* and K_B coefficients in Raoult’s law and Henry’s law. The value of h_B falls in the interval of [K_B, p_B^*] or [p_B^*, K_B] in the case of negative or positive deviation from Raoult’s law, respectively. We uncover an overlapping rule for the ranges of h_A and h_B, which binary mixtures must obey to form azeotropes. We also provide straightforward ways to analyze the characteristic mole fraction and pressure for azeotropes and to understand the relative positions of vapor composition curves with respect to the liquid counterparts. We rely heavily on experimental data available in the literature for representative binary mixtures with both negative and positive deviations from Raoult’s law to illustrate the algebraic derivations. The knowledge gained is useful in the analysis of experimental data from vaporliquid equilibrium measurements and possess pedagogical merit in various relevant fields.

Raoult's law revisited: accurately predicting equilibrium relative humidity points for humidity control experiments

The humidity surrounding a sample is an important variable in scientific experiments. Biological samples in particular require not just a humid atmosphere but often a relative humidity (RH) that is in equilibrium with a stabilizing solution required to maintain the sample in the same state during measurements. The controlled dehydration of macromolecular crystals can lead to significant increases in crystal order, leading to higher diffraction quality. Devices that can accurately control the humidity surrounding crystals while monitoring diffraction have led to this technique being increasingly adopted, as the experiments become easier and more reproducible. Matching the RH to the mother liquor is the first step in allowing the stable mounting of a crystal. In previous work [Wheeler, Russi, Bowler & Bowler (2012). Acta Cryst. F68, 111–114], the equilibrium RHs were measured for a range of concentrations of the most commonly used precipitants in macromolecular crystallography and it was shown how these related to Raoult's law for the equilibrium vapour pressure of water above a solution. However, a discrepancy between the measured values and those predicted by theory could not be explained. Here, a more precise humidity control device has been used to determine equilibrium RH points. The new results are in agreement with Raoult's law. A simple argument in statistical mechanics is also presented, demonstrating that the equilibrium vapour pressure of a solvent is proportional to its mole fraction in an ideal solution: Raoult's law. The same argument can be extended to the case where the solvent and solute molecules are of different sizes, as is the case with polymers. The results provide a framework for the correct maintenance of the RH surrounding a sample.

Raoult’s law revisited: accurately predicting equilibrium relative humidity points for humidity control experiments

The humidity surrounding a sample is an important variable in scientific experiments. Biological samples in particular require not just a humid atmosphere but often a relative humidity (RH) that is in equilibrium with a stabilizing solution required to maintain the sample in the same state during measurements. The controlled dehydration of macromolecular crystals can lead to significant increases in crystal order, which often leads to higher diffraction quality. Devices that can accurately control the humidity surrounding crystals on a beamline have led to this technique being increasingly adopted, as experiments become easier and more reproducible. Matching the relative humidity to the mother liquor is the first step to allow the stable mounting of a crystal. In previous work, we measured the equilibrium relative humidity for a range of concentrations of the most commonly used precipitants and showed how this related to Raoult’s law for the equilibrium vapour pressure of water above a solution. However, a discrepancy between measured values and those predicted by theory could not be explained. Here, we have used a more precise humidity control device to determine equilibrium relative humidity points. The new results are in agreement with Raoult’s law. We also present a simple argument in statistical mechanics demonstrating that the saturated vapour pressure of a solvent is proportional to its mole fraction in an ideal solution: Raoult’s Law. The same argument can be extended to the case where solvent and solute molecules are of different size, as is the case with polymers. The results provide a framework for the correct maintenance of the RH surrounding samples.SynopsisThe equilibrium relative humidity values for a number of the most commonly used precipitants in biological macromolecule crystallisation have been measured using a new humidity control device. A simple argument in statistical mechanics demonstrates that the saturated vapour pressure of a solvent is proportional to its mole fraction in an ideal solution (Raoult’s Law). The same argument can be extended to the case where solvent and solute molecules are of different size.