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

Michael G Bowler,Matthew W Bowler,David R Bowler

doi:10.1107/s1600576717003636

Abstract

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.

Highlights

Sample environments that control relative humidity (RH) are important in many experiments where a wide variety of samples require specific RH values to maintain sample integrity or RH is a parameter to be varied
Substitution yields equation (1) of Bowler, Mueller et al (2015), used in the construction of the curves in Fig. 2. (We have found that the best value of m for polyethylene glycol (PEG) is 38.) Expressed as a function of mass fraction, the RH becomes independent of the polymer molecular mass n as n becomes very large, i.e. for very long chain polymers
We have established that the theoretical RH values we previously calculated (Bowler, Mueller et al, 2015; Wheeler et al, 2012) are in satisfactory agreement with a humidity control device used on protein crystallography beamlines

Summary

Introduction

Sample environments that control relative humidity (RH) are important in many experiments where a wide variety of samples require specific RH values to maintain sample integrity or RH is a parameter to be varied. In order to facilitate this stage we measured the equilibrium RH points for a variety of solutions commonly used for the crystallization of proteins and nucleic acids (Wheeler et al, 2012) This provided a starting point for most experiments and the results obtained were compared with Raoult’s law (Raoult, 1887) for the equilibrium vapour pressure of water above a solution [and for solutions of polymers, with a generalization (Bowler, Mueller et al, 2015)]. We present a simple explanation for Raoult’s law using statistical mechanics and show how this treatment can be extended to polymer solutions, where Raoult’s law breaks down These results illuminate the machinery underlying a long-observed phenomenon and allow the accurate prediction of humid atmospheres for specific sample requirements, applicable to a wide variety of fields

RH measurements

Comparison of measured equilibria and predicted values

Discussion