Relation between adiabatic and pseudoadiabatic compressibility in ultrasonic velocimetry

A fundamental problem in measurements of the adiabatic compressibility of macromolecules in aqueous solution using sound velocity is the unknown heat exchange between the interior of the molecule and the bulk solvent. Usually only a pseudoadiabatic compressibility is measured, which is between isothermal and adiabatic compressibility. The measured compressibility might even be close to the isothermal compressibility, depending on the experimental conditions. In this paper, a model is proposed that enables the deviation of experimental conditions from the requirement of adiabatic measurements to be estimated. This model treats the macromolecule as a heat conductive sphere. Analytical solutions for the temporal and local heat diffusion are obtained. Proteins less than 100 kDa in molecular weight at a sound frequency of less than 10 MHz are found to be essentially in thermal equilibrium with the bulk solvent. In this case, for the aqueous protein solution at temperatures ≤ 25°C and less than 2% protein concentration, the theory predicts that the sound velocity measurement of the interior of protein is close to isothermal conditions.