Improved methods are needed to study sorption of vapor molecules by particles in the gas phase (heterogeneous uptake), which is an important process in both natural and engineered environments. Here, a new measurement system, composed of a differential mobility analyzer (DMA) and drift tube ion mobility spectrometer (DTIMS) in series, is used to examine the heterogeneous uptake of water vapor by 2.85-7.6 nm particles composed of lithium and sodium iodide. The extent of heterogeneous uptake is determined by controlling the relative humidity of the drift region in the DTIMS in the 0-30% range (in air at atmospheric pressure and room temperature), and is quantified via the dimensionless growth factor (GF), i.e. the ratio of the mobility diameter of particles at a prescribed relative humidity relative to their mobility diameter under dry conditions. The precision in GF estimation of the DMA-DTIMS system is shown to be below 0.2%. An analytical equation to calculate the growth factor, based upon predictions of the equilibrium constants for the successive uptake of vapor molecules by particles, is also presented. While the equation is sufficiently general to enable comparison between measured GFs and predictions from any theoretical expression for equilibrium constants, we specifically compare measurements to GF predictions based on the classical Kelvin-Thomson-Raoult (KTR) model for the vapor pressure of a small particle, with consideration of the influence of the ion-dipole potential on water vapor-nanoparticle collisions. It is shown that KTR calculations drastically underpredict the extent of heterogeneous uptake for the examined nanoparticles.