Small systems are known to deviate from the classical thermodynamic description, among other things due to their large surface area to volume ratio compared to corresponding big systems. As a consequence, extensive thermodynamic properties are no longer proportional to the volume, but are instead higher order functions of size and shape. We investigate such functions for second moments of probability distributions of fluctuating properties in the grand-canonical ensemble, focusing specifically on the volume and surface terms of Hadwiger's theorem, explained in Klain, Mathematika, 1995, 42, 329–339. We resolve the shape dependence of the surface term and show, using Hill's nanothermodynamics [Hill, J. Chem. Phys., 1962, 36, 3182], that the surface satisfies the thermodynamics of a flat surface as described by Gibbs [Gibbs, The Scientific Papers of J. Willard Gibbs, Volume 1, Thermodynamics, Ox Bow Press, Woodbridge, Connecticut, 1993]. The Small System Method (SSM), first derived by Schnell et al. [Schnell et al., J. Phys. Chem. B, 2011, 115, 10911], is extended and used to analyze simulation data on small systems of water. We simulate water as an example to illustrate the method, using TIP4P/2005 and other models, and compute the isothermal compressibility and thermodynamic factor. We are able to retrieve the experimental value of the bulk phase compressibility within 2%, and show that the compressibility of nanosized volumes increases by up to a factor of two as the number of molecules in the volume decreases. The value for a tetrahedron, cube, sphere, polygon, etc. can be predicted from the same scaling law, as long as second order effects (nook and corner effects) are negligible. Lastly, we propose a general formula for finite reservoir correction to fluctuations in subvolumes.

Small system properties are known to depend on geometric variables in ways that are insignificant for macroscopic systems. Small system considerations are therefore usually added to the conventional description as needed. This paper presents a thermodynamic analysis of adsorbed films of any size in a systematic and general way within the framework of Hill’s nanothermodynamics. Hill showed how to deal with size and shape as variables in a systematic manner. By doing this, the common thermodynamic equations for adsorption are changed. We derived the governing thermodynamic relations characteristic of adsorption in small systems, and point out the important distinctions between these and the corresponding conventional relations for macroscopic systems. We present operational versions of the relations specialized for adsorption of gas on colloid particles, and we applied them to analyze molecular simulation data. As an illustration of their use, we report results for CO${}_2$ adsorbed on graphite spheres. We focus on the spreading pressure, and the entropy and enthalpy of adsorption, and show how the intensive properties are affected by the size of the surface, a feature specific to small systems. The subdivision potential of the film is presented for the first time, as a measure of the film’s smallness. For the system chosen, it contributes with a substantial part to the film enthalpy. This work can be considered an extension and application of the nanothermodynamic theory developed by Hill. It provides a foundation for future thermodynamic analyses of size- and shape-dependent adsorbed film systems, alternative to that presented by Gibbs.

The ideal gas model is an important and useful model in classical thermodynamics. This remains so for small systems. Molecules in a gas can be adsorbed on the surface of a sphere. Both the free gas molecules and the adsorbed molecules may be modeled as ideal for low densities. The adsorption energy, $U^s$, plays an important role in the analysis. For small adsorbents this energy depends on the curvature of the adsorbent. We model the adsorbent as a sphere with surface area $\Omega =4\pi R^2$, where R is the radius of the sphere. We calculate the partition function for a grand canonical ensemble of two-dimensional adsorbed phases. When connected with the nanothermodynamic framework this gives us the relevant thermodynamic variables for the adsorbed phase controlled by the temperature T, surface area $\Omega$, and chemical potential $\mu$. The dependence of intensive variables on size may then be systematically investigated starting from the simplest model, namely the ideal adsorbed phase. This dependence is a characteristic feature of small systems which is naturally expressed by the subdivision potential of nanothermodynamics. For surface problems, the nanothermodynamic approach is different, but equivalent to Gibbs’ surface thermodynamics. It is however a general approach to the thermodynamics of small systems, and may therefore be applied to systems that do not have well defined surfaces. It is therefore desirable and useful to improve our basic understanding of nanothermodynamics.