On the size- and shape-dependence of integral and partial molar Gibbs energies, entropies, enthalpies and inner energies of solid and liquid nano-particles

Abstract In this paper the size- and shape dependences of 8 different integral and partial molar thermodynamic quantities are derived for solid and liquid nano-phases, starting from the fundamental equation of Gibbs: i) The integral molar Gibbs energies of nano-phases and the partial molar Gibbs energies of components in those nano-phases, ii) The integral molar enthalpies of nano-phases and the partial molar enthalpies of components in those nano-phases, iii) The integral molar entropies of nano-phases and the partial molar entropies of components in those nano-phases, and iv). The integral molar inner energies of nano-phases and the partial molar inner energies of components in those nano-phases. All these 8 functions are found proportional to the specific surface area of the phase, defined as the ratio of its surface area to its volume. The equations for specific surface areas of phases of different shapes are different, but all of them are inversely proportional to the characteristic size of the phase, such as the diameter of a nano-sphere, the side-length of a nano-cube or the thickness of a thin film. Therefore, the deviations of all properties discussed here from their macroscopic values are inversely proportional to their characteristic sizes. The 8 equations derived in this paper follow strict derivations from the fundamental equation of Gibbs. Only the temperature dependent surface energy of solids and surface tension of liquids will be considered as model equations to simplify the final resulting equations. The theoretical equations are validated for the molar Gibbs energy against the experimental values of liquidus temperatures of pure lead. The theoretical equations for the molar enthalpy are validated i). Against the experimental values of dissolution enthalpy differences between nano- and macro cobalt particles in the same liquid alloy and ii). Against the size dependent melting enthalpy of nano-indium particles. In this way, also the theoretical equations for the molar entropy and molar inner energy are validated as they are closely related to the validated equations for the molar Gibbs energy and molar enthalpy.