Thermodynamics of homogeneous nucleation under different constraints

The thermodynamics of nucleation of a pure vapour is investigated under dif ferent constraints. The well-known cases isothermal-isobaric, isothermal-is ochoric and isolated are considered. The systems are composed of a number o f clusters having identical size. The appropriate thermodynamic potentials are calculated from a common point of view. The actual state of the system, clusters and vapour, is compared to the initial supersaturated state. A ve ry large number of such actual states exist if the amount of molecules in t he system is large. They form a surface of the potential function. It is lo oked for relatively stable states of the system compared to neighbouring st ates of the. system. These states are determined by partial derivation of t he potential functions or the determination of gradient curves on the surfa ces. In all three cases investigated critical states can be found. They bel ong to maxima of the potential function in the isothermal-isobaric and the isothermal-isochoric cases and to minima in the isolated case. No stable st ate can be found in the isothermal-isobaric system. On the contrary in the remaining two cases the stable equilibrium of the bulk condensed-phase at s aturation pressure can be reached by the system. In the isothermal-isobaric and in the isothermal-isochoric system there exists a bottom line of the p otential function surface and in the isolated case a crest line can be foun d. Both these curves belong to relatively stable states of the system. Nucl eation and ripening phases of the system may be attributed to these graphs. It is shown that the Kelvin equation gives a proper description of critica l or stable states only in special regions of cluster size and cluster conc entrations. It is furthermore demonstrated that the constraint of constant pressure is responsible for the significant differences between the isother mal-isobaric case and the remaining two cases in a one component system, If the presence of an inert substance is allowed the stable equilibrium of th e condensed phase at saturation vapour pressure may also be attainable unde r constant pressure. The considerations presented for the transition vapour condensed phase can easily be applied to phase transitions in condensed ph ases, respectively.