Volume term of work of critical nucleus formation in terms of chemical potential difference relative to equilibrium one

The work of formation of a critical nucleus is sometimes written as W=n{\Delta}{\mu}+{\gamma}A. The first term W_{vol}=n{\Delta}{\mu} is called the volume term and the second term {\gamma}A the surface term with {\gamma} being the interfacial tension and A the area of the nucleus. Nishioka and Kusaka [J. Chem. Phys. 96 (1992) 5370] derived W_{vol}=n{\Delta}{\mu} with n=V_{\beta}/v_{\beta} and {\Delta}{\mu}={\mu}_{\beta}(T,p_{\alpha})-{\mu}_{\alpha}(T,p_{\alpha}) by rewriting W_{vol}=-(p_{\beta}-p_{\alpha})V_{\beta} by integrating the isothermal Gibbs-Duhem relation for an incompressible {\beta} phase, where {\alpha} and {\beta} represent the parent and nucleating phases, V_{\beta} is the volume of the nucleus, v_{\beta}, which is constant, the molecular volume of the {\beta} phase, {\mu}, T, and p denote the chemical potential, the temperature, and the pressure, respectively. We note here that {\Delta}{\mu}={\mu}_{\beta}(T,p_{\alpha})-{\mu}_{\alpha}(T,p_{\alpha}) is, in general, not a directly measurable quantity. In this paper, we have rewritten W_{vol}=-(p_{\beta}-p_{\alpha})V_{\beta} in terms of {\mu}_{re}-{\mu}_{eq}, where {\mu}_{re} and {\mu}_{eq} are the chemical potential of the reservoir (equaling that of the real system, common to the {\alpha} and {\beta} phases) and that at equilibrium. Here, the quantity {\mu}_{re}-{\mu}_{eq} is the directly measurable supersaturation. The obtained form is similar to but slightly different from W_{vol}=n{\Delta}{\mu}.