Further Development of Scaled Particle Theory for Rigid Spheres: Application of the Statistical Thermodynamics of Curved Surfaces

The thermodynamics and statistical thermodynamics of curved surfaces are used to investigate scaled particle theory as an extension of macroscopic ideas into the molecular domain. A thermodynamic expression for $\mathrm{G(r)}$ , the central function of scaled particle theory, is derived which is exact down to values of $r$ approximating molecular diameters but which contains parameters which can only be specified on the basis of molecular information. These parameters are the pressure $p$ ; ${\gamma }_s$ , the surface tension corresponding to the surface of tension; and ${\delta }_1$ , the distance of the surface of tension from the surface of radius $r$ belonging to the “r‐cule” or solute hard sphere. ${\gamma }_s$ and ${\delta }_1$ depend upon both $r$ and the density $\rho$ , while $p$ depends on $\rho$ . It is shown that the form for $G$ chosen in the original (1959) scaled particle theory represented an inconsistent but fortunate extension of macroscopic ideas into the molecular domain. Upon the introduction of yet another parameter, $\delta$ , the distance between the surface of tension and the so‐called equimolecular surface (also dependent on $r$ and $\rho$ ), it proves possible to develop a system of three simultaneous partial differential equations on the three unknowns ${\gamma }_s{\mathrm{,\; \delta }}_1$ , and $\delta$ for which there exist enough exact boundary conditions to allow a unique determinate solution. In the development of these differential equations, one must invoke the statistical mechanics of surfaces. Use is made of the existing literature, and some new relations are developed along with a critical adjustment of some of the older relations. Unfortunately, one of the three partial differential equations is exact only through terms in $\mathrm{a\hspace{0.17em}/\hspace{0.17em}r}$ in ${\gamma }_s$ , where $a$ is the diameter of a hard sphere molecule. In this sense the present theory remains approximate, but the path towards extending its precision to higher orders of $\mathrm{a\hspace{0.17em}/\hspace{0.17em}r}$ is clearly delineated; and so scaled particle theory is provided with a well‐defined program of self‐improvement and possibly closure. It proves possible to derive exact relations for both $\delta$ and ${\delta }_1$ , for a plane surface, in terms of the dependences of ${\gamma }_s$ (for a plane surface) and of $p$ upon $\rho$ . Thus, with a good approximation to the equation of state (of which several exist for hard spheres), very reliable values for ${\delta }_1$ and $\delta$ at $\mathrm{r\hspace{0.17em}\to \hspace{0.17em}\infty }$ , at any density can be computed. These are probably the first estimates of parameters like ${\delta }_1$ and $\delta$ which are quantitative beyond order‐of‐magnitude considerations. Both ${\delta }_1$ and $\delta$ prove to be exceedingly small even when compared to $a$ . Furthermore, ${\delta }_1\mathrm{\hspace{0.17em}+\hspace{0.17em}\delta }$ at $\mathrm{r\hspace{0.17em}\to \hspace{0.17em}\infty }$ vanishes for infinite compressibility (possibly at a phase transition), a result suggested by some earlier, less quantitative work by Harris and Tully‐Smith. If it is assumed that ${\delta }_1$ and $\delta$ are as small at all values of $r$ as they are at $\mathrm{r\hspace{0.17em}=\hspace{0.17em}\infty }$ , a new...