Reply to ‘Comment on “On the Theory of Electrostatic Interactions in Suspensions of Charged Colloids”’ by Willem H. Mulder

SSSAJ: Volume 74: Number 5 • September –October 2010 I AM HAPPY TO HAVE BEEN given the opportunity to respond to the “Comment” by Drs. Ise and Sogami (2010) regarding my recent paper, Mulder (2010). My critics assert that my analysis is a rehash of Overbeek’s (1987, 1993). This is misleading because I did criticize Overbeek’s argument as well as correct it. The fi rst error that I rectifi ed (my Eq. [6]) does not seem to have been spotted before, and was reproduced by at least one other researcher (Schmitz, 1996, Eq. [13]). Ise and Sogami refer to textbooks on statistical thermodynamics, such as those by McQuarrie (2000) or Hill (1960), to support their claim that the electrical parts of Helmholtz (F) and Gibbs (G) free energies of electrolyte solutions are different and, more specifi cally, are related via the expression el el el G F V Π = + , rather than el el el el G F PV F = + ≅ . They are quite correct in pointing out that this is the crucial difference between the two approaches. To elucidate the origin of the confl icting views I shall refer to Hill’s textbook on Statistical Thermodynamics (1960), Chapter 18. The fi rst thing to note is that the “system” that Hill analyzes is not an electrolyte solution per se, but rather, an electrolyte solution in osmotic equilibrium with a reservoir containing the pure solvent. He makes only oblique reference to this in a statement following his Eq. [18–2]: “The pressure p is the pressure of a “gas mixture” of ions. In reality, of course, if the continuum is a solvent and not a vacuum, p is the osmotic pressure of the electrolyte”. After a discussion of standard Debye-Hückel theory, Hill derives the usual expression for the excess (electrical) chemical potential of an ion. This is his Eq. [18–23], which differs from that given by Ise and Sogami in their textbook (2005), and which I highlighted in my paper. Hill’s thermodynamic analysis then picks up again from the defi nition of a “free energy” F (Eq. [18-29]), which is however neither a Gibbs nor a Helmholtz energy because it lacks the solvent term, s s N μ . Instead, it defi nes a quantity that is more correctly referred to as a semi-grand potential. This is the appropriate thermodynamic potential (the one that tends to a minimum) when describing any solution in osmotic equilibrium with the pure solvent because this time, the chemical potential of the solvent, rather than the number of solvent molecules, is held fi xed. This subtle point is insuffi ciently emphasized by Hill, and appears to have been overlooked. A second “free energy” is dei ned by Hill in his Eq. [18–32] and, to add to the confusion, is denoted by the symbol A, and introduced as a “Helmholtz free energy”. A careful analysis, however, reveals that the actual dei nition is rather dif erent: is time we are dealing with a dif erence between a semi-grand canonical potential, Ω, of the solution and that of N s molecules in pure solvent, taken to be the same number as is present in solution, evaluated at the local pressure P (P0 in the solvent compartment of the osmotic cell, P0 + Π in solution), which is dei ned as s s G N PV Ω μ = − − , that is, a Legendre transform of the Helmholtz energy proper. e dif erence with 0 0 PV Ω = − ,