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**Additional resources for A. Wilhelm Neumann, Robert David, and Yi Zuo (eds): Applied Surface Thermodynamics, 2nd Edition**

**Example text**

The outer wall may have arbitrary shape, however, to insure that the total internal energy Ut is unambiguously determined it is necessary to place certain geometric constraints on the manner in which internal surfaces, lines, and points contact the outer wall. , no amount of its area), lie on the outer wall. , no amount of its length), may lie on the outer wall. Finally, it is necessary to require that a dividing point not be an outer wall point. If any of these conditions are violated, then one would obtain a constrained variation or the mechanical equilibrium conditions for the dividing surfaces, lines, or points would be connected to the geometric shape of the imaginary bounding surface of the composite, fluid system.

The positive sign on the right-hand side denotes the fact that work done by © 2011 by Taylor and Francis Group, LLC 34 John Gaydos, et al. 5 Schematic of the time-averaged variation of the stress across a deformable liquid–vapor interface, where λ is a coordinate directed normal to the interface, σT and σTE are the tangential and excess tangential stresses, and P1 and P2 are the pressures in the two adjacent bulk phases. 123 when the position of the dividing surface is selected as λ = 0. Alternative positions of the dividing surface are denoted by λ1 and λ2.

Mathematically, the difference between the total energy and the total free energy extremum formulations is that the constraints in the first definition (total entropy and masses remaining constant) are replaced by the subsidiary conditions T = T (a constant) and µj + φ = µ j (a constant) in the second definition, such that both problems yield the same solution. The transformations between such conjugate extremum problems are known as involutory transformations [83]. Finally, the advantage of employing the free energy ΩΣ is that there is a direct connection between the variation δΩΣ and the virtual work, as demonstrated later.