By Lieb, Yngvason.

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A half mole of oxygen in a container with a piston and in a magnetic field (two work coordinates, the volume and the magnetization). Systems (a) and (b) joined by a copper thread (three work coordinates). A mixture consisting of 7 moles of hydrogen and one mole of oxygen (one work coordinate). , in chemistry, material science and in astrophysics) we can regard a nonreacting, metastable mixture as capable of being in an equilibrium state, as long as one is careful not to bump the container with one’s elbow.

It should be remarked that the experimental realization of the simple system with volume and surface as independent work coordinates described above might not be easy in practice. In fact, the usual procedure would be to compare measurments on the liquid in bulk and on drops of liquid, and then, by inverting the data, infer the properties of the system where volume and surface are independent variables. The claim that scaling and convexity are compatible with the inclusion of surface effects amounts to saying that these properties hold after such a ‘disentanglement’ of the coordinates.

T)½ by the convex combination axiom A7. Fig. 3 illustrates this theorem in the case " . 7 (Convexity of S ). ¸et the sets S L be defined as in Eq. 12) and assume the state H H space satisfies the convex combination axiom A7 in addition to A1—A5. ¹hen: (i) (ii) S is convex. H . t)½3S H RH > \R H H Proof. (i) This follows immediately from the scaling, splitting and convex combination axioms A4, A5 and A7. (ii) This is proved by splitting, moving the states of the subsystems into forward sectors and bringing the subsystems together at the end.

### Physics and mathematics of the 2nd law of thermodynamics by Lieb, Yngvason.

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