Thermodynamic Venn diagrams: Sorting out forces, fluxes, and Legendre transforms
(a) The thermodynamic square presented by Max Born in 1929 (flipped and rotated to match the top half of Fig. 2) (b) The key to constructing Maxwell relations is to follow around the square’s sides through three corners, then follow around its sides again in a reverse direction ending on the same side of the square. In this example the “shared second side” is the S–V side. Arrows within the thermodynamic square aid in obtaining the correct signs in the differential relations and the Maxwell relation.
(Color online) The thermodynamic Venn diagram overlaid with two concentric “shading disks” that help draw the eye to the three thermodynamic forces (P, , and T, outer disk) and flows (V, N, and S, inner disk). The subscript denotes an N-to- Legendre transform of the indicated potential (U, H, F, or G).
(Color online) The thermodynamic Venn diagram in its differential form. The eight regions each contain a fundamental thermodynamic equation, including the Gibbs–Duhem equation for the outermost region. Students can see that although (see Fig. 2), , and no information about the VP and ST contributions to the internal energy of the system is lost by the Legendre transforms.
(Color online) A mnemonic-free diagram to construct Maxwell relations. In Fig. 2, the negative signs on the variables P, S, and N are arbitrary. Placing them on , V, and T works equally well to obtain the correct signs in the Maxwell relations. There are no negative signs on the variables; instead a diagonal line is used to determine whether a Maxwell relation requires a negative sign. If the diagonal line intersects the “shared arc” (see text) used to build the Maxwell relation, then that relation requires a negative sign.
(Color online) The thermodynamic truncated octahedron, labeled to match the thermodynamic Venn diagram. For best results, this template should be reproduced on cardstock, cut, folded at each intersection, and taped from the five free sides of the hexagon to the , T, G, P, and regions. When the U hexagon is facing up and the G hexagon is facing forward, this thermodynamic truncated octahedron can be easily compared to the thermodynamic Venn diagram (Fig. 2). To find the Maxwell relations, rotate the truncated octahedron until it rests on one of the squares (, P, or T work best); then the four squares lying along the equator corresponds to the four variables lying along one of the three circles in Fig. 2 (chemical, mechanical, or thermal, if resting on the , P, or T squares), and the same procedure to obtain Maxwell relations described for Fig. 2 can be used. If the “shared second side” (see text) is along a dark black edge, the Maxwell relation should have a negative sign. These dark edges are all in the equatorial position when the truncated octahedron rests on the hexagon, as is dictated by the physics that governs the signs of the Maxwell relations (see text).
(Color online) The Venn diagram for the Massieu functions.
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