The manifold of equilibrium states for a fixed number of moles of ideal gas, when provided with a Riemannian metric based on the second derivatives of internal energy (studied by F. Weinhold), is found to have zero intrinsic curvature, in fact to be isometric to the Riemann surface of the natural logarithmic function. This and three other closely related flat spaces associated with an ideal gas are studied by means of explicit isometries.
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Nulton et al. (1985) studied this question.
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