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We derive exact results for the rate of change of thermodynamic quantities,in particular the configurational specific heat at constant volume, $C_V$,along configurational adiabats (curves of constant excess entropy$S_{\textrm{ex}}$). Such curves are designated isomorphs for so-called Roskildeliquids, in view of the invariance of various structural and dynamicalquantities along them. Their slope in a double logarithmic representation ofthe density-temperature phase diagram, $\gamma$ can be interpreted as one thirdof an effective inverse power-law potential exponent. We show that in liquidswhere $\gamma$ increases (decreases) with density, the contours of $C_V$ havesmaller (larger) slope than configurational adiabats. We clarify also theconnection between $\gamma$ and the pair potential. A fluctuation formula forthe slope of the $C_V$-contours is derived. The theoretical results aresupported with data from computer simulations of two systems, the Lennard-Jonesfluid and the Girifalco fluid. The sign of $d\gamma/d\rho$ is thus a third keyparameter in characterizing Roskilde liquids, after $\gamma$ and thevirial-potential energy correlation coefficient $R$. To go beyond isomorphtheory we compare invariance of a dynamical quantity, the self-diffusioncoefficient along adiabats and $C_V$-contours, finding it more invariant alongadiabats.

Statistical mechanics of Roskilde liquids: Configurational adiabats, specific heat contours and density dependence of the scaling exponent