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The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): = = Here is the thermal expansion coefficient: = is the isothermal compressibility (the inverse of the bulk modulus):
See relations between specific heats for the derivation of the thermodynamic relations between the heat capacities. The above definition is the approach used to develop rigorous expressions from equations of state (such as Peng–Robinson ), which match experimental values so closely that there is little need to develop a database of ratios or ...
Also for substances that are nearly incompressible, such as solids and liquids, the difference between the two specific heats is negligible. As the absolute temperature of the system approaches zero, since both heat capacities must generally approach zero in accordance with the Third Law of Thermodynamics , the difference between C P ,m and C V ...
This relation was built on the reasoning that energy must be supplied to raise the temperature of the gas and for the gas to do work in a volume changing case. According to this relation, the difference between the specific heat capacities is the same as the universal gas constant. This relation is represented by the difference between Cp and Cv:
Maxwell's relations are a set of equations in thermodynamics which are derivable from the symmetry of second derivatives and from the definitions of the thermodynamic potentials. These relations are named for the nineteenth-century physicist James Clerk Maxwell .
Clausius–Clapeyron relation; Departure functions; Duhem–Margules equation; Ehrenfest equations; Gibbs–Helmholtz equation; Phase rule; Kopp's law; Noro–Frenkel law of corresponding states; Onsager reciprocal relations; Stefan number; Thermodynamics; Timeline of thermodynamics; Triple product rule; Exact differential
A derivation is discussed in the article Relations between specific heats. For an ideal gas, if is expressed as molar density in the above equation, this equation reduces simply to Mayer's relation, ,, =
The relation is generally expressed as a microscopic change in internal energy in terms of microscopic changes in entropy, and volume for a closed system in thermal equilibrium in the following way. d U = T d S − P d V {\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V\,}