Material properties (thermodynamics)

\quad = -\frac{T}{N}\,\frac{\partial^2 G}{\partial T^2} :* Specific heat at constant volume ::c_V=\frac{T}{N}\left(\frac{\partial S}{\partial T}\right)_V \quad = -\frac{T}{N}\,\frac{\partial^2 A}{\partial T^2} - **Coefficient of thermal expansion** ::\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P \quad = \frac{1}{V}\,\frac{\partial^2 G}{\partial P\partial T} where *P* is pressure, *V* is volume, *T* is temperature, *S* is entropy, and *N* is the number of particles. For a single component system, only three second derivatives are needed in order to derive all others, and so only three material properties are needed to derive all others. For a single component system, the "standard" three parameters are the isothermal compressibility \kappa_T, the specific heat at constant pressure c_P, and the coefficient of thermal expansion \alpha. For example, the following equations are true: :c_P=c_V+\frac{TV\alpha^2}{N\kappa_T} :\kappa_T=\kappa_S+\frac{TV\alpha^2}{Nc_P} The three "standard" properties are in fact the three possible second derivatives of the Gibbs free energy with respect to temperature and pressure. Moreover, considering derivatives such as \frac{\partial^3 G}{\partial P \partial T^2} and the related Schwartz relations, shows that the properties triplet is not independent. In fact, one property function can be given as an expression of the two others, up to a reference state value. The second principle of thermodynamics has implications on the sign of some thermodynamic properties such isothermal compressibility.Israel, R. (1979). Convexity in the Theory of Lattice Gases. Princeton, New Jersey: Princeton University Press. doi:10.2307/j.ctt13x1c8g ## References - {{cite book ## References 1. S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States", [https://arxiv.org/abs/2105.04845 Link to Archiv e-print] [https://hal.archives-ouvertes.fr/hal-03216379/ Link to Hal e-print] ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Material_properties_(thermodynamics)) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Material_properties_(thermodynamics)?action=history). ::