Thermodynamics with the Grüneisen parameter: Fundamentals and applications to high pressure physics and geophysics

Abstract The Gruneisen parameter, γ, conventionally written as a dimensionless combination of familiar properties, expansion coefficient, bulk modulus, density and specific heat, can also be presented in terms of elastic moduli and their pressure derivatives, providing a quantitative link between thermal and mechanical parameters. Useful relationships include the adiabatic variation of temperature with density, ( ∂ l n T / ∂ l n ρ ) S = γ , an interestingly close analogue of the ideal gas equation, from which the efficiency of a heat engine operating between states 1 and 2 can be written in terms of the density ratio corresponding to the adiabatic temperature ratio for the states, η = ( ρ 1 / ρ 2 ) γ - 1 . This applies to convectively driven tectonic activity in the Earth, yielding a value of efficiency in terms of γ and density only, without requiring knowledge of absolute temperatures. A further development gives the melting point variation, d l n T M / d l n ρ = 2 γ , in terms of density on the melting curve, showing why convecting planets solidify from the inside outwards. Recent developments include the properties and uses of derivatives of γ, especially λ = ( ∂ l n q / ∂ l n V ) T , where q = ( ∂ l n γ / ∂ l n V ) T . Its infinite pressure extrapolation, λ ∞ , is used to impose constraints on high pressure equations of state. One application is to the search for a ‘universal’ equation, written as a Taylor expansion about the infinite pressure limit, but it yields coefficients that are indeterminate, inviting the inference that there can be no such general form of high pressure equation. Other thermodynamic problems that can be presented in a new light by applying γ are the theory of thermal expansion and the pressure dependence of the Debye temperature.