Thermodynamically, the Earth is a heat engine
described by a variety of parameters that can be determined
from equations of state (EOS) and models of condensed media. A
great progress has been achieved in the development of such
models [e.g., * Jeanloz,* 1983;
* Hemley et al.,* 1985, 1987;
* Wall et al.,* 1986;
* Catti,* 1986;
* Cohen,* 1987a, 1987b;
* Dovesi et al.,* 1987;
* Wolf and Bukowinski,* 1987, 1988;
* Wall and Price,* 1988;
* Matsui et al.,* 1987;
* Matsui,* 1988, 1989;
* Price et al.,* 1989;
* Catlow and Price,* 1990;
* Isaak et al.,* 1990;
* Reynard and Price,* 1990;
* Agnon and Bukowinski,* 1990a;
* Matsui and Price,* 1991;
* D'Arco et al.,* 1991;
* Walzer,* 1992;
* Silvi et al.,* 1993;
* Catti et al.,* 1993;
* Boison and Gibbs,* 1993]. Nevertheless, practical studies
in
geophysics are based, to a large extent, on the use of
semi-empirical EOS's [*Birch,* 1952, 1986;
* O. Anderson,* 1966b, 1995;
* Pankov and Ullmann,* 1979a, 1979b;
* D. Anderson,* 1967, 1987, 1989;
* Stacey,* 1981;
* Leliva-Kopystynski,* 1991;
* Bina and Helffrich,* 1992;
* Wall et al.,* 1993]. The properties of
geomaterials directly determined from laboratory measurements at
high pressures and temperatures are necessary for solving many
geophysical problems and provide important constraints on the
EOS structure.

Since the fundamental paper of * Birch* [1952], a great
deal of
information has been accumulated on the properties of
geomaterials and their geophysical implication
[e.g., * Stacey,* 1977a, 1977b,
1992, 1994;
* Jeanloz and Thompson,* 1983;
* Brown and Shankland,* 1981;
* Zharkov and Kalinin,* 1971;
* Zharkov,* 1986;
* Jeanloz and Knittle,* 1989;
* O. Anderson et al.,* 1992a, 1992b;
1993;
* Kuskov and Panferov,* 1991;
* D. Anderson,* 1989;
* O. Anderson,* 1988, 1995].

This paper is devoted to the review of relationships
between the basic thermodynamic characteristics and of their
variation with pressure and temperature. First, we deal
with eight parameters of the second order. We emphasize their
self-consistent determination and the relations to EOS's and
give a summary of approaches used to find the empirically based
EOS's. An example of the thermodynamically consistent database
for mantle minerals is presented. Then, each of the second-order
parameters is treated separately: the identities involving their
*P*-*T* derivatives (third-order parameters) are established and
practically useful approximations are analyzed, including some
explicit
*P*-*T* dependences of the second-order parameters. Some
estimates for the fourth-order parameters are also given. The
relations between various quantities are represented in the form
convenient for practical use of experimental data and for
theoretical analysis. Finally, a number of estimates are given
for the low-mantle properties. Our analysis serves as an
addition to the reviews of * O. Anderson* [1995]
and * Stacey* [1994].

In classical thermodynamics, simple systems experiencing
reversable changes of state are described by a variety of
parameters including the hydrostatic pressure
*P*, temperature
*T*, volume
*V* (or density
*r* ), and entropy
*S*. The starting
point of thermodynamic analysis is the standard expresions for
the total thermodynamic differentials
[e.g., * Callen,* 1960;
* Morse,* 1969;
* Kelly,* 1973]

(1) |

(2) |

(3) |

(4) |

where
*E* is the internal energy,
*F* is the free energy
(Helmholtz potential),
*G* is the free enthalpy (Gibbs
potential), and
*H* is the enthalpy.

Eight second-order parameters are largely used in geophysics:
the volume coefficient of thermal expansion
*a*, the
isobaric
*C*_{P} and isochoric
*C*_{V} heat capacities, the isothermal
*K*_{T} and adiabatic
*K*_{S} bulk moduli, the thermal pressure
coefficient
*t*, and the adiabatic pressure derivative of
temperature (adiabatic temperature gradient in pressure)
*t*_{S}. The respective
definitions of these parameters are

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

By equating the cross derivatives of the four thermodynamic
potentials, we obtain the Maxwell relations
[see, e.g., * Stacey,* 1977a]

(11) |

(12) |

(13) |

(14) |

Moreover, it is easily shown that the second derivative of each of these potentials can be expressed in terms of the above parameters or coefficients; i.e., we may write the matrix

(15) |

The number of independent second-order parameters is obviously three, and consequently, the eight second-order parameters introduced above must satisfy five relations. Four of them are (11)-(14), and the fifth can be derived by changing from one pair of characteristic variables to another; specifically,

(16) |

(17) |

Thus, if the parameters
*a*,
*C*_{P}, and
*K*_{S} (or
*K*_{T} ), as
it usually is, are determined experimentally, then the
remaining five parameters can be found from the identities

(18) |

We recall two examples of using the thermodynamic relations in
geophysics. The first concerns the Williamson-Adams-Birch
equation for the density gradient within the Earth
[*Birch,* 1952;
* D. Anderson,* 1989]. We quote this equation in the
form

(19) |

where
*r* and
*P* are the density and pressure in the
Earth's interior, respectively,
*F* = *K*_{S}/*r* is the
seismic parameter, and
*t* = *dT*/*dP* - *t*_{S}
is the
superadiabatic temperature gradient. Equation (19) is easily
obtained from

with reference to (5), (16) and (14) or (18).

Another example is the adiabatic temperature gradient in depth
*l* within the Earth (see (14)) [e.g., * Quareni and Mulargia,* 1989]

(20) |

where
*g* is the gravitational acceleration, and furthermore, the
mechanical equalibrium equation
*dP*/*dl*= *r**g* is used.

According to the PREM model [*D. Anderson,* 1989],
*F* = 50,
80, and 117 km
^{2} /s
^{2} at
*l* = 400, 1071, and 2740 km,
respectively. Assuming that
*g* = 1-1.5 and
*T* = 1700,
2200, and 3000 K sequently at the indicated depths
[*D. Anderson,* 1989; * Pankov,* 1989],
(20) yields
(*dT*/*dl*)_{S} = 0.3-0.5 K/km,
the value usually cited in geophysical literature.