An observation by O. L. Anderson that for close-packed minerals, (∂KT/∂T)V=0, approximately, implies without further assumption the Birch relationship (αKT)T=const, to a similar approximation. At sufficiently high temperatures (and in materials that are not too near to phase transitions) we may also make the classical assumption CV=const, and then we have the thermal Gr¿neisen parameter &ggr; inversely proportional to density &rgr;. The fact that the free volume formulation for &ggr; is also believed to be a good approximation for close-packed materials leads immediately to an equation of state. In integrated form this is P= (Ko/2&ggr;o)(&rgr;/&rgr;o)4/3{exp<2&ggr;o(1-&rgr;o/&rgr;)>-1}. Earth model data for the lower mantle, extrapolated to zero temperature, give only modest agreement with the new equation and indicate that in the lower mantle, &ggr; is less dependent upon &rgr; than the equation implies, although doubt about the model data is such that this is not a secure conclusion. Nevertheless, the agreement is sufficient to give reasonable zero-pressure parameters for the lower mantle, &rgr;o=4180¿30 kg m-3, Ko= (2.42¿0.05) ¿1011 Pa, and &ggr;o=1.17¿0.05, and to indicate a mean temperature gradient (1.3¿0.3) times adiabatic with a value extrapolated to the core-mantle boundary (3200¿900) K. However, the model data are consistent with a much steeper gradient in the lowest 200 km of the mantle, and our preferred fit to the data omits this range. The implied interatomic potential function has a Coulomb attractive term and a repulsion term in the form of an incomplete gamma function.