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I want to ask how the pressure in a solid affects its temperature. If there are a broad range of relationships, perhaps what subject should I search for to learn more? The context for this is I want to know how hot the core of a ball of matter will be, based on the fact it will increase due to the pressure from the surrounding mass.

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    $\begingroup$ This is a rather broad question. When you pressurize a material, it may heat up (if there is internal friction) or it may simply store the energy elastically. You really need to narrow down your question if you want any helpful answers. And "based on the fact it will increase due to the pressure from the surrounding mass" is a statement without any context or reference - and not in general true... Please elaborate where you got that from, and narrow your question. $\endgroup$ – Floris Feb 8 '17 at 14:13
  • $\begingroup$ It's an ill-posed question, but pressure might have a slight effect on temperature in an adiabatic compression, such as that which occurs in a sound wave. $\endgroup$ – Bert Barrois Feb 9 '18 at 13:02
  • $\begingroup$ @BertBarrois It could be based on the observation that temperatures inside the Earth increase with depth. Some of that is from radioactivity, but is the rest due solely to heat left over from the Earth's formation? If you created a large enough ball of solid iron (bigger than a star) would the core be hotter than the surface due to pressure, and by how much? $\endgroup$ – Michael Sep 13 '18 at 2:02
  • $\begingroup$ @ Michael -- The heat (power) source was once thought to be radioactivity, but thorium is chemically litho- rather than siderophilic. Current thinking says it’s the separation and settling of heavier elements, which releases gravitational potential energy. The power is transported through the liquid outer core mainly by convection, and the temperature gradient in convective systems is roughly the adiabatic gradient. The convection currents have the fortuitous side-effect of sustaining the geomagnetic field. $\endgroup$ – Bert Barrois Sep 13 '18 at 11:53
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To a good first approximation, the equation of state for an isotropic solid can be expressed as:

$$\frac{V}{V_0}=e^{\left[\alpha(T-T_0)-\frac{(P-P_0)}{K}\right]}$$ where $\alpha$ is the volumetric coefficient of thermal expansion, K is the bulk modulus, and V is the volume of the solid at temperature $T_0$ and pressure $P_0$. As with a gas, this equation can be combined with the first law of thermodynamics to consider the type of specific situation (e.g., adiabatic reversible compression) you are interested in. To do this, you also need to also know the relationship between the internal energy and specific volume and temperature of a solid.

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