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I'm familiar with the ideal gas law:

$PV = nRT$,

Which is valid for ideal gases. One can move forward and write EOS for non-ideal gases, liquids and gases. But what is the equation of state for a perfect solid. If possible can you provide a reference for further reading?

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  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$
    – Kyle Kanos
    Commented Sep 10, 2018 at 12:36
  • $\begingroup$ The link you give is for real solids not perfect solids. $\endgroup$
    – wander95
    Commented Sep 10, 2018 at 16:58
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    $\begingroup$ Will you define a 'perfect solid' in the question? $\endgroup$
    – Paul T.
    Commented Sep 10, 2018 at 18:48
  • $\begingroup$ A mathematical definition would answer my question. But by an ideal solid, I mean an object that does not deform or shear under the influence of a force of any magnitude and over any time-scale. $\endgroup$
    – wander95
    Commented Sep 11, 2018 at 12:24

2 Answers 2

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It's common to write an equation of state (EOS) as the density as a function of pressure (and other things): $\rho(P,...)$. In this form the ideal gas EOS is:

$$\rho(P,T) = \frac{P}{RT},$$ where $\rho$ is the number density $n/V$. The EOS tells you how squishy something is. If you raise the pressure (squeeze), how much does the density increase (compress). For the ideal gas, it also tells us that when we increase the temperature the density goes down (gas expands).

The answer to your questions, depends on what you mean by perfect solid. To me, a perfect solid would be incompressible, so $$\rho(P) = \rho_0 = \mathrm{constant}.$$ No matter how hard you squeeze the solid doesn't compress, and the density stays the same.

In a comment @kyle-kanos links to the Wikipedia article Equation of State, where a list of EOSs for solids is given.

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  • $\begingroup$ I thought that $\rho=\rho_0$ and $V=V_0$ would follow from continuity equation $\endgroup$
    – wander95
    Commented Sep 10, 2018 at 17:16
  • $\begingroup$ A perfect solid can still expand and contract with temperature. $\endgroup$
    – John Alexiou
    Commented Sep 11, 2018 at 0:16
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This is a very open-ended and subjective question with many possible answers. The ideal gas model is not used necessarily because it models a 'perfect' gas, but because it is very easy to analyze and yet encapsulates some of the important thermodynamic properties of real gases. So it depends on what you are interested in modeling or understanding. If you are interested in phase transitions, for example, then the ideal gas model is no longer very helpful, and the right 'ideal' model might be something like the Ising or Lennard-Jones models.

So to answer your question, it depends on what you are interested in modeling. I don't believe the answer given by Paul T. is actually physical. If a solid is not compressible at all, it would among other things violate relativity.

A better example of an ideal solid might be a perfect classical crystal at near-absolute-zero. This could be modeled very simply as ideal point particles connected by ideal springs in a crystal lattice structure. Models of this sort are useful for things like analysis of vibration modes, stress-strain behavior, and heat capacity. In this model, the solid resists shear and has an elastic regime, as you would expect from a solid as opposed to a liquid.

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  • $\begingroup$ Yes, lets make the approximation that we are studying a classical Newtonian phenomena. Say the motion of an object with a fluid. $\endgroup$
    – wander95
    Commented Sep 11, 2018 at 12:21

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