A How was the Birch-Murnaghan equation derived for high pressure volume behavior?

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The discussion centers around the mathematical derivation of an equation used to describe volume behavior at high pressures above 1 GPa, with Vo representing ambient volume and Bo as the bulk modulus. Participants suggest that Birch's 1947 paper, "Finite Elastic Strain of Cubic Crystals," may contain the derivation needed. A PDF link was shared, which one participant found helpful for understanding the topic. There is a clarification regarding the variable F, which is identified as the Helmholtz free energy rather than force. The conversation highlights the challenges in finding comprehensive literature on this specific derivation.
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Hello, physicists! Does anyone know how was this equation (below) mathematically derived? It is basically used to describe the behavior of volume at very high pressures (above 1 GPa). Vo designates the volume at ambient pressure. Bo is the bulk modulus and B’ its pressure derivative. I can't find anything in the literature where there is a step-by-step or at least a clue on how to come up with this expression.
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I found this with google
https://mcbrennan.github.io/BMderivation.pdf

I would guess that Birch‘s paper has a derivation
Birch,F., “Finite Elastic Strain of Cubic Crystals,” Physical Review 71 (11),809–824 (1947).

Notice that it is isothermal.
 
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Frabjous said:
I found this with google
https://mcbrennan.github.io/BMderivation.pdf

I would guess that Birch‘s paper has a derivation
Birch,F., “Finite Elastic Strain of Cubic Crystals,” Physical Review 71 (11),809–824 (1947).

Notice that it is isothermal.
Hey Frabjous! Thanks for the material. Just now, I briefly checked the pdf file and I think that's more than enough. Again, thank you so much for the help!
 
Sylvester said:
Hey Frabjous! Thanks for the material. Just now, I briefly checked the pdf file and I think that's more than enough. Again, thank you so much for the help!
You are welcome.

BTW, I could be wrong, but I think F is the Helmholtz free energy, not force.
 
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