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

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Discussion Overview

The discussion centers on the mathematical derivation of the Birch-Murnaghan equation, which describes the volume behavior of materials under high pressure, specifically above 1 GPa. Participants seek clarity on the derivation process and relevant literature.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about the mathematical derivation of the Birch-Murnaghan equation, emphasizing its application at high pressures.
  • Another participant suggests that Birch's original paper may contain a derivation and provides a reference to it, noting that the derivation is isothermal.
  • Participants share a link to a PDF that purportedly contains relevant derivation information.
  • There is a clarification regarding the notation used in the equation, with one participant suggesting that F refers to Helmholtz free energy rather than force.

Areas of Agreement / Disagreement

There is no consensus on the derivation process itself, and multiple views regarding the sources of information and interpretations of notation are present.

Contextual Notes

The discussion does not resolve the specific steps of the derivation, and participants express uncertainty about the definitions and interpretations of terms used in the equation.

Sylvester
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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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