Birch–Murnaghan equation of state explained

The Birch–Murnaghan isothermal equation of state, published in 1947 by Albert Francis Birch of Harvard,^[1] is a relationship between the volume of a body and the pressure to which it is subjected. Birch proposed this equation based on the work of Francis Dominic Murnaghan of Johns Hopkins University published in 1944,^[2] so that the equation is named in honor of both scientists.

Expressions for the equation of state

The third-order Birch–Murnaghan isothermal equation of state is given by$$P(V)=\backslash frac\backslash left[\backslash left(\backslash frac\{V\_0\}\{V\}\backslash right)^\{7/3\}\; -\; \backslash left(\backslash frac\{V\_0\}\{V\}\backslash right)^\{5/3\}\backslash right]\backslash left\backslash .$$where P is the pressure, V₀ is the reference volume, V is the deformed volume, B₀ is the bulk modulus, and B₀' is the derivative of the bulk modulus with respect to pressure. The bulk modulus and its derivative are usually obtained from fits to experimental data and are defined as$$B\_0\; =\; -V\; \backslash left(\backslash frac\backslash right)\_$$and$$B\_0\text{'}\; =\; \backslash left(\backslash frac\backslash right)\_$$The expression for the equation of state is obtained by expanding the Helmholtz free energy in powers of the finite strain parameter f, defined as$$f\; =\; \backslash frac\backslash left[\backslash left(\backslash frac\{V\_0\}\{V\}\backslash right)^\{\{2/3\}\}\; -\; 1\backslash right]\; \backslash ,,$$in the form of a series.^[3] This is more evident by writing the equation in terms of f. Expanded to third order in finite strain, the equation reads,^[3] $$P(f)\; =\; 3\; B\_0\; f\; (1\; +\; 2\; f)^\; (1\; +\; a\; f\; +\; \backslash mathit)\backslash ,,$$with

a = (3)/(2) (B _{0'} - 4 ) .

The internal energy,, is found by integration of the pressure:$$E(V)\; =\; E\_0\; +\; \backslash frac\backslash left\backslash .$$

See also

• Albert Francis Birch
• Francis Dominic Murnaghan
• Murnaghan equation of state

Notes and References

1. 10.1103/PhysRev.71.809 . Finite Elastic Strain of Cubic Crystals . 1947 . Birch . Francis . Physical Review . 71 . 11 . 809–824. 1947PhRv...71..809B .
2. 87468 . 244–247 . Murnaghan . F. D. . The Compressibility of Media under Extreme Pressures . 30 . 9 . Proceedings of the National Academy of Sciences of the United States of America . 1944 . 16588651 . 1078704. 1944PNAS...30..244M . 10.1073/pnas.30.9.244 . free .
3. Book: Poirier, J.-P. . "Introduction to the Physics of the Earth's Interior" . 2000 . Cambridge . 9781139164467.