Birch–Murnaghan equation of state

The Birch–Murnaghan isothermal equation of state, published in 1947 by Francis Birch of Harvard, is a relationship between the volume of a body and the pressure to which it is subjected. This equation is named after Albert Francis Birch and Francis Dominic Murnaghan. Birch proposed this equation in a publication in 1947, based on the work of Murnaghan of Johns Hopkins University published in 1944.

Expressions for the equation of state

The third-order Birch–Murnaghan isothermal equation of state is given by:

P(V)={\frac {3B_{0}}{2}}\left[\left({\frac {V_{0}}{V}}\right)^{{\frac {7}{3}}}-\left({\frac {V_{0}}{V}}\right)^{{\frac {5}{3}}}\right]\left\{1+{\frac {3}{4}}\left(B_{0}^{\prime }-4\right)\left[\left({\frac {V_{0}}{V}}\right)^{{\frac {2}{3}}}-1\right]\right\}.

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\left({\frac {\partial P}{\partial V}}\right)_{{P=0}}

and

B_{0}'=\left({\frac {\partial B}{\partial P}}\right)_{{P=0}}

The expression for the equation of state is obtained by expanding the free energy f in the form of a series:

f={\frac {1}{2}}\left[\left({\frac {V}{V_{0}}}\right)^{{-{\frac {2}{3}}}}-1\right]\,.

The internal energy, E(V), is found by integration of the pressure:

E(V)=E_{0}+{\frac {9V_{0}B_{0}}{16}}\left\{\left[\left({\frac {V_{0}}{V}}\right)^{{\frac {2}{3}}}-1\right]^{3}B_{0}^{\prime }+\left[\left({\frac {V_{0}}{V}}\right)^{{\frac {2}{3}}}-1\right]^{2}\left[6-4\left({\frac {V_{0}}{V}}\right)^{{\frac {2}{3}}}\right]\right\}.

References

^ Birch, Francis (1947). "Finite Elastic Strain of Cubic Crystals". Physical Review. 71 (11): 809–824. Bibcode:1947PhRv...71..809B. doi:10.1103/PhysRev.71.809.
^ Murnaghan, F. D. (1944). "The Compressibility of Media under Extreme Pressures". Proceedings of the National Academy of Sciences of the United States of America. 30 (9): 244–247. Bibcode:1944PNAS...30..244M. doi:10.1073/pnas.30.9.244. JSTOR 87468. PMC 1078704. PMID 16588651.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Statistical mechanics
Theory	Principle of maximum entropy ergodic theory
Statistical thermodynamics	Ensembles partition functions equations of state thermodynamic potential: U H F G Maxwell relations
Models	Ferromagnetism models Ising Potts Heisenberg percolation Particles with force field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean field theory and conformal field theory
Critical phenomena	Phase transition Critical exponents correlation length size scaling
Entropy	Boltzmann Shannon Tsallis Rényi von Neumann
Applications	Statistical field theory elementary particle superfluidity condensed matter physics complex system chaos information theory Boltzmann machine

Birch–Murnaghan equation of state

Expressions for the equation of state

See also

References