Editing Murnaghan equation of state

The '''Murnaghan equation of state''' was developed by Francis D. Murnaghan of John Hopkins University. He presented an [[Equations of state | equation of state]] suitable for representing solids <ref>[http://www.jstor.org/stable/2371405 F. D. Murnaghan "Finite Deformations of an Elastic Solid", American Journal of Mathematics '''59''' pp. 235-260 (1937)]</ref><ref>[http://www.pnas.org/content/30/9/244.full.pdf+html F. D. Murnaghan. "The Compressibility of Media under Extreme Pressures", Proceedings of the National Academy of Sciences of the United States of America '''30''' pp. 244-247 (1944)]</ref>.    Having high energy dependence on volume, the equation of state has found considerable use in condensed phase media.  

Three [[Thermodynamic relations |derivative relations]] are utilised to lead to the formulation, namely:

:<math>p=-\left(\frac{\partial U}{\partial V}\right)_S</math>

:<math>B=-V\left(\frac{\partial p}{\partial V}\right)_T</math>

:<math>B'=\left(\frac{\partial B}{\partial p}\right)_T</math>

Where <math>p</math> is the [[pressure]], <math>U</math> is the [[internal energy]], <math>V</math> is the volume, <math>B</math> is the [[Compressibility | bulk modulus]] and  <math>B'</math> is the pressure derivative of the bulk modulus at constant [[temperature]].  

Making the assumption that <math>B'</math> is constant allows one to assign a linear dependence of the bulk modulus on the pressure, which allows specifying one bulk modulus and derivative represented as <math>B_0</math> and <math>B_0'</math> as the material constants.  This leads to a relationship for pressure:

:<math>p=\frac{B_0}{B_0'}\left(\left(\frac{V_0}{V}\right)^{B_0'}-1\right)</math>

That can be integrated to represent the energy of the solid:

:<math>U=U_0+\frac{B_0V}{B_0'}\left(\frac{(V_0/V)^{B_0'}}{B_0'-1}+1\right)-\frac{B_0V_0}{B_0'-1}</math>

==Regions of Applicability==
Many papers have since been published on the applicability of the Murnaghan equation of state, many of which have presented alternate equation of state forms for solids.  In general, it has been shown that the Murnaghan equation of state breaks down for compression ratios greater than ~0.7-0.8 times the original volume, which occurs as a consequence of the linear dependence of the bulk modulus on pressure and constant bulk modulus pressure derivative.  Several popular forms presented to address this issue are the [[Birch-Murnaghan equation of state]], the [[Vinet (Universal) equation of state]], the [[Holzapfel equation of state]], the [[Kumari-Dass equation of state]], and the [[Baonza equation of state]].

==References==
<references/>
[[category: equations of state]]