Murnaghan equation of state - Revision history

Proxima Centari: Fixed red-link

2021-04-07T03:20:52Z

Fixed red-link

← Older revision		Revision as of 05:20, 7 April 2021
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	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.		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:		Three [[Thermodynamic relations \|derivative relations]] are utilised to lead to the formulation, namely:
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	==Regions of Applicability==		==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]].		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 [[Rose-Vinet (Universal) equation of state\|Vinet (Universal) equation of state]], the [[Holzapfel equation of state]], the [[Kumari-Dass equation of state]], and the [[Baonza equation of state]].

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

Carl McBride: Added another reference

2011-11-03T11:09:02Z

Added another reference

← Older revision		Revision as of 13:09, 3 November 2011
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	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.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.		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:		Three [[Thermodynamic relations \|derivative relations]] are utilised to lead to the formulation, namely:

Carl McBride: Added some internal links + slight tidy.

2011-11-03T10:13:48Z

Added some internal links + slight tidy.

← Older revision		Revision as of 12:13, 3 November 2011
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	Francis D. Murnaghan of John Hopkins University presented an equation of state suitable for representing solids <ref> [http://www.~~pubmedcentral~~.~~nih~~.~~gov/articlerender~~.~~fcgi?tool=pmcentrez&artid=1078704~~ F.D. Murnaghan. "The Compressibility of Media under Extreme Pressures~~. ''~~Proceedings of ~~hte~~ National Academy of Sciences of the United States of America'', 30~~(9):224~~-247, 1944] </ref>. ~~The equation has become known as the '''Murnaghan''' equation of state.~~ Having high energy dependence on volume, the equation of state has found considerable use in condensed phase media.		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.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 derivative relations are ~~assumed that~~ lead to the formulation:		Three [[Thermodynamic relations \|derivative relations]] are utilised to lead to the formulation, namely:

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

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

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

	Where <math>P</math>, <math>E</math>, <math>V</math>, <math>K</math> and <math>K'</math> ~~are the pressure, energy, volume, bulk modulus, and~~ the pressure derivative of the bulk modulus at constant temperature.		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>K'</math> is constant allows to assign a linear dependence of the bulk modulus on the pressure, which allows specifying one bulk modulus and derivative represented as <math>~~K_0~~</math> and <math>~~K_0~~'</math> as the material constants. This leads to a relationship for pressure:		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{~~K_0~~}{~~K_0~~'}\left(\left(\frac{V_0}{V}\right)^{~~K_0~~'}-1\right)</math>		:<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:		That can be integrated to represent the energy of the solid:

	:<math>E=~~E_0~~+\frac{~~K_0V~~}{~~K_0~~'}\left(\frac{(V_0/V)^{~~K_0~~'}}{~~K_0~~'-1}+1\right)-\frac{~~K_0V_0~~}{~~K_0~~'-1}</math>		:<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==		==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.		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==
	<references/>		<references/>
			[[category: equations of state]]

Josephrpeterson: Created page with "Francis D. Murnaghan of John Hopkins University presented an equation of state suitable for representing solids [http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pm..."

2011-11-02T16:30:56Z

Created page with "Francis D. Murnaghan of John Hopkins University presented an equation of state suitable for representing solids <ref> [http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pm..."

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Francis D. Murnaghan of John Hopkins University presented an equation of state suitable for representing solids <ref> [http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1078704 F.D. Murnaghan. "The Compressibility of Media under Extreme Pressures. ''Proceedings of hte National Academy of Sciences of the United States of America'', 30(9):224-247, 1944] </ref>. The equation has become known as the '''Murnaghan''' equation of state. Having high energy dependence on volume, the equation of state has found considerable use in condensed phase media.

Three derivative relations are assumed that lead to the formulation:

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

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

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

Where <math>P</math>, <math>E</math>, <math>V</math>, <math>K</math> and <math>K'</math> are the pressure, energy, volume, bulk modulus, and the pressure derivative of the bulk modulus at constant temperature.

Making the assumption that <math>K'</math> is constant allows to assign a linear dependence of the bulk modulus on the pressure, which allows specifying one bulk modulus and derivative represented as <math>K_0</math> and <math>K_0'</math> as the material constants. This leads to a relationship for pressure:

:<math>P=\frac{K_0}{K_0'}\left(\left(\frac{V_0}{V}\right)^{K_0'}-1\right)</math>

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

:<math>E=E_0+\frac{K_0V}{K_0'}\left(\frac{(V_0/V)^{K_0'}}{K_0'-1}+1\right)-\frac{K_0V_0}{K_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/>