Simon–Glatzel equation

: $T\_m\; =\; T\_\backslash text\{ref\}\; \backslash left(\; \backslash frac\{\; P\; -\; P\_\backslash text\{ref\}\; \}\{a\}\; +\; 1\; \backslash right)^\{b\}$

$T\_\backslash text\{ref\}$ and $P\_\backslash text\{ref\}$ are normally the temperature and the pressure of the triple point, but the normal melting temperature at atmospheric pressure are also commonly used as reference point because the normal melting point is much more easily accessible. Typically $P\_\backslash text\{ref\}$ is then set to 0. $a$ and $b$ are component-specific parameters.

The Simon–Glatzel equation can be viewed as a combination of the Murnaghan equation of state and the Lindemann law,{{Cite book|last=Anderson|first=Orson L.|title=Equations of State of Solids for Geophysics and Ceramic Science|publisher=Oxford University Press|year=1995|isbn=0-19-505606-X|location=|pages=281}} and an alternative form was proposed by J. J. Gilvarry (1956):{{Cite journal|last=Gilvarry|first=John James|date=1956|title=Equation of the Fusion Curve|url=https://doi.org/10.1103/PhysRev.102.325|journal=Physical Review|volume=102|issue=2|pages=325–331|doi=10.1103/PhysRev.102.325|bibcode=1956PhRv..102..325G|via=}}

$T\_m\; =\; T\_\backslash text\{ref\}\; \backslash left(K\_0^\{\text{'}\}\; \backslash frac\{P\; -\; P\_\backslash text\{ref\}\; \}\{K\_0\}\; +\; 1\; \backslash right)^\{\backslash frac\{2(\backslash gamma\; -1)\}\{3+f\}\}$

where $K\_0$ is general $K$ at $P\; =\; 0$, $K\_0^\{\text{'}\}$ is pressure derivative $K$ at $P\; =\; 0$, $\backslash gamma$ is Grüneisen ratio, and $f$ is the coefficient in Morse potential.

File:Simon-Glatzel equation.png

For methanol the following parametersDortmund Data Bank can be obtained:

class="wikitable" border="1"
a	188158	kPa
a	188.158	MPa
b−1	5.15905
Tmin	174.61	K
Tmax	228.45	K
Pmax	575000	kPa
Pmax	575.000	MPa

The reference temperature has been T_ref = 174.61 K and the reference pressure P_ref has been set to 0 kPa.

Methanol is a component where the Simon–Glatzel works well in the given validity range.

The Simon–Glatzel equation is a monotonically increasing function. It can only describe the melting curves that rise indefinitely with increasing pressure. It may fail to describe the melting curves with a negative pressure dependence or local maximums. A damping term that asymptotically slopes down under pressure, $D(P)\; =\; \backslash exp[-c(P\; -\; P\_\backslash text\{ref\})]$ (c is another component-specific parameter), is introduced by Vladimir V. Kechin to extend the Simon–Glatzel equationKechin V. V., J. Phys. Condens. Matter, 1995, 7, 531–535 so that all melting curves, rising, falling, and flattening, as well as curves with a maximum, can be described by a unified equation:

$T\_m\; =\; F(P)\backslash cdot\; D(P)$

where $F(P)$ is the Simon–Glatzel equation (rising) and $D(P)$ is the damping term (falling or flattening).

The unified equation may be rewritten as:

$T\_m\; =\; T\_\backslash text\{ref\}\; \backslash left(\; \backslash frac\{\; P\; -\; P\_\backslash text\{ref\}\; \}\{a\}\; +\; 1\; \backslash right)^\{b\}\backslash cdot\; \backslash exp[-c\; (P\; -\; P\_\backslash text\{ref\})]$

This form predicts that all solids have a maximum melting temperature at a positive or (fictitious) negative pressure.

{{reflist}}

{{DEFAULTSORT:Simon-Glatzel equation}}

Category:Phase transitions

Category:Equations

Category:Thermodynamics