General number field sieve: Difference between revisions

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A eutectic system is a mixture of chemical compounds or elements that have a single chemical composition that solidifies at a lower temperature than any other composition made up of the same ingredients. This composition is known as the eutectic composition and the temperature at which it solidifies is known as the eutectic temperature.

On a phase diagram the intersection of the eutectic temperature and the eutectic composition gives the eutectic point.^[1] Non-eutectic mixtures will display solidification of one component of the mixture before the other. Not all binary alloys have a eutectic point; for example, in the silver-gold system the melt temperature (liquidus) and freeze temperature (solidus) both increase monotonically as the mix changes from pure silver to pure gold.^[2]

Eutectic reaction

The eutectic reaction is defined as follows:^[3]

{\text{Liquid}}{\xrightarrow[{\text{cooling}}]{\text{eutectic temperature}}}\alpha \,\,{\text{solid solution}}+\beta \,\,{\text{solid solution}}

This type of reaction is an invariant reaction, because it is in thermal equilibrium; another way to define this is the Gibbs free energy equals zero. Tangibly, this means the liquid and two solid solutions all coexist at the same time and are in chemical equilibrium. There is also a thermal arrest for the duration of the change of phase during which the temperature of the system does not change.^[3]

The resulting solid macrostructure from a eutectic reaction depends on a few factors. The most important factor is how the two solid solutions nucleate and grow. The most common structure is a lamellar structure, but other possible structures include rodlike, globular, and acicular.^[4]

Non-eutectic compositions

Compositions of eutectic systems that are not the eutectic composition are called either hypoeutectic or hypereutectic. Hypoeutectic compositions are those to the left of the eutectic composition while hypereutectic compositions are those to the right. As the temperature of a non-eutectic composition is lowered the liquid mixture will precipitate one component of the mixture before the other.^[3]

Types

Alloys

Eutectic alloys have two or more materials and have a eutectic composition. When a non-eutectic alloy solidifies, its components solidify at different temperatures, exhibiting a plastic melting range. Conversely, when a well-mixed, eutectic alloy melts, it does so at a single, sharp temperature. The various phase transformations that occur during the solidification of a particular alloy composition can be understood by drawing a vertical line from the liquid phase to the solid phase on the phase diagram for that alloy.

Some uses include:

Others

Sodium chloride and water form a eutectic mixture whose eutectic point is −21.2˚C^[5] and 23.3% salt by mass.^[6] The eutectic nature of salt and water is exploited when salt is spread on roads to aid snow removal, or mixed with ice to produce low temperatures (for example, in traditional ice cream making).
Ethanol–water has an unusually biasedTemplate:Huh eutectic point.
Solid-liquid phase change of ethanol water mixtures
"Solar salt", 60% NaNO₃ and 40% KNO₃, forms a eutectic molten salt mixture which is used for thermal energy storage in concentrated solar power plants.^[7] To reduce the eutectic melting point in the solar molten salts calcium nitrate is used in the following proportion: 42% Ca(NO₃)₂, 43% KNO_3, and 15% NaNO₃.
Lidocaine and prilocaine—both are solids at room temperature—form a eutectic that is an oil with a Template:Convert melting point that is used in eutectic mixture of local anesthetic (EMLA) preparations.
Menthol and camphor, both solids at room temperature, form a eutectic that is a liquid at room temperature in the following proportions: 8:2, 7:3, 6:4, and 5:5. Both substances are common ingredients in pharmacy extemporaneous preparations.Template:Clarification neededPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
Minerals may form eutectic mixtures in igneous rocks, giving rise to characteristic intergrowth textures exhibited by granophyre.^[8]
Some inks are eutectic mixtures, allowing inkjet printers to operate at lower temperatures.^[9]

Other critical points

Eutectoid

When the solution above the transformation point is solid, rather than liquid, an analogous eutectoid transformation can occur. For instance, in the iron-carbon system, the austenite phase can undergo a eutectoid transformation to produce ferrite and cementite, often in lamellar structures such as pearlite and bainite. This eutectoid point occurs at Template:Convert and about 0.76% carbon.^[10]

Peritectoid

A peritectoid transformation is a type of isothermal reversible reaction that have two solid phases reacting with each other upon cooling of a binary, ternary, ..., $n\!$ alloy to create a completely different and single solid phase.^[11] The reaction plays a key role in the order and decomposition of quasicrystalline phases in several alloy types.^[12]

Peritectic

Peritectic transformations are also similar to eutectic reactions. Here, a liquid and solid phase of fixed proportions react at a fixed temperature to yield a single solid phase. Since the solid product forms at the interface between the two reactants, it can form a diffusion barrier and generally causes such reactions to proceed much more slowly than eutectic or eutectoid transformations. Because of this, when a peritectic composition solidifies it does not show the lamellar structure that is found with eutectic solidification.

Such a transformation exists in the iron-carbon system, as seen near the upper-left corner of the figure. It resembles an inverted eutectic, with the δ phase combining with the liquid to produce pure austenite at Template:Convert and 0.17% carbon.

Peritectic decomposition. Up to this point in the discussion transformations have been addressed from the point of view of cooling. They also can be discussed noting the changes that occur to some solid chemical compounds as they are heated. Rather than melting, at the peritectic decomposition temperature, the compound decomposes into another solid compound and a liquid. The proportion of each is determined by the lever rule. The vocabulary changes slightly. Just as the cooling of water, which leads to ice, is termed freezing, the warming of ice leads to melting. In the Al-Au phase diagram, for example, it can be seen that only two of the phases melt congruently, AuAl₂ and Au₂Al. The rest peritectically decompose.

Eutectic calculation

The composition and temperature of a eutectic can be calculated from enthalpy and entropy of fusion of each components.^[13]

The free Gibbs enthalpy G depends on its own differential by Eq.

$G=H-TS\Rightarrow {\left\{{\begin{array}{l}H=G+TS\\\\{\left({\frac {\partial G}{\partial T}}\right)_{P}=-S}\end{array}}\right.}\Rightarrow H=G-T\left({\frac {\partial G}{\partial T}}\right)_{P}.$

Thus, the G/T derivative at constant pressure is calculated by equation Eq.

$\left({\frac {\partial G/T}{\partial T}}\right)_{P}={\frac {1}{T}}\left({\frac {\partial G}{\partial T}}\right)_{P}-{\frac {1}{T^{2}}}G=-{\frac {1}{T^{2}}}\left({G-T\left({\frac {\partial G}{\partial T}}\right)_{P}}\right)=-{\frac {H}{T^{2}}}$

The chemical potential $\mu _{i}$ is calculated if we assume the activity is equal to the concentration we suppose the activity equal to the concentration.

$\mu _{i}=\mu _{i}^{\circ }+RT\ln {\frac {a_{i}}{a}}\approx \mu _{i}^{\circ }+RT\ln x_{i}$

At the equilibrium, $\mu _{i}=0$ , thus $\mu _{i}^{\circ }$ is obtained by:

$\mu _{i}=\mu _{i}^{\circ }+RT\ln x_{i}=0\Rightarrow \mu _{i}^{\circ }=-RT\ln x_{i}.$

Using and integrating gives Eq.

${\begin{array}{l}\left({\frac {\partial \mu _{i}/T}{\partial T}}\right)_{P}={\frac {\partial }{\partial T}}\left({R\ln x_{i}}\right)\Rightarrow R\ln x_{i}=-{\frac {H_{i}^{\circ }}{T}}+K\\\\\end{array}}$

The integration constant K may be determined for a pure component with a melting temperature $T^{\circ }$ and an enthalpy of fusion $H^{\circ }$ Eq.

$x_{i}=1\Rightarrow T=T_{i}^{\circ }\Rightarrow K={\frac {H_{i}^{\circ }}{T_{i}^{\circ }}}$

We obtain a relation that determines the molar fraction as a function of the temperature for each component.

$R\ln x_{i}=-{\frac {H_{i}^{\circ }}{T}}+{\frac {H_{i}^{\circ }}{T_{i}^{\circ }}}$

The mixture of n components is described by the system

${\begin{array}{l}\left\{{\begin{array}{*{20}c}{\ln x_{i}+{\frac {H_{i}^{\circ }}{RT}}-{\frac {H_{i}^{\circ }}{RT_{i}^{\circ }}}=0}\\{\sum \limits _{i=1}^{n}{x_{i}=1}}\\\end{array}}\right.\\\\\end{array}}$ ${\begin{array}{l}\left\{{\begin{array}{*{20}c}{\forall i<n\Rightarrow \ln x_{i}+{\frac {H_{i}^{\circ }}{RT}}-{\frac {H_{i}^{\circ }}{RT_{i}^{\circ }}}=0}\\{\ln \left({1-\sum \limits _{i=1}^{n-1}{x_{i}}}\right)+{\frac {H_{n}^{\circ }}{RT}}-{\frac {H_{n}^{\circ }}{RT_{n}^{\circ }}}=0}\\\end{array}}\right.\\\\\end{array}}$

that can be solved by

${\begin{array}{c}\left[{\begin{array}{*{20}c}{\Delta x_{1}}\\{\Delta x_{2}}\\{\Delta x_{3}}\\\vdots \\{\Delta x_{n-1}}\\{\Delta T}\\\end{array}}\right]=\left[{\begin{array}{*{20}c}{1/x_{1}}&0&0&0&0&{-{\frac {H_{1}^{\circ }}{RT^{2}}}}\\0&{1/x_{2}}&0&0&0&{-{\frac {H_{2}^{\circ }}{RT^{2}}}}\\0&0&{1/x_{3}}&0&0&{-{\frac {H_{3}^{\circ }}{RT^{2}}}}\\0&0&0&\ddots &0&{-{\frac {H_{4}^{\circ }}{RT^{2}}}}\\0&0&0&0&{1/x_{n-1}}&{-{\frac {H_{n-1}^{\circ }}{RT^{2}}}}\\{\frac {-1}{1-\sum \limits _{1=1}^{n-1}{x_{i}}}}&{\frac {-1}{1-\sum \limits _{1=1}^{n-1}{x_{i}}}}&{\frac {-1}{1-\sum \limits _{1=1}^{n-1}{x_{i}}}}&{\frac {-1}{1-\sum \limits _{1=1}^{n-1}{x_{i}}}}&{\frac {-1}{1-\sum \limits _{1=1}^{n-1}{x_{i}}}}&{-{\frac {H_{n}^{\circ }}{RT^{2}}}}\\\end{array}}\right]^{-1}.\left[{\begin{array}{*{20}c}{\ln x_{1}+{\frac {H_{1}^{\circ }}{RT}}-{\frac {H_{1}^{\circ }}{RT_{1}^{\circ }}}}\\{\ln x_{2}+{\frac {H_{2}^{\circ }}{RT}}-{\frac {H_{2}^{\circ }}{RT_{2}^{\circ }}}}\\{\ln x_{3}+{\frac {H_{3}^{\circ }}{RT}}-{\frac {H_{3}^{\circ }}{RT_{3}^{\circ }}}}\\\vdots \\{\ln x_{n-1}+{\frac {H_{n-1}^{\circ }}{RT}}-{\frac {H_{n-1}^{\circ }}{RT_{n-1i}^{\circ }}}}\\{\ln \left({1-\sum \limits _{i=1}^{n-1}{x_{i}}}\right)+{\frac {H_{n}^{\circ }}{RT}}-{\frac {H_{n}^{\circ }}{RT_{n}^{\circ }}}}\\\end{array}}\right]\end{array}}$

References

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Bibliography

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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Template:Cite web

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↑ Iron-Iron Carbide Phase Diagram Example
↑ IUPAC Compendium of Chemical Terminology, Electronic version. "Peritectoid Reaction" Retrieved May 22, 2007.
↑ Numerical Model of Peritectoid Transformation. Peritectoid Transformation Retrieved May 22, 2007. Template:Dead link
↑ International Journal of Modern Physics C, Vol. 15, No. 5. (2004), pp. 675-687

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[10] Iron-Iron Carbide Phase Diagram Example

[Gold_Book_PAC,1994,66,588-11] IUPAC Compendium of Chemical Terminology, Electronic version. "Peritectoid Reaction" Retrieved May 22, 2007.

[12] Numerical Model of Peritectoid Transformation. Peritectoid Transformation Retrieved May 22, 2007. Template:Dead link

[13] International Journal of Modern Physics C, Vol. 15, No. 5. (2004), pp. 675-687

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General number field sieve: Difference between revisions

Revision as of 17:43, 19 January 2014

Contents

Eutectic reaction

Non-eutectic compositions