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{{short description|Mixture with a lower melting point than its constituents}}
{{redirect|Eutectic|the sports mascot|St. Louis College of Pharmacy#Mascot}}
[[File:Eutectic system phase diagram.svg|thumb|upright=1.4|A phase diagram for a fictitious binary chemical mixture (with the two components denoted by ''A'' and ''B'') used to depict the eutectic composition, temperature, and point. (''L'' denotes the liquid state.)]]

A '''eutectic system''' or '''eutectic mixture''' ({{IPAc-en|j|uː|ˈ|t|ɛ|k|t|ɪ|k}} {{respell|yoo|TEK|tik}})<ref>{{MerriamWebsterDictionary|eutectic}}</ref> is a type of a [[homogeneous mixture]] that has a [[melting point]] lower than those of the constituents.<ref name="Guthrie">{{cite journal |last1=Guthrie |first1=Frederick |title=LII. On eutexia |journal=[[The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science]] |date=June 1884 |volume=17 |issue=108 |pages=462–482 |doi=10.1080/14786448408627543 |series=5th series |quote=The main argument of the present communication hinges upon the existence of compound bodies, whose chief characteristic is the lowness of their temperatures of fusion. This property of the bodies may be called Eutexia, the bodies possessing it eutectic bodies or eutectics (''εὖ τήκειν''). |quote-page=462 |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015024088166;seq=506}}</ref> The lowest possible melting point over all of the [[mixing ratio]]s of the constituents is called the ''eutectic temperature''. On a [[phase diagram]], the eutectic temperature is seen as the eutectic point (see plot on the right).<ref>{{harvnb|Smith|Hashemi|2006|pp=326–327}}</ref>

Non-eutectic mixture ratios have different melting temperatures for their different constituents, since one component's [[Crystal structure|lattice]] will melt at a lower temperature than the other's. Conversely, as a non-eutectic mixture cools down, each of its components [[solid|solidifies]] into a lattice at a different temperature, until the entire mass is solid.

Not all [[binary alloy]]s have eutectic points, since the [[valence electron]]s of the component species are not always compatible{{clarify|compatible electrons?|date=November 2014}} in any mixing ratio to form a new type of joint crystal lattice. For example, in the silver-gold system the melt temperature ([[liquidus]]) and freeze temperature ([[solidus (chemistry)|solidus]]) "meet at the pure element endpoints of the atomic ratio axis while slightly separating in the mixture region of this axis".<ref>{{cite web|url=http://www.crct.polymtl.ca/fact/phase_diagram.php?file=Ag-Au.jpg&dir=SGTE2007|title=Collection of Phase Diagrams|website=www.crct.polymtl.ca}}</ref>

In the real world, eutectic properties can be used to advantage in such processes as [[eutectic bonding]], where [[silicon chip]]s are bonded to gold-plated substrates with [[ultrasound]], and eutectic alloys prove valuable in such diverse applications as soldering, brazing, metal casting, electrical protection, fire sprinkler systems, and nontoxic mercury substitutes.

The term {{em|eutectic}} was coined in 1884 by British physicist and chemist [[Frederick Guthrie (scientist)|Frederick Guthrie]] (1833–1886). The word originates {{ety|el|''εὐ''- (eû)|well||''τῆξῐς'' (têxis)|melting}}.<ref name="Guthrie" /> Before his studies, chemists assumed "that the alloy of minimum fusing point must have its constituents in some simple atomic proportions", which was indeed proven to be not the case.<ref>{{Cite book |last=Britain) |first=Chemical Society (Great |url=https://books.google.com/books?id=FJ0eAQAAMAAJ&pg=PA329 |title=Journal of the Chemical Society |date=1885 |publisher=Chemical Society. |language=en}}</ref>

==Eutectic phase transition==
[[File:Various eutectic structures.png|thumb|Four eutectic structures: A) lamellar B) rod-like C) globular D) acicular.]]

The eutectic solidification is defined as follows:<ref name="smith327">{{harvnb|Smith|Hashemi|2006|p=327}}.</ref>

:<math>\text{Liquid} \quad \xrightarrow[\text{cooling}]{\text{eutectic} \atop \text{temperature}} \quad \alpha \text{ solid solution} \ + \ \beta \text{ solid solution}</math>

This type of reaction is an invariant reaction, because it is in [[thermal equilibrium]]; another way to define this is the change in [[Gibbs free energy]] equals zero. Tangibly, this means the liquid and two [[solid solution]]s all coexist at the same time and are in [[chemical equilibrium]]. There is also a [[thermal arrest]] for the duration of the [[phase transition|phase change]] during which the temperature of the system does not change.<ref name="smith327"/>

The resulting solid [[wikt:macrostructure|macrostructure]] from a eutectic reaction depends on a few factors, with the most important factor being 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 (crystal habit)|acicular]].<ref>{{harvnb|Smith|Hashemi|2006|pp=332–333}}.</ref>

==Non-eutectic compositions==
Compositions of eutectic systems that are not at the eutectic point can be classified as ''hypoeutectic'' or ''hypereutectic'': 

* ''Hypoeutectic'' compositions are those with a greater composition of species α and a smaller percent composition of species β than the eutectic composition (E) 

* ''Hypereutectic'' compositions are characterized as those with a higher composition of species β and a lower composition of species α than the eutectic composition. 

As the temperature of a non-eutectic composition is lowered the liquid mixture will precipitate one component of the mixture before the other.  In a hypereutectic solution, there will be a ''[[#proeutectoid|proeutectoid]]'' phase of species β whereas a hypoeutectic solution will have a ''[[#proeutectic|proeutectic]]'' α phase.<ref name="smith327"/>

==Types==

===Alloys===
Eutectic [[alloy]]s 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 for eutectic alloys include:
*NEMA eutectic alloy overload relays for [[power-system protection|electrical protection]] of three-phase motors for pumps, fans, conveyors, and other factory process equipment.<ref>{{cite web |url= http://www.industrial-electronics.com/output_devices_amplifiers_valves_relays_variable-frequency_drives_stepper_motors_servomotors/Operation_Overloads.html |title= Operation of the Overloads |access-date= 2015-08-05 }}</ref>
*Eutectic alloys for [[soldering]], both traditional alloys composed of [[lead]] (Pb) and [[tin]] (Sn), sometimes with additional [[silver]] (Ag) or [[gold]] (Au) — especially [[Solder alloys#Sn63Pb37|Sn{{sub|63}}Pb{{sub|37}}]] and Sn{{sub|62}}Pb{{sub|36}}Ag{{sub|2}} alloy formula for electronics - and newer lead-free soldering alloys, in particular ones composed of tin, silver, and [[copper]] (Cu) such as Sn{{sub|96.5}}Ag{{sub|3.5}}.
*Casting alloys, such as [[Silumin|aluminium-silicon]] and [[cast iron]] (at the composition of 4.3% carbon in iron producing an [[austenite]]-[[cementite]] eutectic)
*[[Silicon chip]]s are [[eutectic bonding|eutectic bonded]] to gold-plated substrates through a silicon-gold eutectic by the application of [[ultrasound|ultrasonic]] energy to the chip. 
*[[Brazing]], where diffusion can remove alloying elements from the joint, so that eutectic melting is only possible early in the brazing process
*Temperature response, e.g., [[Wood's metal]] and [[Field's metal]] for [[fire sprinkler]]s
*Non-toxic [[Mercury (element)|mercury]] replacements, such as [[galinstan]]
*Experimental [[amorphous metal|glassy metals]], with extremely high strength and [[corrosion]] resistance
*Eutectic alloys of [[sodium]] and [[potassium]] ([[NaK]]) that are liquid at room temperature and used as [[coolant]] in experimental [[Fast neutron reactor|fast neutron nuclear reactor]]s.

===Others===
[[File:Phase diagram ethanol water s l en.svg|thumb|upright=1.3|Solid–liquid phase change of ethanol–water mixtures]]
* [[Sodium chloride]] and [[water]] form a eutectic mixture whose eutectic point is −21.2&nbsp;°C<ref>{{cite web | last = Muldrew | first = Ken | author2 = Locksley E. McGann | year = 1997 | url = https://www.ucalgary.ca/~kmuldrew/cryo_course/cryo_chap6_1.html | title = Phase Diagrams | work = Cryobiology—A Short Course | publisher = University of Calgary | access-date = 2006-04-29 | archive-url = https://web.archive.org/web/20060615124330/http://www.ucalgary.ca/~kmuldrew/cryo_course/cryo_chap6_1.html | archive-date = 2006-06-15 | url-status = dead }}</ref> and 23.3% salt by mass.<ref>{{cite web | last = Senese | first = Fred | year = 1999 | url = http://antoine.frostburg.edu/chem/senese/101/solutions/faq/saltwater-ice-volume.shtml | title = Does salt water expand as much as fresh water does when it freezes? | work = Solutions: Frequently asked questions | publisher = Department of Chemistry, Frostburg State University | access-date = 2006-04-29}}</ref> 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 biased eutectic point, i.e. it is close to pure ethanol, which sets the maximum proof obtainable by [[fractional freezing]]. 
* "Solar salt", 60% NaNO<sub>3</sub> and 40% KNO<sub>3</sub>, forms a eutectic molten salt mixture which is used for [[thermal energy storage]] in [[concentrated solar power]] plants.<ref>{{cite web |title=Molten salts properties |url=http://www.archimedesolarenergy.com/molten_salt.htm |work=Archimede Solar Plant Specs |access-date=2011-07-15 |archive-date=2020-04-20 |archive-url=https://web.archive.org/web/20200420183555/http://www.archimedesolarenergy.com/molten_salt.htm |url-status=dead }}</ref> To reduce the eutectic melting point in the solar molten salts, [[calcium nitrate]] is used in the following proportion: 42% Ca(NO<sub>3</sub>)<sub>2</sub>, 43% KNO<sub>3,</sub> and 15% NaNO<sub>3</sub>.
* [[Lidocaine]] and [[prilocaine]]—both are solids at room temperature—form a eutectic that is an oil with a {{convert|16|C|abbr=on}} 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.<ref>{{Cite journal |last1=Phaechamud |first1=Thawatchai |last2=Tuntarawongsa |first2=Sarun |last3=Charoensuksai |first3=Purin |date=October 2016 |title=Evaporation Behavior and Characterization of Eutectic Solvent and Ibuprofen Eutectic Solution |journal=AAPS PharmSciTech |volume=17 |issue=5 |pages=1213–1220 |doi=10.1208/s12249-015-0459-x |issn=1530-9932 |pmid=26669887|doi-access=free }}</ref>
* [[Mineral]]s may form eutectic mixtures in [[igneous]] rocks, giving rise to characteristic [[Rock microstructure#Graphic and other intergrowth textures|intergrowth textures]] exhibited, for example, by [[granophyre]].<ref>{{cite web | last = Fichter | first = Lynn S. | year = 2000 | url = http://csmres.jmu.edu/geollab/Fichter/IgnRx/Phasdgrm.html | title = Igneous Phase Diagrams | work = Igneous Rocks | publisher = James Madison University | access-date = 2006-04-29 | archive-date = 2011-06-28 | archive-url = https://web.archive.org/web/20110628195112/http://csmres.jmu.edu/geollab/Fichter/IgnRx/Phasdgrm.html | url-status = dead }}</ref>
* Some inks are eutectic mixtures, allowing [[inkjet printer]]s to operate at lower temperatures.<ref>{{cite patent |country=US |number=5298062A |title=Eutectic compositions for hot melt jet inks |pubdate=1994-03-29 |gdate=1994-03-29 |fdate=1992-08-19 |pridate=1992-08-19 |inventor1-last=Davies |inventor1-first=Nicholas A. |inventor2-last=Nicholas |inventor2-first=Beatrice M.}}</ref>
* [[Choline]] chloride produces eutectic mixtures with many natural products such as [[citric acid]], [[malic acid]] and [[sugars]]. These liquid mixtures can be used, for example, to obtain antioxidant and antidiabetic extracts from [[natural product]]s.<ref>{{Cite journal |last1=Socas-Rodriguez |first1=Bárbara |last2=Torres-Cornejo |first2=Mónica Vanesa |last3=Álvarez-Rivera |first3=Gerardo |last4=Mendiola |first4=Jose A. |date=May 2021 |title=Deep Eutectic Solvents for the Extraction of Bioactive Compounds from Natural Sources and Agricultural By-Products |journal=Applied Sciences  |volume=11 |issue=1 |pages=4897 |doi=10.3390/app11114897 |issn=2076-3417 |doi-access=free |hdl=10261/253199 |hdl-access=free }}</ref>

== Strengthening mechanisms ==

=== Alloys ===
The primary strengthening mechanism of the eutectic structure in metals is [[Composite material|composite]] strengthening (See [[strengthening mechanisms of materials]]). This deformation mechanism works through load transfer between the two constituent phases where the more compliant phase transfers stress to the stiffer phase.<ref>{{Cite book |last=Courtney |first=T. H. |title=Mechanical behavior of materials |publisher=McGraw-Hill |year=1990 |edition=2nd |location=New York}}</ref> By taking advantage of the strength of the stiff phase and the ductility of the compliant phase, the overall toughness of the material increases. As the composition is varied to either hypoeutectic or hypereutectic formations, the load transfer mechanism becomes more complex as there is a load transfer between the eutectic phase and the secondary phase as well as the load transfer within the eutectic phase itself.

A second tunable strengthening mechanism of eutectic structures is the spacing of the secondary phase. By changing the spacing of the secondary phase, the fraction of contact between the two phases through shared phase boundaries is also changed. By decreasing the spacing of the eutectic phase, creating a fine eutectic structure, more surface area is shared between the two constituent phases resulting in more effective load transfer.<ref name=":0">{{Cite book |last=Callister |first=W. D. |title=Materials science and engineering : an introduction |year=2010}}</ref> On the micro-scale, the additional boundary area acts as a barrier to [[dislocation]]s further strengthening the material. As a result of this strengthening mechanism, coarse eutectic structures tend to be less stiff but more ductile while fine eutectic structures are stiffer but more brittle.<ref name=":0" /> The spacing of the eutectic phase can be controlled during processing as it is directly related to the cooling rate during solidification of the eutectic structure. For example, for a simple lamellar eutectic structure, the minimal lamellae spacing  is:<ref>{{Cite book |last1=Porter |first1=D. A. |title=Phase transformations in metals and alloys |last2=Easterling |first2=K. E. |last3=Sherif |first3=M. Y. |year=2009}}</ref>

<math>\lambda^*=\frac{2\gamma V_m T_E }{\Delta H * \Delta T_0}</math>

Where  is <math>\gamma</math> is the [[surface energy]] of the two-phase boundary, <math>V_m</math>'' ''is the [[molar volume]] of the eutectic phase, <math>T_E</math>  is the solidification temperature of the eutectic phase, <math>\Delta H</math> is the [[Enthalpy of Formation|enthalpy of formation]] of the eutectic phase, and <math>\Delta T_0</math> is the undercooling of the material. So, by altering the undercooling, and by extension the cooling rate, the minimal achievable spacing of the secondary phase is controlled.

Strengthening metallic eutectic phases to resist deformation at high temperatures (see [[Creep (deformation)|creep deformation]]) is more convoluted as the primary deformation mechanism changes depending on the level of stress applied. At high temperatures where deformation is dominated by dislocation movement, the strengthening from load transfer and secondary phase spacing remain as they continue to resist dislocation motion. At lower strains where Nabarro-Herring creep is dominant, the shape and size of the eutectic phase structure plays a significant role in material deformation as it affects the available boundary area for vacancy diffusion to occur.<ref>{{Cite journal |last1=Wu |first1=T. |last2=Plotkowski |first2=A. |last3=Shyam |first3=A. |last4=Dunand |first4=D.C. |title=Microstructure and creep properties of cast near-eutectic Al-Ce-Ni alloys |journal=Mater. Sci. Eng. A |volume=833 |pages=12}}</ref>

== Other critical points==
[[File:Iron carbon phase diagram.svg|thumb|upright=1.75|Iron–carbon phase diagram, showing the eutectoid transformation between austenite (γ) and pearlite.]]

===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 [[Allotropes of iron|ferrite]] and [[cementite]], often in lamellar structures such as [[pearlite]] and [[bainite]].  This eutectoid point occurs at {{convert|723|C|abbr=on}} and 0.76 wt% carbon.<ref>{{cite web |first=Kim |last=Ballentine |date=28 April 1996 |url=https://nptel.ac.in/content/storage2/courses/112108150/pdf/PPTs/MTS_07_m.pdf|title=Iron-Iron Carbide Phase Diagram Example|archive-url=https://web.archive.org/web/20080216023642/http://www.sv.vt.edu/classes/MSE2094_NoteBook/96ClassProj/examples/kimcon.html|archive-date=16 February 2008}}</ref>

===Peritectoid===
A ''peritectoid'' transformation is a type of [[isothermal]] [[reversible reaction]] that has two solid [[Phase (matter)|phase]]s reacting with each other upon cooling of a binary, ternary, ..., ''n''-ary [[alloy]] to create a completely different and single solid phase.<ref name="Gold Book PAC,1994,66,588">{{goldbookref|title=Peritectoid reaction|file=P04501}}</ref>  The reaction plays a key role in the order and [[decomposition]] of [[quasicrystalline]] phases in several alloy types.<ref>{{cite journal |last1=Das |first1=Amit |last2=Manna |first2=Indranil |last3=Pabi |first3=S. K. |title=A numerical model of peritectoid transformation |journal=Metallurgical and Materials Transactions A |date=October 1999 |volume=30 |issue=10 |pages=2563–2573 |doi=10.1007/s11661-999-0295-2 |publisher=[[The Minerals, Metals & Materials Society]], [[ASM International]]|bibcode=1999MMTA...30.2563D |s2cid=95279944 }}</ref> A similar structural transition is also predicted for [[Cylinder sphere packing#Columnar structures created by rapid rotations|rotating columnar crystals.]]

===Peritectic===
[[File:Phasendiagramm Gold-Aluminium.svg|upright=1.25|thumb|Gold–aluminium [[phase diagram]]]]
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 {{convert|1495|C|abbr=on}} and 0.17% carbon.

At the peritectic decomposition temperature the compound, rather than melting, decomposes into another solid compound and a liquid. The proportion of each is determined by the [[lever rule]]. In the [[Gold-aluminium intermetallic|Al-Au]] phase diagram, for example, it can be seen that only two of the phases melt congruently, [[Gold-aluminium intermetallic|AuAl<sub>2</sub>]] and [[Gold-aluminium intermetallic|Au<sub>2</sub>Al]], while the rest peritectically decompose.

==="Bad solid solution"===
Not all minimum melting point systems are "eutectic". The alternative of "poor solid solution" can be illustrated by comparing the common precious metal systems Cu-Ag and Cu-Au.
Cu-Ag, source for example https://himikatus.ru/art/phase-diagr1/Ag-Cu.php, is a true eutectic system. The eutectic melting point is at 780 C, with solid solubility limits at fineness 80 and 912 by weight, and eutectic at 719.
Since Cu-Ag is a true eutectic, any silver with fineness anywhere between 80 and 912 will reach solidus line, and therefore melt at least partly, at exactly 780 C. The eutectic alloy with fineness exactly 719 will reach liquidus line, and therefore melt entirely, at that exact temperature without any further rise of temperature till all of the alloy has melted.
Any silver with fineness between 80 and 912 but not exactly 719 will also reach the solidus line at exactly 780 C, but will melt partly. It will leave a solid residue with fineness of either exactly 912 or exactly 80, but never some of both. It will melt at constant temperature without further rise of temperature until the exact amount of eutectic (fineness 719) alloy has melted off to divide the alloy into eutectic melt and solid solution residue.
On further heating, the solid solution residue dissolves in the melt and changes its composition until the liquidus line is reached and the whole residue has dissolved away.
Cu-Au source for example https://himikatus.ru/art/phase-diagr1/Au-Cu.php does display a melting point minimum at 910 C and given as 44 atom % Cu, which converts to about 20 weight percent Cu - about 800 fineness of gold. But this is not a true eutectic.
800 fine gold melts at 910 C, to a melt of exact same composition, and the whole alloy will melt at exact same temperature. But the differences happen away from the minimum composition.
Unlike silver with fineness other than 719 (which melts partly at exactly 780 C through a wide fineness range), gold with fineness other than 800 will reach solidus and start partial melting at a temperature different from and higher than 910 C, depending on the alloy fineness. The partial melting does cause some composition changes - the liquid will be closer in fineness towards 800 than the remaining solid, but the liquid will not have fineness of exactly 800 and the fineness of the remaining solid will depend on the fineness of the liquid.
The underlying reason is that for an eutectic system like Cu-Ag, the solubility in liquid phase is good but solubility in solid phase is limited. Therefore when a silver-copper alloy is frozen, it actually separates into crystals of 912 fineness silver and 80 fineness silver - both are saturated and always have the same composition at the freezing point of 780 C. Thus the alloy just below 780 C consists of two types of crystals of exactly the same composition regardless of the total alloy composition, only the relative amount of each type of crystals differs. Therefore they always melt at 780 C until one or other type of crystals, or both, will be exhausted.
In contrast, in Cu-Au system the components are miscible at the melting point in all compositions even in solid. There can be crystals of any composition, which will melt at different temperatures depending on composition.
However, Cu-Au system is a "poor" solid solution. There is a substantial misfit between the atoms in solid which, however, near the melting point is overcome by entropy of thermal motion mixing the atoms. That misfit, however, disfavours the Cu-Au solution relative to phases in which the atoms are better fitted, such as the melt, and causes the melting point to fall below the melting point of components.

==Eutectic calculation==
The composition and temperature of a eutectic can be calculated from enthalpy and entropy of fusion of each components.<ref>{{cite journal |last1=Brunet |first1=Luc E. |last2=Caillard |first2=Jean |last3=André |first3=Pascal |title=Thermodynamic Calculation of ''n''-component Eutectic Mixtures |journal=International Journal of Modern Physics C |date=June 2004 |volume=15 |issue=5 |pages=675–687 |doi=10.1142/S0129183104006121 |publisher=World Scientific|bibcode=2004IJMPC..15..675B }}</ref>

The Gibbs free energy ''G'' depends on its own differential:
: <math>
G = H - TS
\Rightarrow 
\begin{cases}
 H = G + TS \\
 \left(\frac{\partial G}{\partial T}\right)_P = -S
\end{cases}
\Rightarrow
H = G - T \left(\frac{\partial G}{\partial T}\right)_P.
</math>

Thus, the ''G''/''T'' derivative at constant pressure is calculated by the following equation:
: <math>
    \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}.
</math>

The chemical potential <math>\mu_i</math> is calculated if we assume that the activity is equal to the concentration:
: <math>
\mu_i = \mu_i^\circ + RT\ln \frac{a_i}{a} \approx \mu_i^\circ + RT\ln x_i.
</math>

At the equilibrium, <math>\mu_i = 0</math>, thus <math>\mu_i^\circ</math> is obtained as
: <math>
\mu _i = \mu _i^\circ + RT\ln x_i = 0 \Rightarrow \mu_i^\circ = -RT\ln x_i.
</math>

Using{{clarify|date=January 2019}} and integrating gives
: <math>
 \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.
</math>

The integration constant ''K'' may be determined for a pure component with a melting temperature <math>T^\circ</math> and an enthalpy of fusion <math>H^\circ</math>:
: <math>
x_i = 1 \Rightarrow T = T_i^\circ \Rightarrow K = \frac{H_i^\circ}{T_i^\circ}.
</math>

We obtain a relation that determines the molar fraction as a function of the temperature for each component:
: <math>
R\ln x_i = -\frac{H_i^\circ}{T} + \frac{H_i^\circ}{T_i^\circ}.
</math>

The mixture of ''n'' components is described by the system
: <math>
\begin{cases}
 \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{cases}
</math>
: <math>
\begin{cases}
 \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{cases}
</math>

which can be solved by
: <math>
\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}}} \\
 \vdots & \ddots & \ddots & \ddots & \ddots & { \vdots} \\
 0 & 0 & 0 & 0 & {1 / x_{n - 1} } & { - \frac{H_{n - 1}^\circ }{RT^{2}}}
\\
 {\frac{ - 1}{1 - \sum\limits_{i = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{i = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{i = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{i = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{i = 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 - 1}^\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}
</math>

==See also==
*[[Azeotrope]], or constant boiling mixture
*[[Freezing-point depression]]
*[[Low melting alloy]]

==References==
{{Reflist}}

===Bibliography===
*{{Citation | last1 = Smith | first1 = William F. | last2 = Hashemi | first2 = Javad | title = Foundations of Materials Science and Engineering | edition = 4th | year = 2006 | publisher = McGraw-Hill | isbn = 978-0-07-295358-9 | postscript =.}}

==Further reading==
{{Wiktionary|eutectic}}

*{{cite book
 | last = Askeland
 | first = Donald R.
 |author2=Pradeep P. Phule
 | year = 2005
 | title = The Science and Engineering of Materials
 | publisher = Thomson-Engineering
 | isbn = 978-0-534-55396-8
}}
*{{cite book
 | last = Easterling
 | first = Edward
 | year = 1992
 | title = Phase Transformations in Metals and Alloys
 | publisher = CRC
 | isbn = 978-0-7487-5741-1
}}
*{{cite book
 | last = Mortimer
 | first = Robert G.
 | year = 2000
 | title = Physical Chemistry
 | publisher = Academic Press
 | isbn = 978-0-12-508345-4
}}
*{{cite book
 | last = Reed-Hill
 | first = R. E.
 |author2=Reza Abbaschian
 | year = 1992
 | title = Physical Metallurgy Principles
 | publisher =Thomson-Engineering
 | isbn = 978-0-534-92173-6
| author2-link = Reza Abbaschian
 }}
*{{cite web
 |last=Sadoway 
 |first=Donald 
 |year=2004 
 |url=http://ocw.mit.edu/NR/rdonlyres/Materials-Science-and-Engineering/3-091Fall-2004/6ECFB930-9D59-4DD5-A872-FE48002587B0/0/notes_10.pdf 
 |title=Phase Equilibria and Phase Diagrams 
 |work=3.091 Introduction to Solid State Chemistry, Fall 2004 
 |publisher=MIT Open Courseware 
 |access-date=2006-04-12 
 |author-link=Donald Sadoway 
 |archive-url=https://web.archive.org/web/20051020162911/http://ocw.mit.edu/NR/rdonlyres/Materials-Science-and-Engineering/3-091Fall-2004/6ECFB930-9D59-4DD5-A872-FE48002587B0/0/notes_10.pdf 
 |archive-date=2005-10-20 
 |url-status=dead 
}}

{{Chemical solutions}}

[[Category:Materials science]]
[[Category:Chemistry]]
[[Category:Phase transitions]]