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Regular solution

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Chemistry" title="Chemistry">chemistry</a>, a regular solution is a solution whose <a href="Entropy_of_mixing" title="Entropy of mixing">entropy of mixing</a> is equal to that of an ideal solution with the same composition, but is non-ideal due to a nonzero <a href="Enthalpy_of_mixing" title="Enthalpy of mixing">enthalpy of mixing</a>.<a href="#cite_note-Atkins-1">[1]</a><a href="#cite_note-Rock-2">[2]</a> Such a solution is formed by random mixing of components of similar molar volume and without strong specific interactions,<a href="#cite_note-Atkins-1">[1]</a><a href="#cite_note-Rock-2">[2]</a> and its behavior diverges from that of an <a href="Ideal_solution" title="Ideal solution">ideal solution</a> by showing <a href="Phase_separation" title="Phase separation">phase separation</a> at intermediate compositions and temperatures (a <a href="Miscibility_gap" title="Miscibility gap">miscibility gap</a>).<a href="#cite_note-3">[3]</a> Its <a href="Entropy_of_mixing" title="Entropy of mixing">entropy of mixing</a> is equal to that of an ideal solution with the same composition, due to random mixing without strong specific interactions.<a href="#cite_note-Atkins-1">[1]</a><a href="#cite_note-Rock-2">[2]</a> For two components

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<annotation encoding="application/x-tex">{\displaystyle \Delta S_{mix}=-nR(x_{1}\ln x_{1}+x_{2}\ln x_{2})\,}</annotation>

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</math><img src="./cacdf9eaa8962e447e35e019eaa1340f67d20a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.608ex; height:2.843ex;" alt="{\displaystyle \Delta S_{mix}=-nR(x_{1}\ln x_{1}+x_{2}\ln x_{2})\,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\,}">

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<annotation encoding="application/x-tex">{\displaystyle R\,}</annotation>

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</math><img src="./293563891196765d2d51e0dd54e1ae1000ba9def.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle R\,}" loading="lazy"> is the <a href="Gas_constant" title="Gas constant">gas constant</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\,}">

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<annotation encoding="application/x-tex">{\displaystyle n\,}</annotation>

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</math><img src="./205e33e6845813cc72ca346b896a7945f90ca373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle n\,}" loading="lazy"> the total number of <a href="Mole_(unit)" title="Mole (unit)">moles</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\,}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}\,}</annotation>

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</math><img src="./7beb41cde977577a7aa598a9defd58dc8d529bb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.516ex; height:2.009ex;" alt="{\displaystyle x_{i}\,}" loading="lazy"> the <a href="Mole_fraction" title="Mole fraction">mole fraction</a> of each component. Only the <a href="Enthalpy_of_mixing" title="Enthalpy of mixing">enthalpy of mixing</a> is non-zero, unlike for an ideal solution, while the volume of the solution equals the sum of volumes of components.

<div class="mw-heading mw-heading2"><h2 id="Features">Features</h2></div>

A regular solution can also be described by <a href="Raoult's_law" title="Raoult's law">Raoult's law</a> modified with a <a href="Margules_function" class="mw-redirect" title="Margules function">Margules function</a> with only one parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle \ P_{1}=x_{1}P_{1}^{*}f_{1,M}\,}</annotation>

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</math><img src="./ead153eca0120d3c88fd92a3b2f929a0e7369db3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.25ex; height:2.843ex;" alt="{\displaystyle \ P_{1}=x_{1}P_{1}^{*}f_{1,M}\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \ P_{2}=x_{2}P_{2}^{*}f_{2,M}\,}</annotation>

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</math><img src="./3e22efa349ca259e1ee82b456b909ec4384292df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.25ex; height:2.843ex;" alt="{\displaystyle \ P_{2}=x_{2}P_{2}^{*}f_{2,M}\,}" loading="lazy"></dd></dl>

where the Margules function is

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<annotation encoding="application/x-tex">{\displaystyle \ f_{1,M}={\rm {exp}}(\alpha x_{2}^{2})\,}</annotation>

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</math><img src="./69bc534a970b5386300f78f7647f283817cfdd9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.677ex; height:3.176ex;" alt="{\displaystyle \ f_{1,M}={\rm {exp}}(\alpha x_{2}^{2})\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \ f_{2,M}={\rm {exp}}(\alpha x_{1}^{2})\,}</annotation>

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</math><img src="./7d6a25edec7a3089fab4617b25f8b134d878cd69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.677ex; height:3.176ex;" alt="{\displaystyle \ f_{2,M}={\rm {exp}}(\alpha x_{1}^{2})\,}" loading="lazy"></dd></dl>

Notice that the Margules function for each component contains the mole fraction of the other component. It can also be shown using the <a href="Gibbs-Duhem_relation" class="mw-redirect" title="Gibbs-Duhem relation">Gibbs-Duhem relation</a> that if the first Margules expression holds, then the other one must have the same shape. A regular solutions internal energy will vary during mixing or during process.

The value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> can be interpreted as W/RT, where W = 2U12 - U11 - U22 represents the difference in interaction energy between like and unlike neighbors.

In contrast to ideal solutions, regular solutions do possess a non-zero enthalpy of mixing, due to the W term. If the unlike interactions are more unfavorable than the like ones, we get competition between an entropy of mixing term that produces a minimum in the <a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> at x1 = 0.5 and the enthalpy term that has a maximum there. At high temperatures, the entropic term in the free energy of mixing dominates and the system is fully miscible, but at lower temperatures the G(x1) curve will have two minima and a maximum in between. This results in phase separation. In general there will be a temperature where the three extremes coalesce and the system becomes fully miscible. This point is known as the <a href="Upper_critical_solution_temperature" title="Upper critical solution temperature">upper critical solution temperature</a> or the upper consolute temperature.

In contrast to ideal solutions, the volumes in the case of regular solutions are no longer strictly additive but must be calculated from <a href="Partial_molar_volume" class="mw-redirect" title="Partial molar volume">partial molar volumes</a> that are a function of x1.

The term was introduced in 1927 by the American physical chemist <a href="Joel_Henry_Hildebrand" title="Joel Henry Hildebrand">Joel Henry Hildebrand</a>.<a href="#cite_note-4">[4]</a>

<ul><li><a href="Solid_solution" title="Solid solution">Solid solution</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Atkins-1">^ <a href="#cite_ref-Atkins_1-0">a</a> <a href="#cite_ref-Atkins_1-1">b</a> <a href="#cite_ref-Atkins_1-2">c</a> P. Atkins and J. de Paula, Atkins' Physical Chemistry (8th ed. W.H. Freeman 2006) p.149

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<li id="cite_note-Rock-2">^ <a href="#cite_ref-Rock_2-0">a</a> <a href="#cite_ref-Rock_2-1">b</a> <a href="#cite_ref-Rock_2-2">c</a> P.A. Rock, Chemical Thermodynamics. Principles and Applications (Macmillan 1969) p.263

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> Simon & McQuarrie Physical Chemistry: A molecular approach

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a rel="nofollow" class="external text" href="http://www.nature.com/nature/journal/v168/n4281/abs/168868a0.html">The Term 'Regular Solution'</a> Nature, v.168, p.868 (1951)

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