Entropy of mixing
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2008-06-13T18:00:00Z
Potekhin
7213893
/* Proof */ fixed typo
The '''entropy of mixing''' (also known as '''configurational entropy''') is the change in the [[entropy]], an [[extensive quantity|extensive]] [[thermodynamics|thermodynamic]] quantity, when two different [[chemical substance]]s or [[component]]s are mixed. This entropy change must be positive since there is more [[Information entropy#Formal definitions|uncertainty]] about the [[space|spatial]] locations of the different kinds of [[molecule]]s when they are interspersed. We assume that the mixing process has reached [[thermodynamic equilibrium]] so that the mixture is uniform and homogeneous. If the substances being mixed are initially at different temperatures and pressures, there will, of course, be an additional entropy increase in the mixed substance due to these differences being equilibrated, but if the substances being mixed are initially at the same temperature and pressure, the entropy increase will be entirely due to the entropy of mixing.
The entropy of mixing may be calculated by '''Gibbs' Theorem''' which states that when two different substances mix, the entropy increase upon mixing is equal to the entropy increase that would occur if the two substances were to expand alone into the mixing volume. (In this sense, then the term "entropy of mixing" is a misnomer, since the entropy increase is not due to any "mixing" effect.) Nevertheless, the two substances must be different for the entropy of mixing to exist. This is the [[Gibbs paradox|Gibbs paradox]] which states that if the two substances are identical, there will be no entropy change, yet the slightest detectable difference between the two will yield a considerable entropy change, and this is just the entropy of mixing. In other words, the entropy of mixing is not a continuous function of the degree of difference between the two substances.
The entropy of mixing <math>\Delta S_m\,</math> is given by:
:<math>\Delta S_m = -nR(x_1\ln x_1 + x_2\ln x_2)\,</math>
where <math>R\,</math> is the [[gas constant]], <math>n\,</math> the total number of [[mole (unit)|moles]] and <math>x_i\,</math> the [[mole fraction]] of each of the mixed components
==Proof==
Assume that the molecules of two different substances are approximately the same size, and regard space as subdivided into a [[lattice (group)#Lattices in two dimensions: detailed discussion|square lattice]] whose cells are the size of the molecules. (In fact, any lattice would do, including [[close-packing|close packing]].) This is a [[crystal]]-like [[mathematical model|conceptual model]] to identify the molecular [[center of mass|centers of mass]]. If the two [[phase (matter)|phases]] are [[liquid]]s, there is no spatial uncertainty in each one individually.{{Ref|1}} Everywhere we look in component 1, there is a molecule present, and likewise for component 2. After they are intermingled (assuming they are miscible), the liquid is still dense with molecules, but now there is uncertainty about what kind of molecule is in which location. Of course, any idea of identifying molecules in given locations is a [[thought experiment]], not something one could do, but the calculation of the uncertainty is well-defined.
We can use [[Boltzmann's entropy formula|Boltzmann's equation]] for the entropy change as applied to the ''mixing'' process
:<math>\Delta S_m= k_B \ln\Omega\,</math>
where <math>k_B \,</math> is [[Boltzmann constant|Boltzmann’s constant]]. We then calculate the number of ways <math>\Omega \,</math> of arranging <math>N_1\,</math> molecules of component 1 and <math>N_2\,</math> molecules of component 2 on a lattice, where
:<math>N = N_1 + N_2 \,</math>
is the total number of molecules, and therefore the number of lattice sites.
Calculating the number of [[Permutation#Counting permutations|permutations]] of <math>N\,</math> objects, correcting for the fact that <math>N_1\,</math> of them are ''identical'' to one another, and likewise for <math>N_2\,</math>,
:<math>\Omega = N!/N_1!N_2!\,</math>
After applying [[Stirling's approximation]], the result is
:<math>\Delta S_m = -k_B[N_1\ln(N_1/N) + N_2\ln(N_2/N)]\,</math>
This expression can be generalized to a mixture of <math>r\,</math> components, <math>N_i\,</math>, with <math>i = 1, 2, 3,... r\,</math>
:<math> \Delta S_m =-k_B\sum_{i=1}^r N_i\ln(N_i/N) = -k_B\sum_{i=1}^r N_i\ln x_i\,\!</math>
where we have introduced the [[mole fraction]]s, which are also the [[probability|probabilities]] of finding any particular component in a given lattice site.
::<math>x_i = N_i/N = p_i\,\!</math>
A more direct and logically transparent derivation, not requiring Stirling's approximation, is to start with the [[entropy#Mathematical description|Shannon entropy]] or [[Information entropy#Formal definitions|compositional uncertainty]]{{Ref|2}}
:<math> -k_B\,\sum_{i=1}^r p_i \ln (p_i)</math>
The summation is over the various chemical species, so this is the uncertainty about which kind of molecule is in any one site. It must be multiplied by the number of sites <math>N\,</math> to get the uncertainty for the whole system. The entropy of mixing from above can be rearranged as
:<math> \Delta S_m = -k_BN\sum_{i=1}^r (N_i/N)\ln(N_i/N)\,\!</math>
The equivalence of the two follows immediately.
Reverting to two components, we obtain
:<math>\Delta S_m = - R(n_1\ln x_1 + n_2\ln x_2) = -nR(x_1\ln x_1 + x_2\ln x_2)\,</math>
where <math>R\,</math> is the [[gas constant]], equal to <math>k_B \,</math> times [[Avogadro's number]], <math>n_1\,</math> and <math>n_2\,</math> are the numbers of moles of the components, and <math>n\,</math> is the total number of moles.
Since the mole fractions are necessarily less than one, the values of the [[logarithm]]s are negative. The minus sign reverses this, giving a positive entropy of mixing, as expected.
==Gibbs free energy of mixing==
In an [[ideal gas]] or [[ideal solution]] (no enthalpy term) the '''Gibbs free energy change of mixing''' is given by:
:<math>\Delta G_m = nRT(x_1\ln x_1 + x_2\ln x_2)\,</math>
where <math>\ \Delta G_m</math> is the [[Gibbs free energy]] and <math>\ T</math> the [[absolute temperature]] <ref>Graph as function of temperature, number of moles and composition: [http://www.whfreeman.com/elements/content/livinggraphs/E3012.html Link].</ref>
The Gibbs energy is always negative meaning that mixing as ideal solutions is always spontaneous. The lowest value is when the mole fraction is 0.5 for a mixture of two components or 1/n for a mixture of n components.
==Solutions==
If the [[solute]] is a [[crystal]]line [[solid]], the argument is much the same. A crystal has no spatial uncertainty at all, except for [[crystallographic defect]]s, and a (perfect) crystal allows us to localize the molecules using the crystal [[symmetry group]]. The fact that volumes do not add when dissolving a solid in a liquid is not important for condensed [[phase (matter)|phase]]s. If the solute is not crystalline, we can still use a spatial lattice, as good an approximation for an amorphous solid as it is for a liquid.
The [[Flory-Huggins solution theory]] provides the entropy of mixing for [[polymer]] solutions, in which the [[macromolecule]]s are huge compared to the solute molecules. In this case, the assumption is made that each [[monomer]] subunit in the polymer [[Chain (sequence)|chain]] occupies a lattice site.
Note that solids in contact with each other also slowly [[diffusion|interdiffuse]], and solid mixtures of two or more components may be made at will ([[alloy]]s, [[semiconductor]]s, etc.). Again, the same equations for the entropy of mixing apply, but only for homogeneous, uniform phases.
==Gases==
In gases there is a lot more spatial uncertainty because most of their volume is merely empty space. We can regard the mixing process as simply conjoining the two containers. The two lattices which allow us to conceptually localize molecular [[center of mass|centers of mass]] also join. The total number of empty cells is the sum of the numbers of empty cells in the two components prior to mixing. Consequently, that part of the spatial uncertainty concerning whether ''any'' molecule is present in a lattice cell is the sum of the initial values, and does not increase upon mixing.
Almost everywhere we look, we find empty lattice cells. But we do find molecules in those few cells which are occupied. For each one, there is a ''contingent'' uncertainty about which kind of molecule it is. Using [[Information theory#Conditional Entropy (Equivocation)|conditional probabilities]], it turns out that the analytical problem for the small [[subset]] of occupied cells is exactly the same as for mixed liquids, and the ''increase'' in the entropy, or spatial uncertainty, has exactly the same form as obtained previously. Obviously the subset of occupied cells is not the same at different times. But only when an occupied cell is found do we ask which kind of molecule is there.
See also: [[Gibbs paradox]], in which it would seem that mixing two samples of the ''same'' gas would produce entropy.
==Notes==
* {{Note|1}} 1. This is, of course, an approximation. Liquids have a “free volume” which is why they are (usually) less [[density|dense]] than [[solid]]s.
* {{Note|2}} 2. [[Claude Elwood Shannon|Claude Shannon]] introduced [[Information entropy#Formal definitions|this expression]] for use in [[information theory]], but similar formulas can be found as far back as the work of [[Ludwig Boltzmann]] and [[Josiah Willard Gibbs|J. Willard Gibbs]]. Shannon uncertainty is completely unrelated to the [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]] in [[quantum mechanics]].
== External links ==
* [http://www.phys.uri.edu/~gerhard/PHY525/tln25.pdf Online lecture]
* [http://www.msm.cam.ac.uk/phase-trans/mphil/MP4-3.pdf Online lecture]
==References==
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[[Category:Thermodynamic entropy]]
[[Category:Thermodynamics]]
[[uk:Ентропія змішування]]
[[zh:混合熵]]