User:David Shear/Entropy of Mixing
The entropy of mixing is the uncertainty about the spatial locations of the various kinds of molecules in a mixture. In a pure condensed phase, there is no spatial uncertainty: everywhere we look, we find the same kind of molecule. A single-component gas is mostly empty space, but when we do encounter a molecule, the is no doubt about what kind it is. When two or more substances are interdispersed, we may know the various proportions, but we have no way of knowing which kind of molecule is where. The notion of "finding" a molecule in a given location is a thought experiment, since we can not actually examine spatial locations the size of molecules. Moreover, individual molecules of a given kind are all identical, so we never ask which one is where. "Interchanging" two identical objects is not a real process; it does not lead to a physically distinct condition.
Some liquids will mix while others are
Derivations of the entropy of mixing usually begin with
Strategy
Imagine space to be subdivided into a
![](http://upload.wikimedia.org/wikipedia/commons/1/17/5x5checkerboard.png)
Picture an enormous checkerboard. Relax the condition that the black and white squares are equal in number. Generalize to three dimensions. For a gas, imagine that a great excess of white squares represents empty space. For a mixture, give each component its own color. Now imagine all possible spatial rearrangements. This is our model of the different configurations for the molecules in a system.
A pure
A gas has a huge amount of spatial uncertainty because most of the volume is empty space, which plays the role of “solvent”. For a single-component gas, the only question is: does a lattice site contain the center of mass of a gas molecule, or is it empty? The entropy increase accompanying the free expansion of a gas into a vacuum may be regarded as the entropy of mixing of the gas with empty space. In a mixture of gases, there is a second question, which arises only for occupied sites: which kind of molecule is present?
Boltzmann's method
The fundamental assumption of
in which is the number of (unobservable) microscopic "ways" the molecules can be assigned to different conditions or states consistent with the overall
The justification for splitting position-momentum phase space into a position part, which we will use, and a momentum (energy) part, which we will ignore, is that for all molecular materials at room temperature, the thermal de Broglie wavelength is much less than intermolecular distances; in fact, it is less than actual molecular diameters. In this classical limit Heisenberg's uncertainty principle is irrelevant. We can talk about a classical gas expanding from one corner of an enclosure to fill an entire enclosure, a process which has no sensible meaning using the Schrödinger time-independent wave equation.
Consider a mixture of molecules of two kinds.
in which is the total number of lattice sites, is the number of molecules of component , is the number of molecules of component , and is the number of empty lattice sites — which is zero for a crystal, and small for other condensed phases. The total number of molecules is
Shannon's method
A shorter and more logically transparent method, not requiring require Stirling's approximation, is to use
We employ the same (real or) conceptual lattice, where
is the probability that a molecule of is in any given lattice site, equal to the number of molecules of , , divided by the number of lattice sites, . The summation is over all the chemical species present, so this is the uncertainty about which kind of molecule (if any) is in any one site. It must be multiplied by the total number of sites to get the spatial uncertainty for the whole system. Comment: All this description is not correct. In Gibbs (Shannon) entropy the summation is over the microstates and not over the states. In equilibrium all microstates have the same probability. I recommend to read "entropy God's dice game" www.entropy-book.com
Condensed phases
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Consider a mixture of molecules of two kinds. We are after the number of possible patterns or configurations achievable with molecules of component and molecules of component arranged on a lattice with total sites. This is given by the formula for the permutations of things subject to the condition that of them are identical, and likewise for and .
where is the number of empty lattice sites — zero for a crystal () and a small fraction for other condensed phases, but by far the greatest part of in a gas. It can also be taken as the number of
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Let us proceed first by using the traditional Boltzmann formula. The simplest case is a mixture with only two components, and . We are after the number of possible patterns or configurations achievable with molecules of and molecules of arranged on a lattice with sites. For a condensed phase, the number of sites is equal to the total number of molecules, .
The number of distinct configurations is given by the formula for the permutations of things subject to the condition that of them are identical, and likewise for .
Using this algebraic form in Boltzmann's equation and applying Stirling's approximation for the logarithms of factorials, the configurational uncertainty, or entropy of mixing, turns out to be
which has been written using the conventional notation ( denotes a change), suggesting that the mixture has been formed by a mixing process from two separate pure phases, each of which originally had no spatial uncertainty. This expression can be generalized to a mixture of components, with
We have introduced the mole fractions, which are also the probabilities of finding any particular component in a given lattice site.
For the two-component case,
where is the gas constant, equal to times
Shannon's formula yields the desired result directly.
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 to get the uncertainty for the whole system. Doing this, and using the fact that , we obtain
which is the same as the result obtained using Boltzmann's formula. The two methods are essentially equivalent. (But see the Discussion.)
Solutions
If the
The
Note that solids in contact with each other also slowly interdiffuse, and solid mixtures of two or more components may be made at will (alloys, semiconductors, etc.). Again, the same equations for the entropy of mixing apply, but only for homogeneous, uniform phases.
Gases
In order to get the total entropy of a gas, we must also calculate the contingent uncertainty about the momentum of a molecule for each lattice site that is found to contain one. We obtain a Boltzmann distribution over energies and a partition function which depend on (and define) the temperature, and add this to the spatial uncertainty. But in regard to mixing, we are concerned only with spatial entropy.
If we have a pure gas consisting of molecules, we want to calculate the number of ways, or occupancy patterns, of arranging occupied sites and empty sites on a lattice with total sites.
and
But we have just performed this calculation above, although with a different interpretation of = . Clearly, the spatial uncertainty in gas entropy is just the entropy of mixing of gas molecules and empty space. For a pure gas, considering just the spatial uncertainty part of the entropy,
The simplification is possible because is just slightly less than one and its log is negligible; most of the space in a gas is empty lattice sites. Note that is the molecular concentration, or number density, of the gas molecules, where is the volume of a single lattice site and is the total volume of the system. The reciprocal of this quantity is the volume per molecule, . So long as this is large with respect to , the cube of the thermal de Broglie wavelength, we can be sure that the "wave packets" for the molecules hardly ever touch, and the classical mechanical treatment is the appropriate one. For all real gases at room temperature, this condition is more than satisfied.
In the ideal gas approximation, which is pretty good for dilute gases at normal temperatures, volumes are additive for two samples of different gases combined at constant and . In any case, let be the number of molecules of a second type of gas in a mixture. The spatial part of the entropy of the mixture is times the log of - - - - - - - - - - - - - - - Consider a mixture of molecules of two kinds.
in which is the total number of lattice sites, is the number of molecules of component , is the number of molecules of component , and is the number of empty lattice sites — which is zero for a crystal, and small for other condensed phases. The total number of molecules is
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We can regard the mixing of two kinds of gas (at constant and ) as simply conjoining the two containers. The two lattices which allow us to conceptually localize molecular 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 sites. But for those few sites which are occupied, there is a contingent uncertainty about which kind of molecule it is. Using 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.
See also:
Discussion
The preceding analysis is only an approximation, except for dilute gases. It is not too bad for mixtures of denser gases, or for liquids or amorphous solids with molecules of about the same size. Likewise for crystalline mixtures. We have not considered intermolecular forces (energies). Mixing substances whose molecules cross-react differently than they do in their pure phases results in a (positive or negative) heat (or enthalpy) of mixing, in addition to considerations of entropy. We have ignored any correlations in the dispositions of neighboring molecules, including angular orientations due to molecular shapes, or due to any other geometrical or energetic reason, such as clouds of counter-ions surrounding charged colloidal particles. The fundamental assumption is that all occupancy patters, or spatial "microstates", are counted as equally likely. But biasing effects of near neighbor interactions could perhaps be incorporated into the theory.
It is desirable to maintain the form of the equations derived above, even if correction factors (
There is a tacit mathematical assumption involved in using the Shannon entropy which might have escaped notice, which makes it differ in an interesting way from the Boltzmann formula. If we "find" a molecule of type in the first location we examine, there are only molecules of left to be found in the remaining lattice sites. That is, one site and one molecule of have each been "used up" and we should proceed only after taking that into account. This is possible but algebraically messy. However, it is not a problem in the thermodynamic limit of large systems, where we can regard our system as a smaller subsystem defined by geometrical "walls" through which molecules can pass. In this case, is a time-average value and not a rigid constraint. This is the idea behind Gibbs' grand canonical ensemble. But for systems of finite size, it is the original Boltzmann formulation for entropy in terms of factorials which is really correct, since it uses the actual particle numbers, making the presentation of the "long way" instructive. The use of Stirling's approximation also eliminates any mathematical distinction between the two ensembles, producing the same final results and making epistemological arguments about their inner meanings moot. For systems with a small number of particles, if it is possible to use the Boltzmann formula without Stirling's approximation, that would give the more accurate result. However, the idea that the canonical ensemble represents an external heat bath which maintains constant by maintaining an average internal energy still stands.
Notes
- ^ In addition to spatial uncertainty, all substances also have uncertainty regarding the energies of their molecules (or degrees of freedom). This part of the entropy can be determined by integrating the specific heat over the absolute temperature from up to the ambient temperature. (See Measuring entropy.)
- Flory-Huggins theory.
- ^ 1a. There is often a volume change on mixing: the initial volumes don't quite sum to the volume of the mixture due to interstitial packing and specific molecular interactions.
- this expression for use in information theory, but similar formulas can be found as far back as the work of Ludwig Boltzmann and J. Willard Gibbs. Shannon uncertainty is completely unrelated to the Heisenberg uncertainty principle in quantum mechanics. Note that "uncertainty" in the entropic sense is unrelated to Heisenberg uncertainty.
External links
While this reasoning yields the correct ideal gas equation of state, it also leads to Gibbs paradox, in which it (erroneously) appears that mixing two samples of the same kind of gas leads to an increase in entropy.
only to the different spatial arrangements: will be