Entropy of isotope mixing

For those who are familiar with the statistical mechanical interpretation of entropy, which asserts that at 0 K substances are nonnally restricted to a single quantum state, and hence have zero entropy, it should be pointed out that the conventional thennodynamic zero of entropy is not quite that, since most elements and compounds are mixtures of isotopic species that in principle should separate at 0 K, but of course do not. The thennodynamic entropies reported in tables ignore the entropy of isotopic mixing, and m some cases ignore other complications as well, e.g. ortho- and para-hydrogen. 

A special case of opposing reactions is the one in which chemical equilibrium has been attained, but not isotopic equilibrium. Isotopic equilibration reactions are termed exchange reactions. They occur with virtually no net driving force i.e., AG 4 is very nearly zero, save for that provided by the entropy of isotopic mixing. 

The quantity in the square brackets is called the separative duty. The separative duty is a function only of the product, waste, and feed concentrations and amounts. It is a quantitative measure of the separative work done by the cascade. The function (2x — 1) In x/l — x is termed the separation potential. It has a minimum at x = 1/2, where it is equal to zero. At all other values of x the separation potential is positive. The separative duty is always positive. This statement is consistent with the fact that the entropy of isotope mixing is positive. 

In Chapter 12 we discussed randomization over translational and configurational states and also, to some extent, over rotational, vibrational and electronic states. Two other factors which are known to contribute small amounts to the entropy are (a) the number of possible orientations of the nuclear spins and (6) the entropy of isotope mixing. (The latter factor arises if the substance in question contains two or more isotopes.). However, this leaves quite untouched the possibility of randomness within the nncUus about which nothing whatever is known. It is evident that even when we have allowed for all the known factors our computed entropy may still be incomplete. 

Statistical mechanics predicts that at absolute zero the various vibrations of a crystal lattice all fall into a single lowest-energy state, as do the electronic motions. If there is no entropy of isotopic mixing,... 

It is noteworthy that in the above discussion it has been unnecessary to mention or consider the entropy of mixing of the para and ortho molecules. This is simply because this contribution to the entropy has the same value in all the states considered and may, like the mixing entropy of isotopes, be ignored. 

In brief, it is not possible to calculate absolute entropies. Instead it is necessary to adopt some convention concerning what factors are to be included in Q, in view of the present state of knowledge. The convention which is usually adopted in physical chemistry is that the entropy is taken as zero when the substance is in a physical state such that translational, configurational, rotational, vibrational and electronic contributions to the entropy are all zero. Contributions due to the nucleus, including its spin, are ignored and also the effect of isotope mixing. The justification for omitting these factors lies in the fact that nuclei are conserved in chemical processes and also because the isotopic composition usually remains almost constant. 

The entropy of mixing of very similar substances, i.e. the ideal solution law, can be derived from the simplest of statistical considerations. It too is a limiting law, of which the most nearly perfect example is the entropy of mixing of two isotopic species. 

The values of S° represent the virtual or thermal entropy of the substance in the standard state at 298.15 K (25°C), omitting contributions from nuclear spins. Isotope mixing effects are also excluded except in the case of the H—system. 

In summary, the absolute entropies we calculate and tabulate are, in fact, not so absolute, since they do not include isotopic entropies of mixing nor nuclear spin alignment entropies. The entropies we tabulate are sometimes called practical absolute entropies. They can be used to correctly calculate AS for a chemical process, but they are not true" absolute entropies. 

In contrast, the acid-catalyzed hydrolysis of alkyl selenates is A-2158. The actual species which undergoes decomposition to alcohol and sulfur trioxide is probably the zwitterion as in the case of phosphate monoester monoanions. Evidence for sulfur trioxide as the reactive initial product of the A-1 solvolysis is obtained from the product compositions arising with mixed alcohol-water solvents. The product distribution is identical to that found for sulfur trioxide solvolysis, with the latter exhibiting a three-fold selectivity for methanol. Although the above entropies of activation and solvent deuterium isotope effects do not distinguish between the conventional A-l mechanism and one involving rate-limiting proton transfer, a simple calculation, based on the pKa of the sulfate moiety and the fact that its deprotonation is diffusion controlled. 

The gap to the perfect random distribution is evaluated, in the case of the isotopic mixed crystal, by comparing the entropy term related to the permutation of the molecules, of the order of kfT, with the elastic energy Ve] introduced by the substitution of one hydrogenated cell by one deuterated cell. Then at the temperature formation, we find kTf 102 Vcl, which is consistent with a random distribution. 

The specific acid-catalyzed reaction behaves in many respects like the water reaction. Thus, unusually large negative entropies of activation are found for all but those substrates which react by the acylium ion path (Schaleger and Long, 1963). At least two water molecules are involved in the transition state, the presence of at least five exchangeable hydrogens in the transition state, is deduced from solvent isotope effects in mixed water-heavy water (Salomaa et al., 1964). From the dependence of both acid-catalyzed exchange and hydrolysis in concentrated sulfuric acid on the activity of water. Lane (1964) concluded that two water molecules must be involved in the transition state. Except for the extra proton, the transition states for the acid-catalyzed and water reactions are very probably similar. 

It may be noted, incidentally, that the practical or virtual entropy also does not include the entropy of mixing of different isotopic forms of a given molecular speciee. This quantity is virtually unchanged in a chemical reaction, and so the entropy change of the process is unaffected by its complete omission. 

There are, however, several ways in which a crystal may fail to be truly perfect. Isotopic mixtures of, say 35C1 and 37C1 give rise to an entropy of mixing different combinations of nuclear spin, as occur in ortho- and para-hydrogen, will cause some randomization. Crystals are seldom perfect, and there are other effects which must be accounted for. 

By "conventional entropy" we mean the sum of all contributions to the entropy from translations, rotations, internal Tbrations and electronic degrees of freedom but excluding nuclear degrees of freedom, in particular nuclear spin, and isotopic mixing. 

The neglect of these two effects results in a practical entropy scale, or conventional entropy scale, on which the crystal that is assigned an entropy of zero has randomly-mixed isotopes and randomly-oriented nuclear spins, but is pure and ordered in other respects. This is the scale that is used for published values of absolute third-law molar entropies. The shift of the zero away from a completely-pure and perfectly-ordered crystal introduces no inaccuracies into the calculated value of AS for any process occurring above 1 K, because the shift is the same in the initial and final states. 

Statistical mechanical theory applied to spectroscopic measurements provides an accurate means of evaluating the molar entropy of a pure ideal gas from experimental molecular properties. This is often the preferred method of evaluating Sm for a gas. The zero of entropy is the same as the practical entropy scale—that is, isotope mixing and nuclear spin interactions are ignored. Intermolecular interactions are also ignored, which is why the results apply only to an ideal gas. 

Equation (5.1) includes only the ideal, combinatorial entropy of mixing and the simplest conceivable regular solution type estimate of the enthalpy of mixing based on completely random mixing of monomers mm ( ) = 1 in the liquid state language i referred to as the bare chi parameter since it ignores all aspects of polymer architecture and Interchain nonrandom correlations. For these reasons, the model blend for which Eq. (5.1) is thought to be most appropriate for is an interaction and structurally symmetric polymer mixture. The latter is defined such that the only difference between A and B chains is a v (r) tall potential, which favors phase separation at low temperatures. The closest real system to this idealized mixture is an isotopic blend, where the A and B... 

Assume that an element exists that has two isotopes, and that each isotope has an abundance of 50.00%. Using the formula for the entropy of mixing for an ideal gas, calculate A5mix for a sample of 1.000 mol of this element. 

From the general point of view of Nemst s theorem there are tw > possibilities to make the isotopic entropy of mixing vanish at sufficiently low temperatures either isotopic phase separation or formation of some regular structure. We see that it is the first possibility which is realized. We shall give a more general derivation of this important result in the next section. 

Some research topics in entropy are the determination of entropy changes in the mixing of very similar species (e.g., isotopes), in very cold systems (i.e., T -> 0), and in highly dispersed systems. 

The normal isotopic abundances for Li are 92.48 mole % for 7Li and 7.52 mole % for 6Li. Making reasonable approximations, determine the entropy, enthalpy, and Gibbs free energy changes on mixing the pure isotopes. Discuss your results in terms of the statements made in Section 1.21 in conjunction with the Third Law of Thermodynamics. 

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