Nuclear spin, entropy

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. 

Fig. 7.2. Entropies (divided by the gas constant R) of liquid and solid3 He along the melting curve. The disorder of nuclear spin entropy, corresponding to S /R = ln(2/ +1) = In 2 is marked. The two curves cross at the minimum of die melting curve at 315 mK and 29 bar [[12] p. 214].

A satisfactory explanation for this discrepancy was not available until the development of statistical thermodynamics with its methods of calculating entropies from spectroscopic data and the discovery of the existence of ortho- and parahydrogen. It then was found that the major portion of the deviation observed between Equations (11.24) and (11.25) is from the failure to obtain a tme equilibrium between these two forms of H2 molecules (which differ in their nuclear spins) during thermal measurements at very low temperatures (Fig. 11.4). If true equilibrium were established at all times, more parahydrogen would be formed as the temperature is lowered, and at 0 K, all the hydrogen molecules would be in the... 

Order-disorder transitions are generally associated with (i) positional disordering, (ii) orientational disordering or (iii) disordering of electronic (or nuclear) spin states. The configurational entropy due to disordering is given by... 

Evidence that the proton lies midway between the fluorine atoms in the crystal KHF has been provided by entropy measurements,28 study of the polarized infrared spectrum,29 neutron diffraction,80 and nuclear spin magnetic resonance.81 The uncertainty in the location of the proton at the midpoint between the fluorine atoms is reported to be 0.10 A for the neutron diffraction study and 0.06 A for the nuclear magnetic resonance study. 

When SP [T] = SP"[0] (condition 2), AS°[T] can be expressed as v (SP [T] — Sp"[0]) that is, in terms of the observed quantities. We use the difference (Sp [T] — SP"[0]) as the absolute value of the entropy, which is equivalent to assigning the value of zero to SP"[0]. The two effects for which this assignment is valid are (1) the nuclear effects including those of nuclear spin, provided that the isothermal change of state does not involve a nuclear reaction and (2) the isotopic effects, provided there is no change in the isotopic composition of the substances. 

White, Friedman, and Johnston (343) have measured the critical constants for normal hydrogen and have found 33.244 K. and 12.797 atmospheres. Woolley, Scott, and Brickwedde have presented data on the dissociation energy and the thermodynamic properties for the ideal diatomic gas, including contributions from nuclear spin. We have omitted the spin entropy in compiling our tables. Thermodynamic properties for the ideal monatomic gas have been computed at the National Bureau of Standards (395). Note that the reference state represents 2 gram atomic weights for this element. 

Since atoms retain their nuclear spins unaltered in all processes except those involving ortho-para conversions, there is no change in the nuclear spin entropy. It is consequently the common practice to omit the nuclear spin contribution, leaving what is called the practical entropy or virtual entropy. ... 

It is of importance to note that, except for hydrogen and deuterium molecules, the entropy derived from heat capacity measurements, i.e., the thermal entropy, as it is frequently called, is equivalent to the practical entropy in other words, the nuclear spin contribution is not included in the former. The reason for this is that, down to the lowest temperatures at which measurements have been made, the nuclear spin does not affect the experimental values of the heat capacity used in the determination of entropy by the procedure based on the third law of thermodynamics ( 23b). Presumably if heat capacities could be measured right down to the absolute zero, a temperature would be reached at which the nuclear spin energy began to change and thus made a contribution to the heat capacity. The entropy derived from such data would presumably include the nuclear spin contribution of R In (2i + 1) for each atom. Special circumstances arise with molecular hydrogen and deuterium to which reference will be made below ( 24n). 

The thermal entropy of normal deuterium was found to be 33.90 E.u. mole . Normal deuterium consists of two parts of ortho- to one part of para-molecules at low temperatures the former occupy six and the latter nine closely spaced levels. The spin of each deuterium nucleus is 1 unit. Show that the practical standard entropy of deuterium gas at 25 C is 34.62 e.u. mole" (Add the entropy of mixing to the thermal entropy and subtract the nuclear spin contribution.) Compare the result with the value which would be obtained from statistical calculations, using moment of inertia, etc. in Chapter VI. 

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. 

This gives the configurational entropy of mixing for any number of components. It can be used to calculate residual entropies at absolute zero due to impurities, imperfections, nuclear spin, isotojjes, etc., simply by considering the imperfections as one component of a mixture. Equation (6.37) applies equally well to ideal mixtures at higher temperatures, as we shall see in Chapter 10. 

Finally, we should point out that, while the exceptions to the Third Law noted above may be a headache for scientists who measiue calorimetric properties of materials, they pose no practical problems in most chemical applications. Chemical reactions alone do not change nuclear spin, and in many cases do not alter isotope ratios significantly, so that configurational contributions to the entropy of reactants are normally balanced by those of the products in a reaction. In most cases these effects are thermodynamically minor or insignificant. 

The nuclear spin can be oriented in two ways this leads to an entropy of In 2 per nucleus or a total of 2R In 2. Since this contribution to the entropy persists through all changes, it is not taken as part of the residual entropy and so must be subtracted from the entropy of mixing above. This yields for the residual entropy of hydrogen at 0 K, 18.38 — 2i In 2 = 6.8 J/K mol, which is in good agreement with the observed value of 6.2 J/K mol. 

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. 

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