Refining Campbell's Rule

The numerical constant in this approximation was obtained by averaging the values of ∆_rS°/∆n_g calculated for ten of the eleven example reactions (excluding the one for which ∆n_g = 0) given in Table 1 of Craig’s paper (1), and then combining this with the average of the values of ∆_rS°/∆n_g for 18 of the 23 reactions given in Tables 1 and 2 of an earlier paper by Campbell (again excluding the cases where ∆n_g = 0) (2). The average for Craig’s data was 148 J (K mol rxn)^–1(mol)^–1, whereas that for Campbell’s data was 136 J (K mol rxn)^–1(mol)^–1, with a combined average of 140 J (K mol rxn)^–1(mol)^–1, as given in the approximation in eq 1.

If, instead of following the above procedure, one plots the values of ∆_rS° versus ∆n_g (including those for which ∆n_g = 0) for each data set, and uses a least squares analysis to obtain the best straight line fit, one obtains the results (rounded to the nearest tenth) summarized in Table 1, where those for Craig’s eleven examples are given in the first row; those for Campbell’s 23 examples¹ are given in the second row; and those for the combined Craig–Campbell data sets are given in the third row.

If one evaluates the correlation equation for Craig’s data (row 1) at ∆n_g = 1, one obtains a value of 149.8 J (K mol rxn)^–1, which is essentially identical to Craig’s reported average of 148 J (K mol rxn)^–1(mol)^–1. If, however, one does the same for Campbell’s data set (row 2), one obtains a value of 120.9 (K mol rxn)^–1 at ∆n_g = 1, which is substantially lower than the value of 136 (K mol rxn)^–1(mol)^–1reported by Craig. This discrepancy is apparently due to the fact that we have included the ∆n_g = 0 cases eliminated by Craig. Because of this discrepancy, a similar divergence is obtained for the case of the combined data sets (row 3), where ∆n_g = 1 gives an average value of 128 J (K mol rxn)^–1 rather than the value of 140 J (K mol rxn)^–1(mol)^–1 reported by Craig. Since the intercept of the correlation equation for the combined case is so small (–0.8 J (K mol rxn)^–1), the full equation may be approximated by the simpler relation:

which gives a more statistically significant result than does the approximation in eq 1.

The question naturally arises as to why the average for Campbell’s data set is so much lower than that for Craig’s data set. In this regard, it is of interest to note that only 6 out of 23, or about 26%, of Campbell’s examples involve solids or liquids as well as gases, whereas 9 out 11, or about 82%, of Craig’s examples do. In other words, most of Campbell’s examples involve only the compensation of ∆n_g terms, whereas many of Craig’s examples involve the additional compensation of ∆n_s and ∆n₁ terms as well. Likewise, 7 out of 23, or about 30%, of Campbell’s examples involve H₂(g) or H(g), whereas only 2 out of 11, or about 18%, of Craig’s examples do. The relevance of this latter observation has to do with the well-known logarithmic dependence of the entropy of translation (∆_trS°) of gases on their molecular weights (MW), as given by the Sackur–Tetrode equation (3):

As Craig emphasizes, Campbell’s rule is based on the fact that ∆_rS° is dominated by the ∆_trS° values of the gaseous species and this logarithmic dependency means, in turn, that ∆_trS° varies more rapidly for gases of low molecular weight, such as H₂ and H, than it does for gases of higher molecular weight,² as may be seen from Figure 4.4 of ref 3.

Notes

1. This correlation incorporates the corrected entropy values given by Craig in footnote 2 of his paper (1).
2. Interestingly, the weighted average of the molecular weights of all of the gaseous species appearing in Craig’s examples is 30.9, which gives a value of ∆_trS° = 151.8 J (K mol)^–1 when substituted into the Sackur–Tetrode equation. This is a reasonable approximation to the average value of 149.8 J (K mol rxn)^–1 for ∆_rS° reported for this data set. Unfortunately, no such similar agreement is obtained with Campbell’s data set.

Literature Cited

1. Craig, N. C. J. Chem. Educ. 2003, 80, 1432–1436.
2. Campbell, J. A. J. Chem. Educ. 1985, 62, 231–232.
3. Dascent, W. E. Inorganic Energetics, 2nd ed.; Cambridge University Press: Cambridge, 1982; pp 137–138.

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