# Distinguishability in Entropy Calculations: Chemical Reactions, Conformational and Residual Entropy

## Abstract

**:**

## 1. Introduction

^{rd}law entropies of small molecules can be reproduced with extreme accuracy if the entropy reduction is employed. It therefore appears that the experimental values can be reproduced only by using this correction, or equivalently, by adopting truly quantum statistics from which the entropy reduction emerges naturally due to the symmetry of the wave function [12].

## 2. Discussion

#### 2.1. Indistinguishable Particles and Third Law Entropies

^{rd}law entropies of small molecules in the gas-phase can be computed easily and with remarkable precision by feeding thermodynamic statistical formulae with molecular properties computed with quantum chemical methods [18]. For example, Table 1 shows both theoretical and experimental entropy values reported in the literature for small alkanes [19]. For simplicity, the examples in Table 1 are selected so that the ω possible conformers for each molecule, if any, are not only isoenergetic, but also chemically identical. Thus, the conformational entropy would be zero, depending whether or not we are considering indistinguishability or distinguishability among identical conformers.

Molecule | ${\sigma}_{ext}$ | Theory | Experiment | Abs. Error |
---|---|---|---|---|

methane | 12 | 186.20 | 186.37 | 0.17 |

ethane | 6 | 228.50 | 229.16 | 0.66 |

propane | 2 | 270.20 | 270.31 | 0.11 |

methylpropane | 3 | 295.50 | 295.70 | 0.20 |

dimethylpropane | 12 | 306.74 | 306.00 | 0.74 |

2,2-dimethylbutane | 1 | 358.70 | 358.40 | 0.30 |

#### 2.2. Distinguishable Particles and Third Law Entropies

^{rd}law holds, i.e.,

**Table 2.**Theoretical (B3LYP/cc-pVTZ) and experimental entropy values in $J{K}^{-1}mo{l}^{-1}$ augmented by $Rln\left(\omega {\sigma}_{ext}\right)$ due to the distinguishability.

Molecule | ω | ${\sigma}_{ext}$ | Theory | Experiment | Abs. Error |
---|---|---|---|---|---|

methane | 1 | 12 | 206.86 | 207.03 | 0.17 |

ethane | 3 | 6 | 252.53 | 253.19 | 0.66 |

propane | ${3}^{2}$ | 2 | 294.23 | 294.34 | 0.11 |

methylpropane | ${3}^{3}$ | 3 | 332.03 | 332.23 | 0.20 |

dimethylpropane | ${3}^{4}$ | 12 | 363.93 | 363.19 | 0.74 |

2,2-dimethylbutane | ${3}^{5}$ | 1 | 404.37 | 404.07 | 0.30 |

#### 2.3. Entropy Changes in Chemical Reactions: Is There Any Difference?

_{3}Cl + Cl

_{2}⇌ CH

_{2}Cl

_{2}+ HCl

## 3. Conclusions

## Acknowledgements

## References

- McQuarrie, D. Statistical Mechanics; University Science Books: Sausalito, CA, USA, 2000. [Google Scholar]
- Gilson, M.K.; Irikura, K.K. Symmetry numbers for rigid, flexible, and fluxional molecules: Theory and applications. J. Phys. Chem. B
**2010**, 114, 16304–16317. [Google Scholar] [CrossRef] [PubMed] - Ben-Naim, A. On the so-called Gibbs paradox, and on the real paradox. Entropy
**2007**, 9, 132–136. [Google Scholar] [CrossRef] - Jaynes, E. The Gibbs paradox. In Maximum Entropy and Bayesian Methods; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992. [Google Scholar]
- Lesk, A. On the Gibbs paradox: What does indistinguishability really mean? J. Phys. A: Math. Gen.
**1980**, 13, L111–L114. [Google Scholar] [CrossRef] - Lin, S.K. Correlation of entropy with similarity and symmetry. J. Chem. Inf. Comput. Sci.
**1996**, 36, 367–376. [Google Scholar] [CrossRef] - Lin, S.K. Gibbs paradox and the concepts of information, symmetry, similarity and their relationship. Entropy
**2008**, 10, 1–5. [Google Scholar] [CrossRef] - Versteegh, M.A.M.; Dieks, D. The Gibbs paradox and the distinguishability of identical particles. Am. J. Phys.
**2011**, 79, 741–746. [Google Scholar] [CrossRef] - Pauli, W. Thermodynamics and the Kinetic Theory of Gases. In Pauli Lectures on Physics; MIT Press: Cambridge, MA, USA, 1973. [Google Scholar]
- Swendsen, R. Statistical mechanics of classical systems with distinguishable particles. J. Stat. Phys.
**2002**, 107, 1143–1166. [Google Scholar] [CrossRef] - Nagle, J. Regarding the entropy of distinguishable particles. J. Stat. Phys.
**2004**, 117, 1047–1062. [Google Scholar] [CrossRef] - Pathria, R. Statistical Mechanics; Butterworth-Heinemann: Oxford, UK, 1996. [Google Scholar]
- Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc.
**1935**, 57, 2680–2684. [Google Scholar] [CrossRef] - Kozliak, E. Consistent application of the boltzmann distribution to residual entropy in crystals. J. Chem. Educ.
**2007**, 84, 493–498. [Google Scholar] [CrossRef] - Kozliak, E.; Lambert, F.L. Residual entropy, the third law and latent heat. Entropy
**2008**, 10, 274–284. [Google Scholar] [CrossRef] - Ercolany, G.; Piguet, C.; Borkovec, M.; Hamacek, J. Symmetry numbers and statistical factors in self-assembly and multivalency. J. Phys. Chem. B
**2007**, 111, 12195–12203. [Google Scholar] [CrossRef] [PubMed] - Cheng, C.H. Thermodynamics of the system of distinguishable particles. Entropy
**2009**, 11, 326–333. [Google Scholar] [CrossRef] - DeTar, D.F. Theoretical ab initio calculation of entropy, heat capacity, and heat content. J. Phys. Chem. A
**1998**, 102, 5128–5141. [Google Scholar] [CrossRef] - NIST Computational Chemistry Comparison and Benchmark Database. Available online: http://cccbdb.nist.gov/ (accessed on 10 June 2011).
- Dunning, T. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys.
**1989**, 90, 1007. [Google Scholar] [CrossRef] - Chang, C.; Chen, C.; Gilson, M.K. Ligand configurational entropy and protein binding. Proc. Nat. Acad. Sci. USA
**2007**, 104, 1534–1539. [Google Scholar] [CrossRef] [PubMed] - Suárez, E.; Díaz, N.; Suárez, D. Entropic control of the relative stability of triple-helical collagen peptide models. J. Phys. Chem. B
**2008**, 112, 15248–15255. [Google Scholar] [CrossRef] [PubMed] - Zhou, H.X.; Gilson, M.K. Theory of free energy and entropy in noncovalent binding. Chem. Rev.
**2009**, 109, 4092–4107. [Google Scholar] [CrossRef] [PubMed]

© 2011 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

## Share and Cite

**MDPI and ACS Style**

Suárez, E.
Distinguishability in Entropy Calculations: Chemical Reactions, Conformational and Residual Entropy. *Entropy* **2011**, *13*, 1533-1540.
https://doi.org/10.3390/e13081533

**AMA Style**

Suárez E.
Distinguishability in Entropy Calculations: Chemical Reactions, Conformational and Residual Entropy. *Entropy*. 2011; 13(8):1533-1540.
https://doi.org/10.3390/e13081533

**Chicago/Turabian Style**

Suárez, Ernesto.
2011. "Distinguishability in Entropy Calculations: Chemical Reactions, Conformational and Residual Entropy" *Entropy* 13, no. 8: 1533-1540.
https://doi.org/10.3390/e13081533