# 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

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**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