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How can the stabilities of the tropylium and cyclopropenyl carbocations be compared?

Structures of carbocations

Both are aromatic according to Hückel's rule. I think that to compare the stabilities, we must either count the number of resonance structures, or perhaps we could say that cyclopropenyl is less stable due to angle strain. Am I on the right track?

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Both ions are indeed strongly stabilized by aromatic coupling, and salts containing both ions are known. But, of course, aromaticity is not the only thing that determines stability. Tropylium ion, with its lesser ring strain and more atoms over which to distribute the positive charge, forms ionic salts with more anions (e.g., ionic $\ce{C_7H_7Br}$ versus covalent $\ce{C_3H_3Br}$) and in that sense is more stable.

The ionic structure of some tropylium salts is documented in Wikipedia[1][2][3][4]

In 1891 G. Merling obtained a water-soluble bromine containing compound from a reaction of cycloheptatriene and bromine.[1] Unlike most hydrocarbyl bromides, this compound, later named tropylium bromide, is a water-soluble solid and is insoluble in hydrocarbons, chloroform, and ether. It is purified by crystallization from hot ethanol. Reaction with aqueous silver nitrate immediately gave a precipitate of silver bromide. The structure of tropylium bromide was deduced to be a salt, C7H7+Br–, by Doering and Knox in 1954 by analysis of its infrared and ultraviolet spectra.[2][3] The ionic structures of tropylium perchlorate (C7H7+ClO4–) and tropylium iodide (C7H7+I–) in the solid state have been confirmed by X-ray crystallography.[4] The bond length of the carbon-carbon bonds were found to be longer (147 pm) than those of benzene (140 pm) but still shorter than those of a typical single-bonded species like ethane (154 pm).

For examples of cyclopropenyl cation salts (e.g. $\ce{C3H3SbCl6}$) see Breslow and Groves[5].

References:

  1. Merling, G. Ueber Tropin. Ber. Dtsch. Chem. Ges. 1891, 24 (2), 3108–3126. DOI: 10.1002/cber.189102402151.

  2. von E.doering, W.; Knox, L. H. The Cycloheptatrienylium (Tropylium) Ion. J. Am. Chem. Soc. 1954, 76 (12), 3203–3206. DOI: 10.1021/ja01641a027.

  3. Balaban, A. T.; Oniciu, D. C.; Katritzky, A. R. Aromaticity as a Cornerstone of Heterocyclic Chemistry. Chem. Rev. 2004, 104 (5), 2777–2812. DOI: 10.1021/cr0306790.

  4. Kitaigorodskii, A. I.; Struchkov, Y. T.; Khotsyanova, T. L.; Vol'pin, M. E.; Kursanov, D. N. Crystal structures of tropylium perchlorate and iodide. Russ Chem Bull 1960, 9 (1), 32–36. DOI: 10.1007/BF01178699.

  5. Ronald Breslow and John T. Groves (1970). "Cyclopropenyl cation. Synthesis and characterization". J. Am. Chem. Soc. 92 (4), 984–987. https://doi.org/10.1021/ja00707a040

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