Computing the relative gas-phase populations of C₆₀ and C₇₀: Beyond the traditional ΔH_f,298^o scale

Computations and experiments have shown that the relative heat of formation (i.e., the heat of formation per carbon atom) of C₇₀ is lower than of C₆₀. Moreover, various computations suggest that this is actually a general trend among fullerene cages. The relationship is particularly important for gas-phase fullerenes. Experiments have shown that C₆₀ is typically more populated than C₇₀ when produced in high-temperature gas-phase synthesis. It is not immediately obvious how to reconcile those two terms, or whether the relative heats of formation and the relative populations are in conflict or in agreement. This article deals with this problem, treating it as a general task of relative stabilities of gas-phase clusters of different dimensions (i.e., nonisomeric clusters) under different types of thermodynamic equilibria. The results are then applied to C₆₀ and C₇₀ and point out that the conventional standard pressure of 1 atm is considerably different from actual fullerene-synthesis conditions. Apparently, we should expect considerably lower cluster pressures in carbon-arc synthesis. At 1 atm, C₇₀ is more populated than C₆₀, but at the conditions of a saturated carbon vapor the stability order is reversed in favor of C₆₀ so that an agreement with experiment is obtained already within the thermodynamic treatment. The pressure effects are modeled using the MNDO, AM1, PM3, and SAM1 quantum-chemical semi-empirical methods as well as the available experimental data. The computations consistently show that, if the pressure effects are considered, C₆₀ becomes more populated than C₇₀. Relationships of the thermodynamic treatment to more sophisticated but impractical kinetic analysis are also discussed.