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Despite recent advances in monitoring nucleation from a vapor at close-to-molecular resolution, the identity of the critical cluster, forming the bottleneck for the nucleation process, remains elusive. During past twenty years, the first nucleationtheorem has been often used to extract the size of the critical cluster from nucleation rate measurements. However, derivations of the first nucleationtheorem invoke certain questionable assumptions that may fail, e.g., in the case of atmospheric new particle formation, including absence of sub-critical cluster losses and heterogeneous nucleation on pre-existing nanoparticles. Here, we extend the kinetic derivation of the first nucleationtheorem to give a general framework to include such processes, yielding sum rules connecting the size dependent particle formation and loss rates to the corresponding loss-free nucleation rate and the apparent critical size from a naïve application of the first nucleationtheorem that neglects them.


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