The traditional understanding is that the Hinshelwood-Lindemann mechanism for thermal unimolecular reactions, and the resulting unimolecular rate constant versus temperature and collision frequency ω (i.e., pressure), requires the Rice-Ramsperger-Kassel-Marcus (RRKM) rate constant k(E) to represent the unimolecular reaction of the excited molecule versus energy. RRKM theory assumes an exponential N(t)/N(0) population for the excited molecule versus time, with decay given by RRKM microcanonical k(E), and agreement between experimental and Hinshelwood-Lindemann thermal kinetics is then deemed to identify the unimolecular reactant as a RRKM molecule. However, recent calculations of the Hinshelwood-Lindemann rate constant kuni(ω,E) has brought this assumption into question. It was found that a biexponential N(t)/N(0), for intrinsic non-RRKM dynamics, gives a Hinshelwood-Lindemann kuni(ω,E) curve very similar to that of RRKM theory, which assumes exponential dynamics. The RRKM kuni(ω,E) curve was brought into agreement with the biexponential kuni(ω,E) by multiplying ω in the RRKM expression for kuni(ω,E) by an energy transfer efficiency factor βc. Such scaling is often done in fitting Hinshelwood-Lindemann-RRKM thermal kinetics to experiment. This agreement between the RRKM and non-RRKM curves for kuni(ω,E) was only obtained previously by scaling and fitting. In the work presented here, it is shown that if ω in the RRKM kuni(ω,E) is scaled by a βc factor there is analytic agreement with the non-RRKM kuni(ω,E). The expression for the value of βc is derived.