Existing positron annihilation lifetime spectroscopy (PALS) uses the orthopositronium components, lifetimes, and intensities observed in molecular substrates, such as in polymers, to determine free-volume properties based on an infinitive potential spherical model originally proposed by Tao in 1972. However, in many molecular systems, positronium is either quenched or inhibited by interacting with chemical functional groups and leads to no or nearly no orthopositronium component in PALS. In this Article, a newly modified equation is developed by following the Tao's quantum model using the positron component (not orthopositronium) of PALS. This modified equation is examined by fitting free-volume results obtained from o-Ps lifetimes with the positron lifetimes in pressure- and temperature-dependent data in polymers and calibrated with known or calculated cavity sizes in zeolite materials. A newly modified positron lifetime-free volume correlation equation is established for the determination of free volumes up to the mean radius of 5 A in polymeric systems, where no orthopositronium component is observed in PALS.