The temperature dependence of magnetic susceptibility and sublattice magnetizations were calculated for a Heisenberg Hamiltonian of an S = 1 and S = 1/2 antiferromagnetic alternating spin chain by means of the many-body Green's function theory to show the possible occurrence of a ferrimagnetic phase transition for genuinely organic molecule-based magnets. The S = 1 site in the chain is composed of two S = 1/2 spins coupled by a finite ferromagnetic interaction. From the calculated results, it is found that the sublattice magnetization at low-spin S = 1/2 sites changes its sign from negative to positive with increasing temperature, giving rise to the spin alignments along the chain changing from anti ferromagnetic to ferromagnetic ones, which indicates that there is a magnetic phase transition occurring. Because of the weak intermolecular anti ferromagnetic interactions, the curves of the magnetic susceptibility multiplied by temperature (chi T) against temperature show a round peak at low temperatures, which is well consistent with recent experimental observations, and the ferrimagnetic phase transition could only be detected at an ultralow-temperature region and under very weak,external magnetic fields in practical organic materials. From the analysis of the sublattice magnetizations, it is uncovered that the appearance of the low-temperature peak in the curves of the chi T originates from the ferromagnetic spin alignments for all the spins along the chain, and the intermolecular antiferromagnetic interactions play a pivotal role in ferrimagnetic spin alignments of the magnetic systems. It is also found that the higher spatial symmetry of the intermolecular antiferromagnetic interactions have contributions to stabilize the ferrimagnetic ordering state in the molecule-based magnetic materials.