A high-resolution differential ac free-standing film calorimeter has been constructed in our laboratories. We have employed this unique calorimeter to investigate the evolution of the temperature dependence of heat-capacity anomalies through the smectic-A-hexatic-B-crystal-E transition sequence of one liquid crystal compound as the film thickness is reduced from a few hundred to just three molecular layers. The existence of free surfaces in the free-standing liquid-crystal films has a very strong effect on the development of the molecular order through the smectic-A-hexatic-B transition and other unexpected results related to the crystal-E order.