Exact expressions are obtained for the heat capacity of a two-particle one-dimensional system interacting with the truncated, unshifted Lennard-Jones potential subject to periodic boundary conditions and the minimum image convention in the Gibbs canonical ensemble. Numerical calculations show that along isotherms the heat capacity exhibits maxima and minima as a function of density comparable to those found for three-dimensional models and experimental systems. For the present system, at very low temperatures, the maximum in the heat capacity arises because of a competition between low energy, which drives the particles towards the potential minimum, and high entropy, which drives the particles past the truncation distance where the force of interaction vanishes. The minimum arises because the range of integration in the partition function no longer is effectively infinite at sufficiently high densities. As the temperature rises, the locus of the maxima and the locus of the minima in the temperature-density plane move towards each other and finally merge at a reduced temperature T approximate to 1.3. Above that temperature, the maxima and minima disappear. The contributions of different parts of the potential energy space are calculated. It is shown that the disappearance of the maxima and minima is related to the increasing probability of penetration of the two particles into the core region where the potential energy takes on large positive values.