We study the first order solid-fluid phase transition of a system of semi-flexible Lennard-Jones chains using molecular dynamics simulations. Thermodynamic integration methods are used to calculate the free energy of the solid and fluid phases. The solid phase free energy per chain can be calculated to an accuracy of +/-0.03k(B)T with relative ease. The Gibbs-Duhem integration technique is used to trace out the complete melting curve, scarring with a single point on the curve obtained from the foe energy calculations. For the short chains studied here, we find that increasing the chain length stabilizes the solid phase; i.e., it raises the melting temperature at fixed pressure, and lowers the density at the transition at fixed temperature. Gibbs-Duhem integration was used also to investigate the effects of chain stiffness on the transition. We find that increasing the stiffness also acts to stabilize the solid phase. At fixed temperature, the transition is shifted to lower pressure and lower density with increasing chain stiffness. Further, we find that the density gap between solid and fluid broadens with increasing chain stiffness.