  Specific heat measurements performed on high purity vapor-grown Nb$_3$Sn
crystals show clear features related to both the martensitic and
superconducting transitions. Our measurements indicate that the martensitic
anomaly does not display hysteresis, meaning that the martensitic transition
could be a weak first or a second order thermodynamic transition. Careful
measurements of the two transition temperatures display an inverse correlation
between both temperatures. At low temperature specific heat measurements show
the existence of a single superconducting energy gap feature.
