Specific Heat of (Ca$_{1-x}$Sr$_{x})_{3}$Ru$_{2}$O$_{7}$ Single Crystals*

We have measured the specific heat of crystals of (Ca$_{1-x}$Sr$_{x})_{3}$Ru$_{2}$O$_{7 }$using ac- and relaxation-time calorimetry. Special emphasis was placed on the characterization of the N\'{e}el (T$_{N}$=56 K) and structural (T$_{c}$ = 48 K) phase transitions in the pure, x=0 material. While the latter is believed to be first order, detailed measurements under different experimental conditions suggest that all the latent heat (with L $\sim $ 0.3 R) is being captured in a broadened peak in the effective heat capacity. The specific heat has a mean-field-like step at T$_{N}$, but its magntitude ($\Delta $c$_{P }\sim $ R) is too large to be associated with a conventional itinerant electron (e.g. spin-density-wave) antiferromagnetic transition, while its entropy is too small to be associated with full ordering of localized spins. The T$_{N}$ transition broadens with Sr substitution while its magnitude decreases slowly. On the other hand, the entropy change associated with the T$_{c}$ transition decreases rapidly with Sr substitution and is not observable for our x=0.58 sample.