Hi all, I'm working with the heat capacities definition and I have got a confusion. I don't understand why we can express them like Cp = T(∂S/∂T)p Cv = T(∂S/∂T)v I know that Cp=(dQ/dT)p = (∂H/∂T)p with H equal to TdS + VdP and Cv=(dQ/dT)v = (∂U/∂T)v with U equal to TdS + PdV, My guess: If I begin for instance with the enthalpy, H, and I constrain it to constant pressure I get just the TdS term, that is the thermodynamic definition of "Heat (Q)" . So I get the definition of the heat capacity if I derive Q respect to T, or TdS respect to T (is the same thing). Doing it, I get: dQ/dT = (TdS)/dT = Td^2S/d^2T + dS/dT. An expression very different of the definition. Can anyone help me? Thanks in advance.