# Connection between heat capacity and the derivative of enthalpy

One can define the heat capacity of isobaric processes as $$c_P = \left( \frac{\partial H}{\partial T} \right)_P .$$ Now, we know that the unit of heat capacity is Joule per Kelvin, i.e., I need to put a certain amount of energy into something to heat it by $x$ degrees. According to Wikipedia, for nearly every system we normally look at, heat capacity is positive (which makes sense from our daily interaction with the world). In my professors notes, there's an exercise which asks (verbatim): ''Show a connection between $c_P$ and the derivative of enthalpy and deduce from the curvature of this derivative the sign of heat capacity''.

How can I make a general assumption about enthalpy which yields the positive sign I instinctively assume?

We know the relationship between enthalpy, energy, pressure and volume $$H = U + P V \, . \tag{1}$$ Differentiation of this expression yields $$dH = T dS + V dP \, .$$ However, $$TdS = dQ$$ where dQ is the amount of heat system received or gave away. Take the derivative of equation $(1)$ with respect to $T$ at constant $P$ to get $$\left( \frac{dH}{dT} \right)_p = \left( \frac{dQ}{dT} \right)_p = C_p \, .$$ If $T$ of the system increases then it received heat and $dT$, $dQ$ and $(dQ/dT)_p$ are all positive. Oppositely, if $T$ decreases then $dT<0$ and $dQ < 0$ but $(dQ/dT)_p$ is still positive. Roughly speaking this means that heat capacity is always positive.