When all the energy is contained in quadratic degrees of freedom, then U = N f kT/2, and CV = N f k/2 = constant. This would apply to a monatomic ideal gas, or a diatomic ideal gas in the range where f=5 or f=7 (but not in between). Also a solid at high T.
An example where CV isn't constant is a diatomic ideal gas in the temperature range where kT is just barely large enough to excite the vibrational mode. In the region we are crossing from f=5 to f=7, so the specific heat is changing with temperature.
The specific heat capacity (or specific heat) is independent of N.
One can increase the enthalpy of a system at constant pressure either by heat flow or by other kinds of work (like a microwave) without heat flow. Either way, CP = (dH/dT) at constant P works fine, so it is more general than just a ``heat'' capacity.
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