Ideal Gas

P V = n R T, P v = \frac{R T}{w}, u = u (t)

d u = \frac{\partial u}{\partial T} \cdot d T = C_{v} d T

h = u + P v = u (T) + \frac{R}{w} \cdot T = h (T)

d h = \frac{\partial h}{\partial T} \cdot d T = C_{p} d T = \frac{d u}{d T} \cdot d T + \frac{R}{w} d T = (C_{v} + \frac{R}{w}) \cdot d T

$C_{v}$ energy increase due to 1K increase at V=const necessary heat
$C_{p}$ enthalpy increase due to 1K increase at P=const