Specific heat

For thermally perfect and calorically perfect gases

$\begin{array}{cc}& C_p=\frac{dh}{dT}\\ & C_v=\frac{de}{dT}\end{array}$

(Eq. 1.1)

From the definition of enthalpy and the equation of state $p=\rho RT$

$h=e+\frac{p}{\rho }=e+RT$

(Eq. 1.2)

Differentiate (Eq. 1.2) with respect to temperature gives

$\frac{dh}{dT}=\frac{de}{dT}+\frac{d(RT)}{dT}$

(Eq. 1.3)

Inserting the specific heats gives

$C_p=C_v+R$

(Eq. 1.4)

Dividing (Eq. 1.4) by $C_v$ gives

$\frac{C_p}{C_v}=1+\frac{R}{C_v}$

(Eq. 1.5)

Introducing the ratio of specific heats defined as

$\gamma =\frac{C_p}{C_v}$

(Eq. 1.6)

Now, inserting (Eq. 1.6) in Eqn. \ref{eq:specificheat:c} gives

$C_v=\frac{R}{\gamma -1}$

(Eq. 1.7)

In the same way, dividing (Eq. 1.4) with $C_p$ gives

$1=\frac{C_v}{C_p}+\frac{R}{C_p}=\frac{1}{\gamma }+\frac{R}{C_p}$

(Eq. 1.8)

and thus

$C_p=\frac{\gamma R}{\gamma -1}$

(Eq. 1.9)