For isentropic processes, they must be adiabatic and reversible. Adiabatic refers to a system where heat does not leave or enter it and a reversible process is when a system can be reversed without leaving any trace on its surroundings. Because of these stipulations, we can write that $ \delta q = 0 $ and $ ds_{irrev} = 0 $, where $ \delta q $ is the net heat transfer in a system and $ ds_{irrev} $ is the change in entropy due to irreversibility in a system.
We can use an equation from thermodynamics that relates the change in entropy with temperature and pressure.
[ 1 ]
$$ s_2 - s_1 = c_p \ln \frac{T_2}{T_1} - R \ln \frac{P_2}{P_1} $$
In Eq. (1), $ s_2 - s_2 $ is the change in entropy of a system, $ c_p $ is the heat capacity at constant pressure of a gas, and $ R $ is the gas constant. Now, since the change in entropy is zero, Eq. (1) can be written as
$$ 0 = c_p \ln \frac{T_2}{T_1} - R \ln \frac{P_2}{P_1} $$
$$ \Rightarrow \ln \frac{P_2}{P_1} = \frac{c_p}{R} \ln \frac{T_2}{T_1} $$
$$ \Rightarrow \frac{P_2}{P_1} = \begin{pmatrix} \frac{T_2}{T_1} \end{pmatrix}^{\frac{c_p}{R}}$$
We can use a relationship between $ c_p $ and $ R $ such that
[ 2 ]
$$ \frac{c_p}{R} = \frac{\gamma}{\gamma - 1} $$
Here, $ \gamma $ is called the heat capacity ratio and $ \gamma = \frac{c_p}{c_v} $. Now, Eq. (2) can then be substituted into our modified Eq. (1) to get
[ 3 ]
$$ \frac{P_2}{P_1} = \begin{pmatrix} \frac{T_2}{T_1} \end{pmatrix}^{\frac{\gamma}{\gamma - 1}} $$
A similar process can be done for another thermodynamic relation relating temperature and specific volume; let's do that below.
$$ s_2 - s_1 = 0 = c_v \ln \frac{T_2}{T_1} + R \ln \frac{v_2}{v_1} $$
$$ \Rightarrow \ln \frac{v_2}{v_1} = - \frac{c_v}{R} \ln \frac{T_2}{T_1} $$
$$ \Rightarrow \frac{v_2}{v_1} = \begin{pmatrix} \frac{T_2}{T_1} \end{pmatrix}^{\frac{-c_v}{R}} $$
Now, there is a relation between $ \gamma $ and $ c_v / R $, and that is
[ 4 ]
$$ \frac{c_v}{R} = \frac{1}{\gamma - 1} $$
We can make that substitution for $ \frac{c_v}{R} $ and we can also relate the specific volumes with density, $ \rho $, because specific volume is the inverse of density. Hence, $ \frac{\rho_1}{\rho_2} = \frac{v_2}{v_1} $. This leaves us with
[ 5 ]
$$ \frac{\rho_2}{\rho_1} = \begin{pmatrix} \frac{T_2}{T_1} \end{pmatrix}^{\frac{1}{\gamma - 1}} $$
Putting both isentropic relations together, Eq. (3) and Eq. (5), we get
[ 6 ]
$$ \frac{P_2}{P_1} = \begin{pmatrix} \frac{\rho_2}{\rho_1} \end{pmatrix}^{\gamma} = \begin{pmatrix} \frac{T_2}{T_1} \end{pmatrix}^{\frac{\gamma}{\gamma - 1}} $$
Now we can use these relations for two points in compressible, inviscid flow!