For an isentropic process of an ideal gas with approximately constant specific heats, entropy remains constant and the pressure, temperature and specific volume satisfy useful relations.

Let

$$k=\frac{c_p}{c_v}.$$

Then

$$pv^k=\text{constant},$$

$$\frac{T_2}{T_1}=\left(\frac{p_2}{p_1}\right)^{(k-1)/k},$$

and

$$\frac{T_2}{T_1}=\left(\frac{v_1}{v_2}\right)^{k-1}.$$

For air approximated by $k=1.4$, an isentropic compression from $100$ to $800\ \mathrm{kPa}$ starting at $300\ \mathrm K$ gives

$$T_2=300\left(8\right)^{0.4/1.4}\approx543\ \mathrm K.$$

These are property relations between equilibrium states; they do not say that every adiabatic process is isentropic. Friction or other irreversibilities generate entropy even when $Q=0$.