Consider the case of isentropic flow through a nozzle.

a. What is the critical temperature, \(T_{c}\), for the flow?

b. Show that the average velocity at any point in the nozzle can be described by:

\[\bar{v}=\sqrt{2 C_{p} T_{1}\left[1-\left(P / P_{1}\right)^{\frac{\gamma-1}{\gamma}}\right]}\]

c. If the Mach number is defined as the ratio of the gas velocity to the velocity of sound at local temperature and pressure conditions, show that the temperature and pressure at any point in the nozzle is a function of the Mach number.