c$)($i$)$ Show that the critical pressure ratio $r_{\text{c}\text{r}\text{i}\text{t}}$ for an isentropic nozzle is given by the expression

$r_{\text{c}\text{r}\text{i}\text{t}}=[\frac{2}{+1}{]}^{\frac{-}{-1}}$

where the usual notation applies.

$($ii$)$ A nozzle expands gas from stagnation pressure $1$ bar and stagnation temperature $400$K$.$ The ambient pressure is $0\mathrm{.}4$ bar. What is the critical pressure ratio for isentropic flow, and can this nozzle expand to supersonic exit conditions for the gas states? Then, what is the ratio of the pressure at the nozzle throat to ambient pressure?

$[15]$

For this question you should use $=1\mathrm{.}4,C_p=1\mathrm{.}004kJk{g}^{-1}{K}^{-1}.$