Question

A thick-walle d insulatin g chambe r contain s n x

 mole s o f 

helium ga s a t a hig h pressur e Px

 an d temperatur e T x

. I t is allowed 

to lea k ou t slowl y t o th e atmospher e a t a pressur e P 0

 throug h a 

small valve. Show that the final temperature o f the n2

 mole s of helium 

left i n the chambe r i s 

(Hint: Conside r a s you r syste m th e ga s tha t i s ultimatel y lef t i n 

the chamber. ) 

Answer 1

For the adiabatic process,

$P_iV_i^\gamma=P_fV_f^\gamma$

So, we can write it as,

$(\frac{P_1}{P_2})=(\frac{V_f}{V_i})^\gamma ...(i)$

From the ideal gas equation,

$PV=nRT$

$\Rightarrow \frac{P_iV_i}{T_1}=\frac{P_fV_f}{T_f}$

$\Rightarrow \frac{T_f}{T_i}=\frac{P_fV_f}{P_iV_i}$

Hence,

$T_f=T_i(\frac{P_f}{P_i})^(\frac{\gamma-1}{\gamma})$