Question

Two vessels A and B of eual volume $V_0$ are connected by a narrow tube which can be closed by a valve. The vessels are fitted with pistions whi ch can be moved to change the volumes. Initially, the valve is open and the vessels contain an ideal gas $(C_p/C_v=\gamma )$ at atmospheric pres sure $P_0$ and atmospheric temperature $T_0$. The walls of the vessels A are diathermic and those of B are adiabatic. The valve is now clo sed and the pistons are slowely pulled out to incr ease the volumes of the vessels to double the original value.

(a) Find the temperatures and pressures in the two vessels.

(b) The valve is now opened for suffiecient time so that the gases acquire a common temperature and pressure. Find the new values of the temper ature and the pressure.

Answer 1

Initial pressure = $P_0$

Initial temperature = $T_0$

Initial volume = $V_0\frac{C_P}{C_v}=\gamma$

(a) For diathermic vessel the temperature inside remains constant.

$P_1{V}_1=P_2{V}_2$

$\Rightarrow P_0{V}_0=P_2\times V_2$

$\Rightarrow P_2=\frac{P_0}{2{}^{\prime }}$

Temperature = $T_0$

For adiabatic vessels the temperature does not remain constant. The process is adiabatic

$T_1{V}_1^{\gamma -1}=T_2{V}_2^{\gamma -1}$

$\Rightarrow T_o{V}_o^{\gamma -1}=T\times {\left(2V_o\right)}^{\gamma -1}$

$\Rightarrow T_2=T_o\times 2^{\gamma -1}$

$\Rightarrow P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$

$P_o{V}_o^{\gamma }=P_2\times {\left(2V_o\right)}^{\gamma }$

$\Rightarrow P_2=\left(\frac{P_o}{2^{\gamma }}\right)$

(b) When the valves are open, the temperature remains $T_o$ through out

$P_1=P_{2}and{T}_1=T_1=T_o$

So, $P_o=P_1+P_2$

= $2P_1=2P_2$

So, $P_1=P_2=\frac{P_o}{2}$