Question

Three samples A, B and C of the same gas ($\gamma =1.5)$ have equal volumes and temperatures. The volu me of each sample is doubled, the process being isother mal for A, adiabatic for B and isobaric for C. If the final pressures are equal for the three samples, Find the ratio of the initial pressures.

Answer 1

$V_A=V_B=V_C$ $T_A=T_B=T_c$

For A the process is isothermal

$P_A{V}_A=P_A^{\prime }2V_A$

$\Rightarrow P_A^{\prime }=P_A\times \frac{1}{2}$

For B the process is adiabatic

$P_B{V}_B^{\gamma }=P_B^{\prime }(2V_B{)}^{\prime }$

$\Rightarrow P_B^{\prime }=\frac{P_B}{2^{1.5}}$

For C the process is adiabatic

$\frac{V_C}{T_C}=\frac{V_C^{\prime }}{T_C^{\prime }}\Rightarrow \frac{V_C}{T_C}=\frac{2V_C}{T_C^{\prime }}$

$\Rightarrow T_c^{\prime }=2T_C$

Final pressure are equal

$\frac{P_A}{2}=\frac{P_B}{2^{1.5}}=P_C$

$\Rightarrow P_A:P_B:P_C=2:2^{1.5}:1$