Question

Figure shows two vessels A and B with rigid walls containing ideal gases. The pressure, temperature and the volume are $p_A,T_A$, V in the vessel A and $p_B,T_B$, V in the vessel B. The vessels are now connected through a small tube. Show that the pressure p and the temperature T satisfy $\frac{p}{T}=\frac{1}{2}(\frac{p_A}{T_A}+\frac{p_B}{T_B})$ when equilibrium is achieved.

Answer 1

Let the final pressure, volume and temperature after connection are

$P_A^{\prime }:\to$ partial pressure of A

$P_B^{\prime }:\to$ partial pressure of B

Now, $\frac{P_A^{\prime }\times 2V}{T}=\frac{P_A^{\prime }\times V}{T_A}$

or, $\frac{P_A^{\prime }}{P_A}$ ...(i)

Similarly $\frac{P_B^{\prime }}{T}=\frac{P_B}{2T_B}$ ...(ii)

Addiding (i) and (ii)

$\frac{P_A^{\prime }}{T}+\frac{P_B^{\prime }}{T}=\frac{P_A}{2T_A}+\frac{P_B}{2T_B}$

= $\frac{1}{2}[\frac{P_A}{T_A}+\frac{P_B}{T_B}]$

$\Rightarrow \frac{P}{T}=\frac{1}{2}[\frac{P_A}{T_A}+\frac{P_B}{T_B}]$

$[\therefore$ P'A + P'B = P]