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 vessel $A$ and $p_{B}, T_{B}, V$ in 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.

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Solution

Given $p_{A}, T_{A}, V, n_{A}$ $p_{B}, T_{B}, V, n_{B}$
Pressure, temperature, volume, no. of moles

Before opening value
${\Rightarrow p}_{A} V = n_{A} R T_{A} \to n_{A} = \frac{p_{A} V}{R T_{A}} ⟶ (1)$
and ${\Rightarrow p}_{B} V = n_{B} R T_{B} \to n_{B} = \frac{p_{B} V}{R T_{B}} ⟶ (2)$

After opening value
$p, T, 2 V, n$

Using the ideal gas equation:
So, $2 p V = n R T$
$\Rightarrow n = \frac{2 p V}{R T} ⟶ (3)$

As $n = n_{A} + n_{B}$

Putting from equation $(1) (2) (3)$
$\Rightarrow \frac{2 p V}{R T} = \frac{p_{A} V}{R T_{A}} + \frac{p_{B} V}{R T_{B}}$

$\Rightarrow \frac{p}{T} = \frac{1}{2} [\frac{p_{A}}{T_{A}} + \frac{p_{B}}{T_{B}}]$
Hence, solve.

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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.