Figure (24 - E3) 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. The ratio between P and T is

\frac{1}{2} (\frac{P_{A}}{T_{A}} + \frac{P_{B}}{T_{B}})

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\frac{1}{2} (\frac{P_{A}}{T_{A}} + \frac{P_{B}}{T_{B}}) R

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(\frac{P_{A}}{T_{B}} + \frac{P_{B}}{T_{A}})

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None

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Solution

The correct option is A $\frac{1}{2} (\frac{P_{A}}{T_{A}} + \frac{P_{B}}{T_{B}})$
$P_{A} V = n_{A} R T_{A}, P_{B} V = n_{B} R T_{B}$
$n_{A} = \frac{P_{A} V}{R T_{A}}, \frac{P_{B} V}{R T_{B}} = n_{B}$
$n_{A} + n_{B}$ = Total no. of molecules after equilibrium
Volume total = 2V.
$P_{e} 2 V = n_{e} R T_{e}$
$P_{e} .2 V = (n_{A} + n_{B}) R T_{e}$
$P_{e} .2 V = (\frac{P_{A} V}{R T_{A}} + \frac{P_{B} V}{R T_{B}}) . R T_{e}$
$P_{e} = \frac{1}{2} (\frac{P_{A}}{T_{A}} + \frac{P_{B}}{T_{B}}) . T_{e}$

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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.