Question

a volume V of a gas at a temeperature T and a pressure p is enclosed in a sphere. it is connected to another sphere of volume V/2 by a tube and stopcock. the second sphere is initially evacuated and the stopcock is closed. if the stopcock is opened the temperature of the gas in the second sphere becomes T2. the first sphere is maintained at a temperature T1 . what is the final pressure p within the appatatus

Answer 1

Use gaseous equation, $PV=nRT$

in case 1st sphere ,
volume = V,
pressure = P,
temperature = $T_1$
so, mole of gas in 1st sphere = $\frac{PV}{RT_1}$
now stop is opened , then pressure of 1st sphere becomes $P_1$
so, mole in 1st sphere = $\frac{P_1V}{RT_1}$

similarly, in case of 2nd sphere,
volume = V/2
pressure = $P_1$
Temperature = $T_2$
so, mole in 2nd sphere = $\frac{P_2V/2}{RT_2}=\frac{P_2V}{RT_2}$

now, total mole of gas = mole of gas in 1st sphere + mole of gas in 2nd sphere.
$\frac{PV}{RT_1}=\frac{P_1V}{RT_1}+\frac{P_1V}{2RT_2}$

$P_1=\frac{2PT_2}{2T_2+T_1}$

hence, pressure within the apparatus is $P_1=\frac{2PT_2}{2T_2+T_1}$