Question

A volume V of a gas at a temperature ${T_1}$ ​, and a pressure p is enclosed in a sphere. It is connected to another sphere of volume $\dfrac{V}{2}$ 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 ${T_2}$ . The first sphere is maintained at a temperature ${T_1}$ ​. What is the final pressure ${p_1}$ ​within the apparatus?

Answer 1

If we take total number of molecules of the gas be n. of which ${n_1}$ are in the larger sphere and ${n_2}$ in the smaller sphere after the stopcock is opened and Using ideal gas equation, $PV = nRT$ . Total moles of gas (n) in first sphere will be $\dfrac{{{P_1}V}}{{R{T_1}}}$ and moles in the second sphere will be $\dfrac{{{P_1}V}}{{2R{T_2}}}$ . Hence, we can find the final pressure ${p_1}$within the apparatus.

Complete step by step answer:
Let the total number of molecules of the gas be n. of which ${n_1}$ are in the larger sphere and ${n_2}$ in the smaller sphere after the stopcock is opened.
Using ideal gas equation,
$PV = nRT$
$n = \dfrac{{PV}}{{RT}}$
Case I - When the stopcock is closed
Pressure enclosed in sphere $= p$
Temperature of sphere $= {T_1}$
Volume $= V$
Therefore, total moles of gas
$(n) = \dfrac{{PV}}{{R{T_1}}}$
Case II - When the stopcock is opened.
Pressure within the apparatus, i.e., final pressure $= {p_1}$
Temperature of first sphere $= {T_1}$
Temperature of second sphere $= {T_2}$
Volume of first sphere $= V$
Volume of second sphere $= \dfrac{V}{2}$
Therefore, Moles of gas in first sphere ${n_1} = \dfrac{{{p_1}V}}{{R{T_1}}}$
and Moles of gas in second sphere
$\Rightarrow {n_2} = \dfrac{{{p_1}\left( {\dfrac{V}{2}} \right)}}{{R{T_2}}}$
$\Rightarrow {n_2} = \dfrac{{{p_1}V}}{{2R{T_2}}}$
Now,
Total moles of gas = moles of gas in first sphere + moles of gas in second sphere
$n = {n_1} + {n_2}$
$\Rightarrow \dfrac{{pV}}{{R{T_1}}} = \dfrac{{{p_1}V}}{{R{T_1}}} + \dfrac{{{p_1}V}}{{2R{T_2}}}$
$\Rightarrow {p_1} = \dfrac{{2p{T_2}}}{{2{T_2} + {T_1}}}$

Note: We can define an ideal gas as a hypothetical gaseous substance whose behaviour is independent of attractive and repulsive forces and can be completely described by the ideal gas law.