Question
Two thermally insulated vessels 1 and 2 are filled with air at temperatures $$\left( {{T_1},{T_2}} \right),$$ volume $$\left( {{V_1},{V_2}} \right)$$ and pressure $$\left( {{P_1},{P_2}} \right)$$ respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be
A.
$${T_1} + {T_2}$$
B.
$$\frac{{\left( {{T_1} + {T_2}} \right)}}{2}$$
C.
$$\frac{{{T_1}{T_2}\left( {{P_1}{V_1} + {P_2}{V_2}} \right)}}{{{P_1}{V_1}{T_2} + {P_2}{V_2}{T_1}}}$$
D.
$$\frac{{{T_1}{T_2}\left( {{P_1}{V_1} + {P_2}{V_2}} \right)}}{{{P_1}{V_1}{T_1} + {P_2}{V_2}{T_2}}}$$
Answer :
$$\frac{{{T_1}{T_2}\left( {{P_1}{V_1} + {P_2}{V_2}} \right)}}{{{P_1}{V_1}{T_2} + {P_2}{V_2}{T_1}}}$$
Solution :
$$\eqalign{
& \frac{{P\left( {{V_1} + {V_2}} \right)}}{T} = \frac{{{P_1}{V_1}}}{{{T_1}}} + \frac{{{P_2}{V_2}}}{{{T_2}}}\,......\left( {\text{i}} \right) \cr
& {\text{Also}}\,\,P\left( {{V_1} + {V_2}} \right) = {P_1}{V_1} + {P_2}{V_2}\,......\left( {{\text{ii}}} \right) \cr} $$
After solving above equations, we get
$$T = \left[ {\frac{{\left( {{P_1}{V_1} + {P_2}{V_2}} \right){T_1}{T_2}}}{{{P_1}{V_1}{T_2} + {P_2}{V_2}{T_1}}}} \right]$$