Three perfect gases at absolute temperatures $${T_1},$$ $${T_2}$$ and $${T_3}$$ are mixed. The masses of molecules are $${m_1},$$ $${m_2}$$ and $${m_3}$$ and the number of molecules are $${n_1},$$ $${n_2}$$ and $${n_3}$$ respectively. Assuming no loss of energy, the final temperature of the mixture is :

Solution :
Number of moles of first gas $$ = \frac{{{n_1}}}{{{N_A}}}$$
Number of moles of second gas $$ = \frac{{{n_2}}}{{{N_A}}}$$
Number of moles of third gas $$ = \frac{{{n_3}}}{{{N_A}}}$$
If there is no loss of energy then
$$\eqalign{ & {P_1}{V_1} + {P_2}{V_2} + {P_3}{V_3} = PV \cr & \frac{{{n_1}}}{{{N_A}}}R{T_1} + \frac{{{n_2}}}{{{N_A}}}R{T_2} + \frac{{{n_3}}}{{{N_A}}}R{T_3} \cr & = \frac{{{n_1} + {n_2} + {n_3}}}{{{N_A}}}R{T_{mix}} \cr & {T_{mix}} = \frac{{{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}}}{{{n_1} + {n_2} + {n_3}}} \cr} $$