JEE MAIN - Physics (2011 - No. 17)
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:
$${{{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}} \over {{n_1} + {n_2} + {n_3}}}$$
$${{{n_1}T_1^2 + {n_2}T_2^2 + {n_3}T_3^2} \over {{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}}}$$
$${{n_1^2T_1^2 + n_2^2T_2^2 + n_3^2T_3^2} \over {{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}}}$$
$${{\left( {{T_1} + {T_2} + {T_3}} \right)} \over 3}$$
Explanation
Number of moles of first gas $$ = {{{n_1}} \over {{N_A}}}$$
Number of moles of second gas $$ = {{{n_2}} \over {{N_A}}}$$
Number of moles of third gas $$ = {{{n_3}} \over {{N_A}}}$$
If there is no loss of energy then
$${P_1}{V_1} + {P_2}{V_2} + {P_3}{V_3} = PV$$
$${{{n_1}} \over {{N_A}}}R{T_1} + {{{n_2}} \over {{N_A}}}R{T_2} + {{{n_3}} \over {{N_A}}}R{T_3}$$
$$ = {{{n_1} + {n_2} + {n_3}} \over {{N_A}}}R{T_{mix}}$$
$$ \Rightarrow {T_{mix}} = {{{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}} \over {{n_1} + {n_2} + {n_3}}}$$
Number of moles of second gas $$ = {{{n_2}} \over {{N_A}}}$$
Number of moles of third gas $$ = {{{n_3}} \over {{N_A}}}$$
If there is no loss of energy then
$${P_1}{V_1} + {P_2}{V_2} + {P_3}{V_3} = PV$$
$${{{n_1}} \over {{N_A}}}R{T_1} + {{{n_2}} \over {{N_A}}}R{T_2} + {{{n_3}} \over {{N_A}}}R{T_3}$$
$$ = {{{n_1} + {n_2} + {n_3}} \over {{N_A}}}R{T_{mix}}$$
$$ \Rightarrow {T_{mix}} = {{{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}} \over {{n_1} + {n_2} + {n_3}}}$$
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