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:

\frac{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}{n_{1} + n_{2} + n_{3}}

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\frac{n_{1} T_{1} + n_{2} {T_{2}}^{2} + n_{3} {T_{3}}^{2}}{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}

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\frac{n_{1} {T_{1}}^{2} + n_{2} {T_{2}}^{2} + n_{3} {T_{3}}^{2}}{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}

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\frac{(T_{1} + T_{2} + T_{3})}{3}

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Solution

The correct option is A $\frac{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}{n_{1} + n_{2} + n_{3}}$
$Given:-$ Absolute temperatures $= T_{1}, T_{2} a n d T_{3}$
Masses of molecules $= m_{1}, m_{2} a n d m_{3}$
Number of molecules $= n_{1}, n_{2} a n d n_{2}$

$Solution:-$ We know that ,

$\frac{F}{2} n_{1} k T_{1} + \frac{F}{2} n_{2} k T_{2} + \frac{F}{2} n_{3} k T_{3} = \frac{F}{2} (n_{1} + n_{2} + n_{3}) K T$

[ $∵$ F is degree of freedom]

Temperature of mixture

$T = \frac{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}{n_{1} + n_{2} + n_{3}}$

$Hence the correct option is A$

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