Three perfect gasses at absolute temperatures $T_{1}$ , $T_{2}$ and $T_{3}$ are mixed. If number of molecules of the gasses are $n_{1}$ , $n_{2}$ and $n_{3}$ respectively then temperature of mixture will be (assume no loss of energy)

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

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

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

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

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Solution

The correct option is C $\frac{n_{1} T_{1} + n_{2} T_{2} + n_{3} T_{3}}{n_{1} + n_{2} + n_{3}}$
For perfect gas,
Kinetic Energy of n molecule, $K . E = n (\frac{1}{2} K_{B} T)$
Where, $K_{B}$ is Boltzmann constant
If there is no loss of energy.
Total kinetic energy of mixture is sum of each gas kinetic energy.
$n_{t o t a l} K . E_{t o t a l} = n_{1} K . E_{1} + n_{2} K . E_{2} + n_{3} K . E_{3}$
$(n_{1} + n_{2} + n_{3}) (\frac{1}{2} K_{B} T) = n_{1} (\frac{1}{2} K_{B} T_{1}) + n_{2} (\frac{1}{2} K_{B} T_{2}) + n_{3} (\frac{1}{2} K_{B} T_{3})$

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

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Three perfect gasses at absolute temperatures $T_{1}$ , $T_{2}$ and $T_{3}$ are mixed. If number of molecules of the gasses are $n_{1}$ , $n_{2}$ and $n_{3}$ respectively then temperature of mixture will be (assume no loss of energy)