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  4. One mole of hydrogen gas at temperature T is mixed with one mole of helium gas at temperature 2 T in an isolated vessel. The equilibrium temperature of the gas mixture is

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Q. One mole of hydrogen gas at temperature $T$ is mixed with one mole of helium gas at temperature $2\, T$ in an isolated vessel. The equilibrium temperature of the gas mixture is

Solution:

Equating internal energy
$\frac{5}{2} R T \times 1+\frac{3}{2} \times 1 \times R(2 T)=U_{i}$
$U_{f}=\frac{5}{2} R T'+\frac{3}{2} R T'=4 R T'$ [final internal energy]
$\therefore \frac{11 R T}{2}=4 R T'$
$\therefore T'=\frac{11}{8} T$