A closed vessel contains 0.1 mole of a monatomic ideal gas at 200 K.

Question:

A closed vessel contains $0.1$ mole of a monatomic ideal gas at $200 \mathrm{~K}$. If $0.05$ mole of the same gas at $400 \mathrm{~K}$ is added to it, the final equilibrium temperature (in K) of the gas in the vessel will be close to______

Solution:

(266.67)

Here work done on gas and heat supplied to the gas are zero. Let $T$ be the final equilibrium temperature of the gas in the vessel.

Total internal energy of gases remain same.

i.e., $u_{1}+u_{2}=u_{1}^{\prime}+u_{2}^{\prime}$

or, $n_{1} C_{v} \Delta T_{1}+n_{2} C_{v} \Delta T_{2}=\left(n_{1}+n_{2}\right) C_{v} T$

$\Rightarrow(0.1) C_{v}(200)+(0.05) C_{v}(400)=(0.15) C_{v} T$

$\therefore T=\frac{800}{3}=266.67 \mathrm{~K}$

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