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  4. Two moles of an ideal monoatomic gas occupy a volume 2 V at temperature 300 K , it expands to a volume 4 V adiabatically, then the final temperature of gas is

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Q. Two moles of an ideal monoatomic gas occupy a volume $ 2\, V $ at temperature $ 300\, K $ , it expands to a volume $ 4 \,V $ adiabatically, then the final temperature of gas is

Thermodynamics

Solution:

Since in an adiabatic process $ T_{1}V_{1}^{\gamma-1} = T_{2}V_{2}^{\gamma-1} $
$ T_{2} = T_{1}\left(\frac{V_{1}}{V_{2}}\right)^{\gamma-1} = 300 \left(\frac{2V}{4V}\right)^{\left(\frac{5}{3}-1\right)} $
$ [\because V_{1}=2V $ and $ V_{2} = 4V $ for monotonic gas $ \gamma=\frac{5}{3}] $
$ = 300\left(\frac{1}{2}\right)^{{2}/{3}} = \frac{300}{2^{{2}/{3}}} $
$ = 188.98 \approx 189 \,K $