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Question: If an ideal gas at 27°C is compressed suddenly to one fourth of its initial volume, then rise in its...

If an ideal gas at 27°C is compressed suddenly to one fourth of its initial volume, then rise in its temperature is (γ\gamma = 7/5)

A

222.33 K

B

233.33 K

C

244.33 K

D

255.33 K

Answer

222.33 K

Explanation

Solution

: Here, T1=27∘C=27+273=300 KT _ { 1 } = 27 ^ { \circ } \mathrm { C } = 27 + 273 = 300 \mathrm {~K}

V2=V14V _ { 2 } = \frac { V _ { 1 } } { 4 }

As the gas is compressed suddenly, then the process is adiabatic, then.

⇒T1V1γ−1=T2V2γ−1\Rightarrow T _ { 1 } V _ { 1 } ^ { \gamma - 1 } = T _ { 2 } V _ { 2 } ^ { \gamma - 1 }

Or T2=T1(V1V2)γ−1T _ { 2 } = T _ { 1 } \left( \frac { V _ { 1 } } { V _ { 2 } } \right) ^ { \gamma - 1 } =300(V1V1/4)75−1=300×(4)2/5=522.33 K= 300 \left( \frac { V _ { 1 } } { V _ { 1 } / 4 } \right) ^ { \frac { 7 } { 5 } - 1 } = 300 \times ( 4 ) ^ { 2 / 5 } = 522.33 \mathrm {~K}

Hence change in temperature

⇒ΔT=T2−T1=522.33−300=222.33 K\Rightarrow \Delta \mathrm { T } = \mathrm { T } _ { 2 } - \mathrm { T } _ { 1 } = 522.33 - 300 = 222.33 \mathrm {~K}