Question 898: Thermodynamics & KTG

Question

A monoatomic ideal gas initially at temperature $$\mathrm{T}_1$$ is enclosed in a cylinder fitted with 8 frictionless piston. The gas is allowed to expand adiabatically to a temperature $$\mathrm{T}_2$$ by releasing the piston suddenly. $$\mathrm{L}_1$$ and $$\mathrm{L}_2$$ are the lengths of the gas columns before and after the expansion respectively. Then $$\frac{\mathrm{T}_2}{\mathrm{~T}_1}$$ is

Accepted Answer

For adiabatic expansion we have
$$\begin{aligned}
& \mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1} \
& 	herefore \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}ight)^{\gamma-1}=\left(\frac{\mathrm{L}_1}{\mathrm{~L}_2}ight)^{\gamma-1} \quad[\because \mathrm{v} \propto \mathrm{L}] \
& =\left(\frac{\mathrm{L}_1}{\mathrm{~L}_2}ight)^{\frac{5}{3}-1} \quad 	ext{[For monoatocomic gas}~ \gamma=\frac{5}{3}]\
& =\left(\frac{L_1}{L_2}ight)^{\frac{2}{3}} \
\end{aligned}$$

Answer

$$\left(\frac{\mathrm{L}_2}{\mathrm{~L}_1}\right)^{2 / 3}$$

Answer

$$\left(\frac{L_1}{L_2}\right)^{2 / 3}$$

Answer

$$\left(\frac{\mathrm{L}_1}{\mathrm{~L}_2}\right)^{1 / 2}$$

Answer

$$\left(\frac{\mathrm{L}_2}{\mathrm{~L}_1}\right)^{1 / 2}$$

Thermodynamics & KTG

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