Heat and Thermodynamics 5 Question 24
25. A monoatomic ideal gas, initially at temperature $T _1$, is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature $T _2$ by releasing the piston suddenly. If $L _1$ and $L _2$ are the lengths of the gas column before and after expansion respectively, then $T _1 / T _2$ is given by
(2000, 2M)
(a) $\left(L _1 / L _2\right)^{2 / 3}$
(b) $\left(L _1 / L _2\right)$
(c) $L _2 / L _1$
(d) $\left(L _2 / L _1\right)^{2 / 3}$
Show Answer
Solution:
- During adiabatic expansion, we know
$$ T V^{\gamma-1}=\text { constant } \text { or } T _1 V _1^{\gamma-1}=T _2 V _2^{\gamma-1} $$
For a monoatomic gas, $\gamma=\frac{5}{3}$
$$ \begin{aligned} \therefore \quad \frac{T _1}{T _2}=\left(\frac{V _2}{V _1}\right)^{\gamma-1} & =\left(\frac{A L _2}{A L _1}\right)^{(5 / 3)-1} \\ (A & =\text { Area of cross-section of piston }) \\ & =\left(\frac{L _2}{L _1}\right)^{2 / 3} \end{aligned} $$
