Q: A monoatomic ideal gas, initially at temperature T1, is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperatrue T2 by releasing the piston suddenly. If L1 and L2 are the lengths of the gas column before and after expansion respectively, then T1/T2 is given by

(a) (L_{1}/L_{2})^{2/3}

(b) (L_{1}/L_{2})

(c) L_{2}/L_{1}

(d) (L_{2}/L_{1})^{2/3}

Ans: (d)

Sol: In adiabatic Expansion ;

$\large T V^{\gamma – 1} = constant $

$\large T_1 V_1^{\gamma – 1} = T_2 V_2^{\gamma – 1} $

For monoatomic gas , γ = 5/3

$\large \frac{T_1}{T_2} = (\frac{V_2}{V_1})^{\gamma – 1}$

$\large \frac{T_1}{T_2} = (\frac{A L_2}{A L_1})^{\frac{5}{3} – 1}$

(Where A = Area of cross-section of piston)

$\large \frac{T_1}{T_2} = (\frac{L_2}{L_1})^{\frac{2}{3}}$