A thermodynamics ideal gas , initially at temperature $\;T_{1}\;$ ia enclosed in a cylinder fitted with a friction less 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 gas column before and after expansion respectively , then $\;\large\frac{T_{1}}{T_{2}}\;$ is given by

Question

$(a)\;(\large\frac{L_{1}}{L_{2}})^{\large\frac{2}{3}}\qquad(b)\;\large\frac{L_{1}}{L_{2}}\qquad(c)\;\large\frac{L_{2}}{L_{1}}\qquad(d)\;(\large\frac{L_{2}}{L_{1}})^{\large\frac{2}{3}}$