A monoatomic ideal gas, initially at temperature $T_1$ is enclosed in a cylinder fitted with a frictionless pistion. The gas is allowed to expand adiabatically to a temperature $T_2$ by releasing the piston suddenly. If $I_1$ and $I_2$ are the lengths of the gas column, before and after the expansion respectively, then the value of $\frac{T_1}{T_2}$ will be :