A monoatomic ideal gas, initially at temperature ${\mathrm{T}}_1$, is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature ${\mathrm{T}}_2$ by releasing the piston suddenly. If ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ are the lengths of the gas column before and after expansion respectively, then $\raisebox{1ex}{${\mathrm{T}}_1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{T}}_2$}\right.$ is given by