One mole of an ideal gas expands adiabatically from an initial state $\left({\mathrm{T}}_{\mathrm{A}},{\mathrm{V}}_0\right)$ to final state $\left({\mathrm{T}}_{\mathrm{f}},5{\mathrm{V}}_0\right)$ . Another mole of the same gas expands isothermally from a different initial state $\left({\mathrm{T}}_{\mathrm{B}},{\mathrm{V}}_0\right)$ to the same final state $\left({\mathrm{T}}_{\mathrm{f}},5{\mathrm{V}}_0\right)$ . The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\mathrm{\gamma }$. What is the ratio $\frac{{\mathrm{T}}_{\mathrm{A}}}{{\mathrm{T}}_{\mathrm{B}}}?$