Thermodynamic process is shown below on a $$P-V$$ diagram for one mole of an ideal gas. If $$V_2 = 2V_1$$, then the ratio of temperature $$\frac{T_2}{T_1}$$ is:

$$\text{Given polytropic process equation: } P V^{1/2} = \text{constant}$$

$$\text{From the ideal gas equation: } P = \frac{nRT}{V} \implies P \propto \frac{T}{V}$$

$$\text{Substituting } P \text{ into the process equation: } \left(\frac{T}{V}\right) V^{1/2} = \text{constant} \implies T V^{-1/2} = \text{constant}$$

$$\text{Setting up the ratio between state 1 and state 2: } T_1 V_1^{-1/2} = T_2 V_2^{-1/2}$$

$$\frac{T_2}{T_1} = \left(\frac{V_1}{V_2}\right)^{-1/2} = \left(\frac{V_2}{V_1}\right)^{1/2}$$

$$\text{Given that } V_2 = 2V_1 \implies \frac{V_2}{V_1} = 2:$$ $$\frac{T_2}{T_1} = (2)^{1/2} = \sqrt{2}$$