1. Use the ideal gas law for one mole: $PV = RT$ Hence, $T \propto PV$ for fixed number of moles.

2. Read the process from the given $P$-$V$ curve.
   The curve shown is a rectangular hyperbola-like relation corresponding to $P \propto \frac{1}{\sqrt V}$ so that $P\sqrt V = \text{constant}$

3. Therefore, between states 1 and 2: $P_1\sqrt{V_1} = P_2\sqrt{V_2}$ which gives $\frac{P_2}{P_1} = \sqrt{\frac{V_1}{V_2}}$

4. Now, $\frac{T_2}{T_1} = \frac{P_2V_2}{P_1V_1} = \frac{P_2}{P_1}\cdot \frac{V_2}{V_1}$ Substitute the pressure ratio: $\frac{T_2}{T_1} = \sqrt{\frac{V_1}{V_2}}\cdot \frac{V_2}{V_1} = \sqrt{\frac{V_2}{V_1}}$

5. Given: $V_2 = 2V_1$ so, $\frac{T_2}{T_1} = \sqrt{2}$

6. Check options:

   • A: $\sqrt 2$ ✅
   • B: $\dfrac{1}{\sqrt 2}$
   • C: $\dfrac{1}{2}$
   • D: $2$

So the correct answer is A.