The cylinder is completely insulated from its surroundings. As a result, no heat is exchanged between the system (cylinder) and its surroundings. Thus, the process is adiabatic.

Initial pressure inside the cylinder $=P_{1}$

Final pressure inside the cylinder $=P_{2}$

Initial volume inside the cylinder $=V_{1}$

Final volume inside the cylinder $=V_{2}$

Ratio of specific heats, $\gamma=1.4$

For an adiabatic process, we have:

$P_{1} V_{1}^{\gamma}=P_{2} V_{2}^{\gamma}$

The final volume is compressed to half of its initial volume.

$\therefore V_{2}=\frac{V_{1}}{2}$

$P_{1}\left(V_{1}\right)^\gamma=P_{2}\left(\frac{V_{1}}{2}\right)^{\gamma}$

$\frac{P_{2}}{P_{1}}=\frac{\left(V_{1}\right)^{\gamma}}{\left(\frac{V_{1}}{\gamma}\right)^{2}}=(2)^{\gamma}=(2)^{1.4}=2.639$

Hence, the pressure increases by a factor of 2.639