An ideal monoatomic gas at 300K expands adiabatically to twice its...

For an ideal monatomic gas, the ratio of specific heat capacity at constant pressure to specific heat capacity at constant volume is

$\gamma = \dfrac{5}{3}$

In an adiabatic process,

$TV^{\gamma - 1} =$ constant

i.e. ${T}_{f}{V}_f^{\gamma - 1} = {T}_{i}{V}_i^{\gamma - 1} ⇒ {T}_{f} = {T}_{i}\left(\dfrac{Vi}{Vf}\right)\gamma - 1 = 300\left(\dfrac{1}{2}\right)\dfrac{5}{3} - 1$

$= 300\left(\dfrac{1}{2}\right)^{2} / 3$