In an adiabatic process, the following relationship holds:
[tex]TV^{\gamma -1} = cost.[/tex]
where T is the gas temperature, V is the volume and [tex]\gamma[/tex] is the adiabatic index, which is equal to [tex]\gamma = \frac{5}{3} [/tex] for a monoatomic gas.
We can re-write the equation as
[tex]T_1 V_1^{\gamma -1} = T_2 V_2^{\gamma -1}[/tex]
where the labels 1,2 refer to the initial and final conditions of the gas.
Let's rewrite it for [tex]T_2[/tex], the final temperature:
[tex]T_2 = T_1 ( \frac{V_1}{V_2} )^{\gamma-1}[/tex]
We can now substitute the initial temperature, T1=300 K, and [tex]V_2 = 2V_1[/tex], because the final volume is twice the initial one. So we find the value of the final temperature:
[tex]T_2 = 300 K( \frac{1}{2})^{ \frac{2}{3} } =189 K[/tex]
