Answer
$\frac{W}{Q} = \frac{2}{5}$
Work Step by Step
The total internal energy of the system is
$\Delta U = \frac{3}{2}n R \Delta T $ since $W = p\Delta V = nR\Delta T$,
$\Delta U = \frac{3}{2} p\Delta V $
$\Delta U = \frac{3}{2} W$
Then $Q = \Delta U + W$
$Q = \frac{3}{2} W + W $
$Q = \frac{5}{2} W$
So, the fraction of energy used to do the expansion work of the gas is
$\frac{W}{Q} = \frac{2}{5}$
