Answer
$\Delta S = -23.04J/K$
Work Step by Step
Isothermal process equation of an ideal gas is $Q = W$
$\int \frac{dQ}{T}= \int \frac{p dV}{(pV/nR)} $
$\Delta S= nR \int \frac{dV}{V}$
$\Delta S = nR ln (\frac{V_f}{V_i}) $
From the question, $n = 4.00 mol, V_2 = 2.00V_1$, but going back to the initial state, so $V_f = 1.00$ and $V_i = 2.00$
$\Delta S = (4.00)(8.31 J/mol.K) ln (\frac{1.0}{2.0}) $
$\Delta S = -23.04J/K$ The negative sign implies the compression state.
