A gas is enclosed in a cylinder with a movable frictionless piston....

In adiabatic process
${P}^{3}{V}^{5} =$ constant
$⇒ P{V}^{\dfrac{5}{3}} =$ constant
$⇒ \gamma = \dfrac{5}{3} ⇒ {C}_{V} = \dfrac{3}{2}Rand{C}_{P} = \dfrac{5}{2}R$
In another process

$∆ Q = n{C}_{P} ∆ T + n{C}_{v} ∆ T$
$= \dfrac{5}{2}nR\left({T}_{B} - {T}_{A}\right) + \dfrac{3}{2}nR\left({T}_{C} - {T}_{B}\right)$
$∆ Q = \dfrac{5}{2}\left({P}_{B}{V}_{B} - {P}_{A}{V}_{A}\right) + \dfrac{3}{2}\left({P}_{C}{V}_{C} - {P}_{B}{V}_{B}\right)$
Putting values
$∆ Q = 587.5J \approx 588J$