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Work is F*d, and here, the F*d is equivalent to P*V. In an irreversible reaction, the pressure instantly drops to a lower value and so the lower value is multiplied by V. In a reversible reaction, the pressure slowly drops, which means that we have to sum all the infinitely small sections of P*V. This includes sections at the beginning where P is still at the high point. This means that over the course of the reaction, the average pressure is higher and so the work done would also be higher.
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