Clarification about the difference of internal energy formula
Your book has driven you crazy with its stupid sign convention.  If the gas is expanding against constant pressure, then the equation is still $$\Delta U=Q+W$$ but, in this case, the work done on the system by the surroundings is negative (i.e., W is a negative number, because the system is doing work on the surroundings).  So not all the heat added to the system is being used to increase its internal energy.  The increase in internal energy is less than if the gas was not pushing back its surroundings.