First Law Of Thermodynamics

By the First Law of Thermodynamics, the heat $Q$ supplied to a system equals the increase in internal energy $\Delta U$ plus the work $W$ done by the system, written as $$\begin{gathered} Q=\Delta \\ cup+W \end{gathered}$$. Rearranging gives $$\begin{gathered} \Delta \\ cup=Q-W \end{gathered}$$, which isolates the change in the gas's stored energy. Substituting $Q=250\text{ J}$ and $W=90\text{ J}$ yields $$\begin{gathered} \Delta \\ cup=250-90=160\text{ J} \end{gathered}$$. The value 340 J wrongly adds the work instead of subtracting it, which would violate energy conservation. The value 90 J mistakes the work output for the internal energy change. The value 250 J ignores the work done entirely and assumes all heat became internal energy, which is only true at constant volume. As a sanity check, internal energy rises by less than the heat absorbed because part of that energy left the system as expansion work, and 160 J is correctly smaller than 250 J.