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(D) The change in internal energy $\Delta U$ is a state function,meaning it depends only on the initial and final states,not on the path taken.For the

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$-43$

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(D) The change in internal energy $\Delta U$ is a state function,meaning it depends only on the initial and final states,not on the path taken.For the path $iaf$,the first law of thermodynamics is given by $\Delta Q = \Delta U + \Delta W$.Given $\Delta Q_{iaf} = 50 \, J$ and $\Delta W_{iaf} = 20 \, J$,we have:$\Delta U_{if} = \Delta Q_{iaf} - \Delta W_{iaf} = 50 \, J - 20 \, J = 30 \, J$.For the return path $fi$,the initial state is $f$ and the final state is $i$. Therefore,the change in internal energy is $\Delta U_{fi} = -\Delta U_{if} = -30 \, J$.Using the first law of thermodynamics for the path $fi$:$\Delta Q_{fi} = \Delta U_{fi} + \Delta W_{fi}$.Given $\Delta W_{fi} = -13 \, J$ and $\Delta U_{fi} = -30 \, J$,we get:$\Delta Q_{fi} = -30 \, J + (-13 \, J) = -43 \, J$.

When a system is taken from state $i$ to a state $f$ along path $iaf$,$Q = 50 \, J$ and $W = 20 \, J$. Along path $ibf$,$Q = 35 \, J$. If $W = -13 \, J$ for the curved return path $fi$,$Q$ for this path is ...... $J$.

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