A system goes from state 1 to state 2 and back to state 1 . (a) What is the relationship between the value of \(\Delta E\) for going from state 1 to state 2 to that for going from state 2 back to state \(1 ?\) (b) Without further information, can you conclude anything about the amount of heat transferred to the system as it goes from state 1 to state 2 as compared to that upon going from state 2 back to state \(1 ?(\mathrm{c})\) Suppose the changes in state are reversible processes. Can you conclude anything about the work done by the system upon going from state 1 to state 2 as compared to that upon going from state 2 back to state \(1 ?\)

The energy change of a system is given by the equation: \[\Delta E = E_{final} - E_{initial}\] For going from state 1 to state 2: \[\Delta E_{1 \to 2} = E_2 - E_1\] For going back from state 2 to state 1: \[\Delta E_{2 \to 1} = E_1 - E_2\] By analyzing these two equations, we can determine the relationship between the energy changes.

By comparing the two equations: \[\Delta E_{1 \to 2} = -(E_1 - E_2)\] \[\Delta E_{2 \to 1} = E_1 - E_2\] We can see that: \[\Delta E_{1 \to 2} = -\Delta E_{2 \to 1}\] #Part (b): Heat Transfer Comparison#

It is not possible to determine anything about the heat transfer for these processes without further information. Heat transfer depends on the specific path taken by a system during the process, not only on its initial and final states. #Part (c): Work Done Comparison (Assuming Reversible Processes)#

In case the changes in state are reversible processes, we can use the fact that \(\Delta E\) depends on the heat, \(Q\), and the work, \(W\). Specifically, \[\Delta E = Q - W\] Since we already found the relationship between the energy changes in part (a), we can rewrite the equation in terms of heat and work: \[Q_{1 \to 2} - W_{1 \to 2} = - (Q_{2 \to 1} - W_{2 \to 1})\] If the processes are reversible, the heat transfers will be the same for both processes but with opposite signs (because reversible processes are path-independent regarding heat transfer). Thus, \[Q_{1 \to 2} = -Q_{2 \to 1}\] Using this information, we can conclude that the work done during the two processes is equal in magnitude but opposite in sign: \[W_{1 \to 2} = -W_{2 \to 1}\]