Consider the following process: a system changes from state 1 (initial state) to state 2 (final state) in such a way that its temperature changes from $300 \mathrm{~K}\( to \)400 \mathrm{~K}$. (a) Is this process isothermal? (b) Does the temperature change depend on the particular pathway taken to carry out this change of state? (c) Does the change in the internal energy, \(\Delta E\), depend on whether the process is reversible or irreversible?

An isothermal process is a process that occurs at a constant temperature. Here, the initial temperature is 300 K, and the final temperature is 400 K. Since the temperature changes during the process, we can conclude that it is not an isothermal process.

The given information tells us that the initial state of the system has a temperature of 300 K and the final state has a temperature of 400 K. The temperature change, which is the difference between final and initial temperatures, depends only on the initial and final states (\( \Delta T = T_{final} - T_{initial} = 400 K - 300 K = 100 K\)). It does not depend on the specific pathway taken to achieve this change in state. Therefore, the temperature change is independent of the particular pathway.

The change in internal energy, denoted as \(\Delta E\), depends on the initial and final states of the system, and not on the pathway taken or the nature of the process (reversible or irreversible). Essentially, the internal energy is a state function, meaning it only depends on the initial and final states and not on the details of the process connecting those states. In conclusion, (a) No, this process is not isothermal as the temperature changes from 300 K to 400 K, (b) The temperature change does not depend on the specific pathway taken, and (c) The internal energy change, \(\Delta E\), does not depend on whether the process is reversible or irreversible as it is only a function of the initial and final states, not the process's nature connecting those states.