To determine where the internal energy of a gas remains unchanged, we must understand the thermodynamic processes involved:
Isothermal Process: In an isothermal process, the temperature of the gas remains constant. According to the first law of thermodynamics, the change in internal energy \(\Delta U\) of an ideal gas is given by: \(\Delta U = nC_v\Delta T\), where \(n\) is the number of moles, \(C_v\) is the molar heat capacity at constant volume, and \(\Delta T\) is the change in temperature. Because \(\Delta T = 0\) in an isothermal process, the change in internal energy \(\Delta U\) is also \(0\). Thus, the internal energy remains unchanged.
Adiabatic Process: In an adiabatic process, no heat is exchanged with the surroundings (\(Q = 0\)). However, the internal energy can change due to work done on or by the system. Therefore, the internal energy does not necessarily remain constant in an adiabatic process.
Reversible Process: A reversible process is an idealized process that occurs infinitely slowly, allowing the system to remain in equilibrium. The internal energy may change depending on the type of reversible process. Thus, the internal energy isn't necessarily unchanged.
Cyclic Process: In a cyclic process, the system returns to its initial state at the end of each complete cycle. Since the internal energy is a state function, it depends only on the initial and final states. Therefore, after one complete cycle, there is no net change in the internal energy of the system.
Based on the analysis above, the internal energy of a gas remains unchanged in an isothermal process (I) and a cyclic process (IV). Therefore, the correct answer is I and IV.
