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In the context of a gas expanding isothermally (at constant temperature), understanding internal energy is crucial. Internal energy, denoted by \( U \), is the total energy contained within a system. It comprises the kinetic energy of molecules due to their motion and the potential energy resulting from intermolecular forces. For an isothermal process, the temperature remains unchanged, and so does the internal energy.According to the first law of thermodynamics, the change in internal energy \( \Delta U \) is given by \( \Delta U = Q - W \), where \( Q \) is the heat added to the system and \( W \) is the work done by the system. During isothermal expansion, any work done by the gas is exactly offset by heat absorbed from the surroundings. This balance ensures that the internal energy stays constant, emphasizing option (d) in the original exercise. Hence, the internal energy of the system does not change because the temperature, which dictates molecular kinetic energy, remains constant.

Entropy is a key concept in thermodynamics, representing the degree of disorder or randomness in a system. During an isothermal expansion, a gas moves to occupy a larger volume, which naturally increases disorder. Thus, entropy increases.The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. This means spontaneous processes, like the expansion of gas into a vacuum or larger area, invariably lead to an increase in entropy. Since the temperature remains steady in an isothermal process, entropy will rise without any subsequent decrease, contrary to some misconceptions.Moreover, entropy change \( \Delta S \) due to heat transfer is calculated using \( \Delta S = \frac{Q}{T} \), where \( T \) is the constant temperature. This highlights that as the gas takes in heat to balance the work done, entropy continuously increases.

The first law of thermodynamics plays a central role in understanding energy changes in thermodynamic processes. In summary, it states that energy cannot be created or destroyed, only transferred or converted. The formal representation is: \( \Delta U = Q - W \).During an isothermal expansion, even though the gas does work (\( W \)) on its surroundings, the first law assures us that the total change in internal energy (\( \Delta U \)) remains zero, as the work done is exactly compensated by the heat absorbed (\( Q \)).In practical terms:

• Work done by the system means the system loses energy in terms of work output.
• Heat absorbed suggests the system gains energy in equivalence to the external work.

In essence, the first law confirms the conservation of energy, such that during an isothermal process, all changes between heat added and work done even out to maintain consistent internal energy.