A system consisting of n moles of an ideal gas with molar...

A system consisting of $n$ moles of an ideal gas with molar specific heat at constant pressure $C_{p}$ undergoes two reversible processes. It starts with pressure $P_{i}$ and volume $V_{i}$ expands isothermally, and then contracts adiabatically to reach a final state with pressure $P_{i}$ and volume $3 V_{i}$. (a) Find its change in entropy in the isothermal process. (The entropy does not change in the adiabatic process.) (b) What If? Explain why the answer to part (a) must be the same as the answer to Problem 46 . (You do not need to solve Problem 46 to answer this question.)

Key Concepts

State Functions

State functions are properties of a system that depend solely on its current state and not on the path used to reach that state. In thermodynamics, quantities like entropy, internal energy, and enthalpy are state functions. This concept is essential because it allows one to calculate the change in these properties by simply considering the initial and final conditions of a process, even when the system undergoes multiple or complex paths.

Entropy

Entropy is a fundamental thermodynamic state function that quantifies the degree of disorder or randomness in a system. It is particularly important in the analysis of reversible processes, where the change in entropy can be determined by integrating the reversible heat transfer divided by temperature. The fact that entropy is a state function means that its total change depends only on the initial and final states of the system, regardless of the process path taken.

Reversible Process

A reversible process is an idealized thermodynamic process that happens infinitely slowly, maintaining the system in continuous equilibrium with its surroundings. Such processes can be reversed without leaving any net change in both the system and the environment. Reversibility ensures that thermodynamic quantities, such as entropy, can be calculated precisely using state functions, facilitating analysis by comparing different process paths between the same states.

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant throughout the process. For an ideal gas undergoing an isothermal change, the internal energy remains unchanged, and any work done by or on the system is exactly compensated by the heat flow. This property allows the entropy change to be calculated explicitly using the integral of the heat exchanged divided by the constant temperature.

Adiabatic Process

An adiabatic process is characterized by the absence of heat exchange between the system and its surroundings. In a reversible adiabatic process, there is no change in entropy since the entire process is insulated and the work done results solely in changes to the system's internal energy. This concept is crucial when analyzing processes where heat transfer is not allowed, ensuring that any entropy change observed in the overall process comes from other steps.

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