A system consisting of n mol of an ideal gas undergoes two...

A system consisting of $n$ mol of an ideal gas 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 56 .

Key Concepts

Entropy as a State Function

Entropy is a state function, meaning its change between two states is independent of the path taken. This property is significant because it allows the calculation of entropy change for complex processes by breaking them down into simpler reversible steps, with the total entropy change being a sum of the changes in each step. This concept underlies the consistency in entropy calculations across different processes leading to the same final state.

Adiabatic Processes

An adiabatic process is one in which no heat is exchanged with the surroundings. In a reversible adiabatic process, this means the process is both thermally isolated and occurs without any change in entropy. For ideal gases undergoing adiabatic changes, specific relations connect the pressure, volume, and temperature of the gas, simplifying the analysis of such processes.

Ideal Gas Behavior

Ideal gas behavior is governed by the ideal gas law, which relates pressure, volume, and temperature in a simple way. In the context of thermodynamic processes, this relationship provides a foundation for deriving expressions for work, heat, and entropy changes during isothermal and adiabatic processes. The simplicity of the ideal gas model allows for clear analytical results that help illustrate fundamental thermodynamic principles.

Reversible Processes

Reversible processes are idealized transformations that occur slowly enough to remain in thermodynamic equilibrium at all times. They are characterized by the fact that the system and its surroundings can be returned to their initial states with no overall change in entropy, except where entropy is generated in each individual step. This concept is crucial for defining precise relations between thermodynamic variables such as pressure, volume, and temperature.

Isothermal Processes

In an isothermal process, the temperature of the system remains constant. For ideal gases, this implies that any change in volume is accompanied by an inversely proportional change in pressure. Energy exchange during such a process occurs in the form of heat transfer, and the change in entropy is determined by the amount of heat added divided by the constant temperature, making the mathematical treatment particularly straightforward.

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