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Step-by-Step Explanation
Step 1: Understand the Nature of the System
The gas is expanding in an isolated system. By definition of an isolated system, there is no exchange of heat or matter with the surroundings. Hence, the total internal energy of the system remains constant throughout the process.
Step 2: Note the Internal Energy Change
For an ideal gas, the internal energy $U$ depends only on temperature $T$. Because the system is isolated, the change in internal energy $ΔU$ is zero. However, the way the expansion occurs (reversible vs. irreversible) affects how work is done and distributed within the system.
Step 3: Compare Reversible and Irreversible (Adiabatic) Expansions
Reversible Adiabatic Expansion: The process proceeds infinitely slowly, allowing the system to perform the maximum possible work on the surroundings. When maximum work is done by the gas, the gas loses more internal energy, thereby its temperature drops more compared to an irreversible process.
Irreversible Adiabatic Expansion: The expansion is sudden (e.g., free expansion), and the gas does less work on the surroundings (or effectively zero in a free expansion). Therefore, the temperature drop is less pronounced, keeping the final temperature higher compared to a reversible process.
Step 4: Conclude the Relationship of Final Temperatures
Because the reversible expansion does maximum work and transfers more internal energy into work, the final temperature is lower in a reversible process than in an irreversible expansion. Symbolically,
$ (T_f)_{\text{irrev}} \;>\; (T_f)_{\text{rev}}. $
This matches the given correct answer:
$(T_f)_{\text{irrev}} > (T_f)_{\text{rev}}.$
