Question

If a closed system would undergo an internally reversible process and an irreversible process between the same end states, how would the changes in entropy for the two processes compare? How would the amounts of entropy produced compare?

Short answer

Entropy change is the same for both processes, but entropy production is greater in the irreversible process.

Step-by-step solution

Understand the Problem

The problem involves comparing entropy changes and entropy production in a closed system undergoing two types of processes: an internally reversible process and an irreversible process between the same end states.

Define Entropy Change

For both processes, the change in entropy \(\triangle S\) depends only on the initial and final states, since entropy is a state function. Thus, \(\triangle S \) will be the same for both processes.

Internally Reversible Process

In an internally reversible process, the system can return to its initial state without leaving any changes in the surroundings. There is no internal entropy production \(\triangle S_{gen} = 0 \).

Irreversible Process

An irreversible process, however, involves entropy generation within the system due to factors like friction, unrestrained expansion, rapid mixing, etc. Thus, the entropy generation \(\triangle S_{gen} > 0 \) for an irreversible process.

Compare Entropy Changes

Since the entropy change depends only on the initial and final states, the change in entropy for both processes will be the same, \(\triangle S_{reversible} = \triangle S_{irreversible}\).

Compare Entropy Produced

Entropy production in an internally reversible process is zero \(\triangle S_{gen} = 0 \), whereas entropy production in an irreversible process is positive \(\triangle S_{gen} > 0 \). Therefore, the amount of entropy produced in the irreversible process will be greater.

Key concepts

reversible process

A reversible process is an idealized concept referring to a process that can be reversed without leaving any trace on the surrounding. In a reversible process, the system and its surroundings can be returned to their original states without any net change. This means that there are no losses, no friction, and everything happens infinitely slowly allowing the system to stay in equilibrium throughout the process.
For example, think of a very slow compression or expansion of a gas in a piston. If done infinitely slowly, each tiny adjustment allows the surroundings and the system to adapt, maintaining equilibrium. No entropy is generated within the system during this type of process because it avoids dissipative factors like friction.
It's important to note that in real life, truly reversible processes do not exist; they are theoretical constructs. However, they are useful for understanding the limits of performance and efficiency in thermodynamic systems.

irreversible process

An irreversible process, as opposed to a reversible one, involves factors that generate entropy and result in dissipative phenomena. These factors can include friction, rapid mixing, unrestrained expansions, and other forms of resistances or gradients. Because of these factors, irreversible processes cannot be exactly reversed to restore both the system and environment to their original states without net changes.
Consider stirring cream into coffee. The process happens quickly and results in a uniform mixture, but you cannot un-stir the mixture to separate the cream from the coffee again. This is a basic example of the irreversibility in thermodynamic processes.
In these processes, entropy is produced within the system. Mathematically, if \(\triangle S_{gen} > 0\), this indicates that there is entropy generation due to irreversibilities. Understanding this helps in analyzing the real-world performance of engines, refrigerators, and other thermodynamic systems which are always less efficient than their ideal reversible counterparts.

entropy generation

Entropy generation is a concept deeply tied to the second law of thermodynamics. It quantifies the amount of entropy produced within a system due to irreversible processes. Unlike reversible processes where \( \triangle S_{gen} = 0 \), in irreversible processes \( \triangle S_{gen} > 0 \), meaning there is a net increase in entropy.
Entropy generation is a measure of the 'lost work' or inefficiencies within a system. Higher entropy generation often correlates with greater inefficiencies. For example, in an engine, minimizing friction, turbulence, and other resistive forces can reduce entropy generation, leading to greater efficiency.
It's useful to understand that while \( \triangle S \) (change in entropy) between two states is the same for both reversible and irreversible processes because entropy is a state function, \( \triangle S_{gen} \) differs. In the case of the exercise, this means that the entropy change for both the internally reversible and irreversible processes might be identical, but entropy generation is zero for the reversible process and positive for the irreversible one. This is crucial for designing efficient thermodynamic systems.