Irreversible Processes

Irreversible thermodynamic processes: entropy generation, Clausius inequality and quantitative spontaneity criterion.

Conceptual context

The second law of thermodynamics establishes that natural processes have a preferred direction: heat flows spontaneously from hot to cold, gases expand into vacuum, and isolated systems tend toward equilibrium. This leaf focuses on the quantification of that irreversibility: given a specific process, how much entropy does the universe generate and what consequences does it have for available useful energy?

Mastering the calculation of $\Delta S_{\mathrm{univ}}$ is essential for evaluating the real efficiency of any heat engine, determining whether a proposed process is thermodynamically possible, and quantifying the degradation of useful energy (exergy) at each stage of an industrial process.

Irreversibility is not a defect that can be completely eliminated: every heat transfer across a finite gradient, every friction force, and every spontaneous mixing generates entropy. The practical goal is to minimize it, and for that it is essential to know how to measure it quantitatively.

🟢 Essential level

An irreversible process is one that cannot undo itself spontaneously: once it has occurred, the universe has changed permanently. The quantity that detects this irreversibility is $\Delta S_{\mathrm{univ}}$, the entropy change of the universe (system + surroundings).

[!TIP] Think of $\Delta S_{\mathrm{univ}}$ as a "degradation counter": every time something spontaneous happens in nature, the counter goes up. It never goes down. If a calculation gives you a negative $\Delta S_{\mathrm{univ}}$, there is a definite error.

The basic reading has three cases: if the global balance increases, the process is irreversible; if it remains zero, it describes the ideal reversible limit; if it decreases, the result does not represent a physically possible process.

To calculate $\Delta S_{\mathrm{univ}}$, compare the system's entropy change with the surroundings' entropy change. The system can lose entropy as long as the surroundings compensate that loss and the combined balance does not decrease.

The main sources of irreversibility in nature are: heat transfer across finite temperature gradients, mechanical friction, spontaneous mixing of substances, spontaneous chemical reactions, and free expansion into vacuum.

🔵 Formal level

Basic formulas

The quantitative criterion of irreversibility is expressed as:

Irreversibility criterion (second law)

Where $\Delta S_{\mathrm{sis}}$ is calculated as a state function (via a reversible path between the system's initial and final states) and $\Delta S_{\mathrm{ent}}$ is calculated from the actual heat exchanged: $\Delta S_{\mathrm{ent}}$ = −$Q_{\mathrm{sis}}$ / $T_{\mathrm{ent}}$ for surroundings modeled as a reservoir at constant temperature.

[!NOTE] The asymmetry is crucial: for the system, a reversible path is used (because $\Delta S_{\mathrm{sis}}$ is a state function). For the surroundings, the real heat is used (because the surroundings, being a reservoir in equilibrium, always behaves reversibly internally).

Real student questions

Why is the reversible process unattainable? Because it requires infinitesimal gradients (ΔT → 0, ΔP → 0), which implies infinite time to complete any finite transfer. In practice, there are always finite gradients, friction, and limited times that generate irreversibility.

Can a system gain entropy in an irreversible process without receiving heat? Yes. Free expansion of an ideal gas involves no heat or work, but the system gains entropy because accessible microstates increase. Irreversibility comes from the spontaneous transition to a new equilibrium, not necessarily from heat flow.

How do I know if the $\Delta S_{\mathrm{sis}}$ I calculated is correct for an irreversible process? The trick is that $\Delta S_{\mathrm{sis}}$ does not depend on the actual process (it is a state function). It is calculated by inventing a reversible path between the same states i and f. If the system goes from (T_i, V_i) to (T_f, V_f), $\Delta S_{\mathrm{sis}}$ is calculated with standard formulas regardless of how it actually got there.

What happens if there are several sources of irreversibility simultaneously? They add up. If there is friction and a thermal gradient, the total entropy generated is the sum of that generated by friction and that generated by the gradient. In industrial design, they are identified and quantified separately to know where to invest in improvements.

Final synthesis

An irreversible process is identified and quantified by $\Delta S_{\mathrm{univ}}$ > 0. The universe's entropy always increases in real processes. Sources of irreversibility (thermal gradients, friction, mixing) are additive and quantifiable. Destroyed exergy is T_0 · $S_{\mathrm{gen}}$, providing the direct connection between thermodynamic irreversibility and economic loss of useful energy. The reversible process ($\Delta S_{\mathrm{univ}}$ = 0) is the ideal limit that establishes maximum theoretical efficiency.