Questions: Entropy and the Second Law: Irreversibility

Question 1

A gas undergoes free (Joule) expansion into a vacuum: ΔU = 0, Q = 0, W = 0. A student concludes that since no energy was transferred, the process might be reversible. Why is this reasoning wrong?

• A. The process is irreversible because the gas temperature decreased
• B. ΔS_system > 0 while ΔS_surroundings = 0, so ΔS_total > 0 — proving irreversibility even without heat or work exchange
• C. The process is reversible because no heat was transferred to the surroundings
• D. The first law is violated when no work is done during an expansion

Answer: (B)

The first law being balanced says nothing about reversibility. Entropy generation is the measure of irreversibility. In free expansion, the gas occupies a larger volume with more accessible microstates, so ΔS_system = nR ln(V₂/V₁) > 0. Since the surroundings are untouched, ΔS_surroundings = 0, and σ = ΔS_total > 0 — the process is irreversible. The tempting misconception is confusing energy conservation with thermodynamic reversibility.

Question 2

In a reversible isothermal expansion, a gas absorbs heat Q from a reservoir at temperature T. The system gains entropy Q/T. What is the total entropy change of the universe?

• A. ΔS_total > 0, because the system absorbed heat
• B. ΔS_total = Q/T, because only the system's entropy counts
• C. ΔS_total = 0, because the reservoir loses exactly Q/T while the system gains Q/T
• D. ΔS_total < 0, because the gas became more ordered after expansion

Answer: (C)

In a reversible process, σ = 0. The gas gains +Q/T, but the reservoir releases that same heat Q at temperature T, so ΔS_surroundings = −Q/T. Thus ΔS_total = Q/T − Q/T = 0. In an irreversible process, the surroundings gain more entropy than the system loses (or vice versa), leaving a positive surplus σ > 0. Zero entropy generation is the defining signature of a reversible process.

Question 3

Irreversible processes violate microscopic physical laws — the time-reversed version of a free gas expansion would be physically very difficult under Newton's equations.

• T. True
• F. False

Answer: False

Microscopic laws (Newton's equations, Schrödinger's equation) are time-symmetric — they look identical run forwards or backwards. The time-reversed process (all gas molecules spontaneously contracting to one corner) is not forbidden by the equations of motion; it is merely overwhelmingly improbable. The number of microstates with the gas expanded vastly outnumbers those with it contracted. Entropy increase is a statistical phenomenon — the near-certainty of what happens to a system with very many degrees of freedom — not a prohibition imposed on top of mechanics.

Question 4

The entropy of an isolated system can remain constant during a real, spontaneous process.

• T. True
• F. False

Answer: False

For any real (irreversible) process in an isolated system, ΔS_total > 0 — entropy strictly increases. Only a perfectly reversible process (a limiting idealization impossible to achieve in practice) has ΔS_total = 0. All real processes generate entropy. The statement ΔS_universe ≥ 0 contains both cases: equality holds only for the reversible ideal.

Question 5

Why does the second law correctly predict the direction of spontaneous change even when the first law is completely silent — as in free expansion where ΔU = 0, Q = 0, and W = 0?

Model answer: Because the second law measures entropy generation, not energy transfer. In free expansion the system's entropy increases (the gas occupies more microstates in a larger volume) while the surroundings are unaffected, so ΔS_total > 0 — the second law identifies the process as spontaneous and irreversible. The first law only tracks energy accounting; it cannot distinguish a spontaneous process from its time-reverse, because both satisfy energy conservation. Irreversibility is a statistical property — the reverse process is mechanically allowed but vanishingly improbable — and entropy quantifies exactly how improbable.

The key insight is that energy conservation and spontaneity are independent questions. The first law tells you whether a process is energetically possible; the second law tells you whether it is thermodynamically allowed — i.e., whether it increases the entropy of the universe. A process can conserve energy perfectly and still be forbidden by the second law (e.g., heat spontaneously flowing from cold to hot), or it can be silent from the first law's perspective (free expansion) yet clearly directed by the second.