Entropy and the second law

Result

Entropy and the second law

T-033Home CU-201Threads thermo

Statement

Entropy \(S\) is a state function, defined for a reversible process by \(dS\equiv dq_{\text{rev}}/T\); because \(S\) depends only on the state, its change for any process (even an irreversible one) can always be computed using a hypothetical reversible path connecting the same initial and final states. The second law states that for any spontaneous, real (irreversible) process, the total entropy of the universe strictly increases, \(\Delta S_{\text{universe}}=\Delta S_{\text{system}}+\Delta S_{\text{surroundings}} > 0\), reaching exactly zero only in the idealised reversible limit.

Why it matters

The first law (first-law-chemistry) conserves energy but says nothing about which direction a process proceeds spontaneously: energy conservation alone permits heat to flow either from hot to cold or from cold to hot, yet only the first ever happens without continuous external work input. Entropy is precisely the quantity that breaks this symmetry and identifies the direction of spontaneous change, making it the single most important criterion for predicting whether a proposed process can occur on its own.

Hypotheses

Entropy is a genuine state function.This is an empirical postulate, on the same logical footing as the first law's postulate of energy conservation, though it is more deeply justified by statistical mechanics (relating entropy to the number of accessible microscopic arrangements of a system, a connection developed more fully in a later unit of this curriculum). Being a state function is what licenses the crucial practical technique used throughout this result: computing \(\Delta S\) for an actual, real (possibly irreversible) process via any convenient reversible path connecting the same two states, exactly the same logical move already used for enthalpy in Hess's law (hess-law). For a sufficiently large surroundings (a thermal reservoir), heat exchanged with the surroundings can itself be treated as occurring reversibly at the surroundings' own fixed temperature, even while the system's own process is irreversible.This is a standard, well-justified simplification: a sufficiently large reservoir's temperature does not measurably change no matter how much heat it absorbs or releases, so from the reservoir's own perspective the exchange is effectively reversible, letting \(\Delta S_{\text{surroundings}}=q_{\text{surroundings}}/T_{\text{surroundings}}\) be computed directly using the actual heat exchanged, without needing a separate reversible-path construction for the surroundings.

Proof

dS \equiv \frac{dq_{\text{rev}}}{T}

The Clausius definition of entropy change for an infinitesimal reversible step; for a finite reversible process, \(\Delta S=\int dq_{\text{rev}}/T\). A

\Delta S_{\text{system}} \text{ (any actual process)} = \Delta S_{\text{system}} \text{ (hypothetical reversible path, same endpoints)}

Since \(S\) is a state function (Hypotheses), its change between two given states does not depend on which path was actually taken; \(\Delta S_{\text{system}}\) can therefore always be computed via Step 1 applied to any convenient reversible path connecting the same initial and final states, even when the real process was irreversible. A

\Delta S_{\text{surroundings}} = \frac{q_{\text{surroundings, actual}}}{T_{\text{surroundings}}} = -\frac{q_{\text{system, actual}}}{T}

Using the second Hypothesis, the surroundings' own entropy change is computed directly from the actual heat exchanged (not a hypothetical reversible value), since the reservoir experiences that exchange reversibly from its own perspective regardless of what the system underwent. A

\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \begin{cases} = 0 & \text{reversible process} \\ > 0 & \text{irreversible (spontaneous) process} \end{cases}

Combining Steps 2 and 3: for a reversible process, the system's and surroundings' entropy changes are computed from the identical heat value (Step 3 of first-law-chemistry, \(q_{\text{rev}}\)) with opposite sign, cancelling exactly. For an irreversible process, less heat is actually exchanged than in the reversible case (first-law-chemistry's Result), so the surroundings' entropy loss (Step 3, using the smaller, actual \(q\)) is smaller in magnitude than the system's entropy gain (Step 2, still computed via the reversible path's larger \(q_{\text{rev}}\)), leaving a strictly positive net total. A

Result

\Delta S_{\text{system}} = \int\frac{dq_{\text{rev}}}{T}\ (\text{any path, via a reversible substitute}), \qquad \Delta S_{\text{universe}} \ge 0\ (\text{equality only if reversible})

Reading. Entropy's status as a state function is what makes it computable at all for real, irreversible processes (via a hypothetical reversible substitute path); the second law's inequality is what makes it useful for predicting spontaneity, since only a strictly positive total entropy change signals a genuinely spontaneous, real-world process.

Scope. \(\Delta S_{\text{universe}}\ge0\) is a statement about the combined system-plus-surroundings; a system's own entropy alone (\(\Delta S_{\text{system}}\)) can decrease during a spontaneous process (Corollaries), so long as the surroundings' entropy increases by at least as much.

Corollaries & converses

A system's own entropy can decrease during a genuinely spontaneous process (water freezing, a molecule crystallising into an ordered solid) so long as the surroundings' entropy increases by a larger amount, keeping \(\Delta S_{\text{universe}}>0\) overall — spontaneity is never a statement about the system alone, only about system plus surroundings together.
Directly building on first-law-chemistry's numeric example (isothermal ideal gas expansion, \(1.00\,\text{mol}\), \(298\,\text{K}\), \(10.0\to20.0\,\text{L}\)): the reversible case gives \(\Delta S_{\text{universe}}=0\) exactly, while the irreversible case (constant external pressure \(P_2\)) gives \(\Delta S_{\text{universe}}\approx+1.61\,\text{J/K}\), a direct, fully quantitative confirmation of the second law using the identical numbers already established there (Worked examples).
Converse: a proposed process for which \(\Delta S_{\text{universe}}\) would be negative cannot occur spontaneously under any circumstances, regardless of how favourable it might appear from energy considerations (first law) alone — the second law is an absolute, not merely a probabilistic-in-practice, constraint on which processes are possible.

Fails without

Drop entropy's state-function property (Hypotheses), attempt to compute \(\Delta S\) using only the actual (possibly irreversible) heat exchanged, \(q_{\text{actual}}/T\): this would give the wrong answer for the system's entropy change whenever the actual process is irreversible, since \(dS=dq_{\text{rev}}/T\) is defined specifically using reversible heat, not whatever heat happens to be exchanged along the real, irreversible path (Worked examples shows exactly this contrast: \(q_{\text{rev}}\ne q_{\text{irrev}}\), yet only \(q_{\text{rev}}/T\) correctly gives \(\Delta S_{\text{system}}\) in either case, since \(\Delta S_{\text{system}}\) is identical regardless of which path actually occurred).
Consider only the system's own entropy change, ignoring the surroundings entirely: this would incorrectly predict that any process with \(\Delta S_{\text{system}}<0\) (like freezing or crystallisation) can never occur spontaneously, directly contradicted by common, everyday observation — the second law's actual criterion is always the combined \(\Delta S_{\text{universe}}\), never the system alone (Corollaries, first bullet).

Common errors

Using the actual (irreversible) heat exchanged, rather than the reversible-path heat, to compute \(\Delta S_{\text{system}}\) (Fails without, first bullet).
Judging spontaneity from \(\Delta S_{\text{system}}\) alone, without also accounting for \(\Delta S_{\text{surroundings}}\) (Fails without, second bullet).
Assuming the second law forbids any local decrease in entropy anywhere; it forbids only a net decrease in the total (system plus surroundings) entropy, a substantially weaker and more commonly satisfied constraint than "entropy never decreases anywhere."

Discussion

Rudolf Clausius introduced both the term "entropy" and its defining relation \(dS=dq_{\text{rev}}/T\) in 1865, building directly on Sadi Carnot's earlier (1824) analysis of the maximum possible efficiency of heat engines — work that had already identified the reversible-versus-irreversible distinction central to this result, years before Clausius formalised it into a single new state function. The deeper, molecular-level meaning of entropy (as a measure of the number of microscopic arrangements consistent with a system's macroscopic state) was established later by Ludwig Boltzmann, connecting this purely macroscopic, empirically defined quantity to statistical mechanics.

The technique of computing \(\Delta S\) for an irreversible process via a hypothetical reversible substitute path (Step 2) is conceptually identical to Hess's law's use of hypothetical intermediate steps to compute an otherwise unmeasurable enthalpy change (hess-law) — both rely on the same underlying logical move, licensed in each case by the relevant quantity (\(H\) there, \(S\) here) being a genuine state function, independent of the specific path actually taken.

Common misconception: that the second law means "the universe is running down" in some vague, pessimistic sense, or that entropy increase forbids the local formation of order (living organisms, crystals, complex molecules). Local order can and does increase spontaneously, provided the surroundings' entropy increases by more than enough to compensate (Corollaries), keeping the total, universe-wide entropy change positive — life, crystallisation, and countless other locally ordering processes are entirely consistent with, not contradictions of, the second law.

Worked examples

\text{Reversible case (first-law-chemistry's numbers): } q_{\text{rev}}=+1717.4\,\text{J}, \quad \Delta S_{\text{sys}}=\frac{1717.4}{298}\approx5.763\,\text{J/K}

Since the actual process here was reversible, \(q_{\text{actual}}=q_{\text{rev}}\), and the surroundings lose exactly this heat: \(\Delta S_{\text{surr}}=-1717.4/298\approx-5.763\,\text{J/K}\). \(\Delta S_{\text{universe}}=5.763+(-5.763)=0\) exactly, confirming the reversible-limit prediction. A

\text{Irreversible case: } q_{\text{irrev}}=+1238.9\,\text{J (actual heat)}, \quad \Delta S_{\text{sys}}=\frac{q_{\text{rev}}}{T}\approx5.763\,\text{J/K (unchanged, state function)}

Crucially, \(\Delta S_{\text{sys}}\) is computed using the reversible \(q_{\text{rev}}\), not the smaller actual \(q_{\text{irrev}}\), since \(S\) depends only on the initial and final states (Step 2). The surroundings, however, only actually receive the smaller \(q_{\text{irrev}}\): \(\Delta S_{\text{surr}}=-1238.9/298\approx-4.157\,\text{J/K}\). A

\Delta S_{\text{universe}}(\text{irreversible}) = 5.763 + (-4.157) \approx +1.606\,\text{J/K} > 0

Reading. The same isothermal expansion gives \(\Delta S_{\text{universe}}=0\) when carried out reversibly and \(\Delta S_{\text{universe}}>0\) when carried out irreversibly — a direct, fully numeric confirmation that only the reversible limit achieves exact equality, with any real (irreversible) process giving a strictly positive total.

Scope. \(\Delta S_{\text{sys}}\) is identical (\(\approx5.763\,\text{J/K}\)) in both cases, since it depends only on the shared initial and final states; only \(\Delta S_{\text{surr}}\) (and hence \(\Delta S_{\text{universe}}\)) differs between the two pathways.

Problems

Using the reversible \(q_{\text{rev}}=+3435\,\text{J}\) computed in first-law-chemistry's Problem 1 (\(2.00\,\text{mol}\), same expansion), find \(\Delta S_{\text{sys}}\) for this larger-scale reversible expansion.
Solution
\(\Delta S_{\text{sys}}=q_{\text{rev}}/T=3435/298\approx11.53\,\text{J/K}\), exactly double the \(1.00\,\text{mol}\) result (\(5.763\,\text{J/K}\)), consistent with entropy being an extensive property scaling directly with the amount of substance.
A different irreversible process has \(q_{\text{actual}}=+900\,\text{J}\) while the corresponding reversible path (same initial and final states) would have \(q_{\text{rev}}=+1500\,\text{J}\), both at a constant surroundings temperature of \(300\,\text{K}\). Compute \(\Delta S_{\text{sys}}\), \(\Delta S_{\text{surr}}\), and \(\Delta S_{\text{universe}}\), and confirm the second law is satisfied.
Solution
\(\Delta S_{\text{sys}}=q_{\text{rev}}/T=1500/300=5.00\,\text{J/K}\) (using the reversible value, Step 2). \(\Delta S_{\text{surr}}=-q_{\text{actual}}/T=-900/300=-3.00\,\text{J/K}\) (using the actual value, Step 3). \(\Delta S_{\text{universe}}=5.00+(-3.00)=+2.00\,\text{J/K}>0\), consistent with this being a genuine, spontaneous irreversible process.
Explain, without further calculation, why \(\Delta S_{\text{sys}}\) is identical in both the reversible and irreversible cases of Worked example 1–2, even though the actual heat exchanged, \(q\), differs substantially between them.
Solution
Entropy is a state function (Hypotheses), so \(\Delta S_{\text{sys}}\) depends only on the system's initial and final states (here, the same starting and ending volume, temperature, and amount of gas in both cases), never on the specific path taken to get from one to the other. Since both the reversible and irreversible processes connect the identical initial and final states, \(\Delta S_{\text{sys}}\) must be identical for both — the difference in actual heat exchanged reflects only how "efficiently" that fixed entropy change was achieved (with more of the theoretically available reversible heat actually exchanged with the surroundings in the reversible case), not any difference in the system's own true entropy change.
Water freezing at \(-5^\circ\text{C}\) is a spontaneous process with \(\Delta S_{\text{system}}<0\) (the system becomes more ordered). Explain, using the Result and Corollaries, why this does not violate the second law.
Solution
The second law requires \(\Delta S_{\text{universe}}=\Delta S_{\text{system}}+\Delta S_{\text{surroundings}}\ge0\), not that \(\Delta S_{\text{system}}\) alone must be non-negative (Fails without, second bullet). Freezing releases heat to the surroundings (an exothermic process), increasing the surroundings' entropy; at \(-5^\circ\text{C}\), this surroundings entropy increase is larger in magnitude than the system's own entropy decrease, giving a net positive \(\Delta S_{\text{universe}}\) overall — entirely consistent with the second law, exactly as the Corollaries' first bullet describes for spontaneous ordering processes generally.