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Gibbs free energy

T-034Home CU-201Threads thermo
Statement

At constant temperature and pressure — the most common laboratory and industrial condition — the second law's requirement \(\Delta S_{\text{universe}}\ge0\) can be rewritten entirely in terms of system properties alone by defining the Gibbs free energy \(G\equiv H-TS\): \(\Delta G\le0\) for any spontaneous process, with \(\Delta G=0\) exactly at equilibrium and \(\Delta G>0\) for a non-spontaneous process, at constant \(T\) and \(P\).

Why it matters

entropy-second-law's criterion, \(\Delta S_{\text{universe}}\ge0\), is exact but practically inconvenient: it requires tracking both the system and the surroundings separately. Gibbs free energy converts this into a single-quantity test using only the system's own properties, which is why \(\Delta G\) (not \(\Delta S_{\text{universe}}\) directly) is the spontaneity criterion actually used in essentially all practical chemistry, from predicting whether a reaction proceeds to determining phase-transition temperatures.

Hypotheses
Temperature and pressure are both held constant throughout the process.This is the specific condition under which \(G\) is the natural, useful state function; at constant temperature and volume instead, an analogous quantity (Helmholtz free energy, \(A\equiv U-TS\)) plays the equivalent role, though it is used less often in ordinary chemistry, where reactions are far more commonly run at constant (atmospheric) pressure than at constant volume. The surroundings act as a large thermal reservoir at fixed temperature \(T\), exactly as already assumed in entropy-second-law.This licenses computing \(\Delta S_{\text{surroundings}}\) directly from the actual heat exchanged with the system, \(\Delta S_{\text{surr}}=q_{\text{surr}}/T=-\Delta H_{\text{sys}}/T\) (using \(q_p=\Delta H\) from calorimetry), the key substitution that makes the derivation below possible.
Proof
1
\Delta S_{\text{universe}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} = \Delta S_{\text{sys}} - \frac{\Delta H_{\text{sys}}}{T} \ge 0
Substituting the surroundings' entropy change (Hypotheses) into the second law's general statement (entropy-second-law). A
2
T\Delta S_{\text{sys}} - \Delta H_{\text{sys}} \ge 0 \quad (\text{multiplying by } T>0)
Multiplying Step 1 through by the (positive) temperature \(T\) preserves the inequality direction. A
3
\Delta H_{\text{sys}} - T\Delta S_{\text{sys}} \le 0 \quad (\text{multiplying by } -1 \text{ flips the inequality})
Rearranging Step 2 into the form that will define \(\Delta G\). A
4
\Delta G \equiv \Delta H - T\Delta S \ \Longrightarrow\ \Delta G \le 0 \ (\text{spontaneous}), \quad \Delta G = 0\ (\text{equilibrium})
Defining \(\Delta G\) as exactly the left-hand side of Step 3 converts the second law's system-plus-surroundings requirement into a criterion using only the system's own enthalpy and entropy changes; equality (\(\Delta G=0\)) corresponds to the reversible limit (\(\Delta S_{\text{universe}}=0\)), while strict inequality (\(\Delta G<0\)) corresponds to a genuinely spontaneous, irreversible process. A
Result
\Delta G = \Delta H - T\Delta S \begin{cases} <0 & \text{spontaneous} \\ =0 & \text{equilibrium} \\ >0 & \text{non-spontaneous} \end{cases} \quad (\text{constant } T,P)

Reading. A single, entirely system-based quantity, built from the same enthalpy and entropy already established in earlier results, reproduces the full second law's spontaneity verdict without ever needing to separately track the surroundings.

Scope. Strictly valid only at constant temperature and pressure (Hypotheses); reactions run under genuinely varying conditions require the more general differential relation \(dG=-S\,dT+V\,dP\), beyond this introductory treatment.

Corollaries & converses
  • The sign combination of \(\Delta H\) and \(\Delta S\) determines whether, and how, spontaneity depends on temperature: \(\Delta H<0,\Delta S>0\) gives \(\Delta G<0\) at every temperature (always spontaneous); \(\Delta H>0,\Delta S<0\) gives \(\Delta G>0\) at every temperature (never spontaneous); \(\Delta H<0,\Delta S<0\) is spontaneous only below a crossover temperature; \(\Delta H>0,\Delta S>0\) is spontaneous only above a crossover temperature.
  • The crossover temperature, where \(\Delta G=0\) exactly, is \(T=\Delta H/\Delta S\) — a directly computable prediction for the temperature at which a process transitions between spontaneous and non-spontaneous, including ordinary phase transitions (Worked examples).
  • Converse: given an experimentally measured phase-transition or reaction-onset temperature and one of \(\Delta H\) or \(\Delta S\), the other can be solved for directly from \(T=\Delta H/\Delta S\), a standard technique for determining one of these quantities when the other is already known.
Fails without
  • Apply \(\Delta G\le0\) at conditions that are not actually constant temperature and pressure: the derivation (Steps 1–4) depends specifically on the constant-\(P\) substitution \(q_p=\Delta H\) and the constant-\(T\) treatment of the surroundings; under genuinely varying \(T\) or \(P\), \(\Delta G\) computed this way no longer corresponds to the correct spontaneity criterion.
  • Treat \(\Delta G<0\) as guaranteeing a fast reaction: the derivation says nothing whatsoever about reaction rate or mechanism, only about whether a process is thermodynamically favourable given enough time — the classic counterexample is diamond converting to graphite, thermodynamically spontaneous (\(\Delta G<0\)) at ordinary conditions, yet proceeding so slowly, due to a large kinetic activation barrier, that diamonds persist essentially indefinitely on any human timescale (Common errors, Discussion).
Common errors
  • Confusing thermodynamic spontaneity (\(\Delta G<0\), a statement about whether a process is favourable) with kinetic speed (how fast it actually proceeds) — two entirely separate questions, directly illustrated by the diamond-to-graphite example (Fails without).
  • Applying the constant-\(T,P\) formula to a process under different or varying conditions without the appropriate modification (Fails without, first bullet).
  • Forgetting to convert \(\Delta S\) to consistent energy units with \(\Delta H\) before subtracting \(T\Delta S\) (a common arithmetic slip: \(\Delta H\) is typically tabulated in \(\text{kJ/mol}\) while \(\Delta S\) is typically tabulated in \(\text{J/(mol}\cdot\text{K)}\), a factor-of-\(1000\) unit mismatch that must be reconciled before combining them).
Discussion

Josiah Willard Gibbs developed this quantity and much of the mathematical foundation of chemical thermodynamics in a remarkable series of papers published between 1873 and 1878 (most notably "On the Equilibrium of Heterogeneous Substances"). Despite the fundamental and lasting importance of this work, it was largely overlooked for years after publication, partly because it appeared in the relatively obscure Transactions of the Connecticut Academy of Sciences and was written in a dense, highly mathematical style; recognition grew substantially only once European scientists, notably Wilhelm Ostwald, championed and helped translate Gibbs's work into wider circulation.

The four-way classification by sign of \(\Delta H\) and \(\Delta S\) (Corollaries) is one of the most practically used pieces of chemical reasoning in the entire discipline: it explains, for example, why some reactions (strongly exothermic, entropy-increasing) proceed spontaneously under essentially any conditions, while others (like the thermal decomposition of many carbonates and other solids into a solid and a gas, entropy-increasing but endothermic) require heating above a specific threshold temperature before becoming thermodynamically favourable at all — directly the physical basis of industrial processes such as lime production from limestone.

Common misconception: that a positive \(\Delta G\) means a reaction "cannot happen" in any sense. It means the reaction is not spontaneous under the stated conditions as written; the identical reaction can become spontaneous under different conditions (a different temperature, pressure, or concentration, the last addressed in gibbs-equilibrium-constant), or can be driven forward non-spontaneously by continuous external energy input (as in electrolysis or many industrial and biological processes), a distinction with real practical consequences.

Worked examples
1
\text{Ice melting: } \Delta H_{\text{fus}}=+6.01\,\text{kJ/mol},\ \Delta S_{\text{fus}}=+22.0\,\text{J/(mol}\cdot\text{K)}: \quad T_{\text{melt}}=\frac{\Delta H}{\Delta S}=\frac{6010}{22.0}\approx273.2\,\text{K}
Using the Corollaries' crossover-temperature formula with real thermodynamic data for water's fusion; the predicted value, \(273.2\,\text{K}\) (\(\approx0.03^\circ\text{C}\)), matches water's actual, well-known melting point almost exactly. A
2
\text{Limestone decomposition: } \text{CaCO}_3(s)\to\text{CaO}(s)+\text{CO}_2(g),\ \Delta H^\circ=+178.3\,\text{kJ/mol},\ \Delta S^\circ=+160.5\,\text{J/(mol}\cdot\text{K)}
Both \(\Delta H>0\) and \(\Delta S>0\) (a solid decomposing to release gas, increasing disorder), placing this reaction in the Corollaries' "spontaneous only above a crossover temperature" category. \(T=\dfrac{178{,}300}{160.5}\approx1111\,\text{K}\ (\approx838^\circ\text{C})\), closely matching the real, industrially used temperatures (typically \(825\)–\(900^\circ\text{C}\)) at which limestone is calcined to produce lime in real kilns. A
T_{\text{melt}}(\text{H}_2\text{O})\approx273.2\,\text{K}\ (\text{matches }0^\circ\text{C}); \qquad T_{\text{decomp}}(\text{CaCO}_3)\approx838^\circ\text{C}\ (\text{matches industrial practice})

Reading. The same simple ratio formula, applied to tabulated thermodynamic data with no other input, predicts both an everyday physical transition temperature and an industrially significant reaction onset temperature, in both cases matching real-world values closely.

Scope. Both examples fall into the temperature-dependent spontaneity categories established in the Corollaries, since both have \(\Delta H\) and \(\Delta S\) of the same sign.

Problems
  1. A reaction has \(\Delta H^\circ=-92.0\,\text{kJ/mol}\) and \(\Delta S^\circ=-198.0\,\text{J/(mol}\cdot\text{K)}\) (the Haber process, \(\text{N}_2+3\text{H}_2\to2\text{NH}_3\), roughly). Determine which of the four Corollaries categories this falls into, and compute the crossover temperature above which the reaction becomes non-spontaneous.
    Solution\(\Delta H<0\) and \(\Delta S<0\): "spontaneous only below a crossover temperature." \(T=\dfrac{92{,}000}{198.0}\approx465\,\text{K}\ (\approx192^\circ\text{C})\) — above this temperature, \(\Delta G\) becomes positive and the reaction is no longer thermodynamically favourable, consistent with the real Haber process being run at only moderately elevated temperature (not arbitrarily hot) to balance thermodynamic favourability against a practically useful reaction rate.
  2. Classify each of the following as "always spontaneous," "never spontaneous," "spontaneous only at low \(T\)," or "spontaneous only at high \(T\)": (a) \(\Delta H=-50\,\text{kJ/mol}\), \(\Delta S=+80\,\text{J/(mol}\cdot\text{K)}\); (b) \(\Delta H=+40\,\text{kJ/mol}\), \(\Delta S=-30\,\text{J/(mol}\cdot\text{K)}\).
    Solution(a) \(\Delta H<0\), \(\Delta S>0\): every term in \(\Delta G=\Delta H-T\Delta S\) is negative or makes \(\Delta G\) more negative as \(T\) increases, so \(\Delta G<0\) at every positive temperature — always spontaneous. (b) \(\Delta H>0\), \(\Delta S<0\): both terms make \(\Delta G\) positive at every positive temperature — never spontaneous.
  3. Explain why the fact that diamond spontaneously converts to graphite (\(\Delta G<0\) at ordinary conditions) does not mean diamonds are expected to visibly turn into graphite on any observable human timescale.
    Solution\(\Delta G<0\) is a purely thermodynamic statement: it establishes that graphite is the lower-free-energy, thermodynamically favoured form of carbon under ordinary conditions, and that the conversion is, in principle, energetically "downhill." It says nothing about the rate or mechanism of that conversion, which is governed by kinetics, not thermodynamics (hess-law's Discussion makes the same distinction for reaction mechanism generally). The activation energy barrier for rearranging diamond's rigid, strongly bonded tetrahedral lattice into graphite's layered structure is extremely large, making the actual conversion rate immeasurably slow at ordinary temperature — thermodynamically favourable, but kinetically inaccessible on any practical timescale.
  4. Given \(\Delta H^\circ_{\text{rxn}}=+130.0\,\text{kJ/mol}\) for a reaction, and knowing from separate experiments that the reaction becomes spontaneous above \(400\,\text{K}\), estimate \(\Delta S^\circ_{\text{rxn}}\), assuming the crossover occurs at exactly this temperature.
    SolutionUsing \(T=\Delta H/\Delta S\) at the crossover point: \(\Delta S=\dfrac{\Delta H}{T}=\dfrac{130{,}000}{400}=325\,\text{J/(mol}\cdot\text{K)}\) — a positive value, consistent with the reaction being endothermic yet becoming spontaneous only above a threshold temperature (the Corollaries' "spontaneous only at high \(T\)" case).