18.3 Gibbs Free Energy and the Relationship between ΔG, ΔH, & ΔS
PRE-HEALTH PREP
Gibbs Free Energy and Spontaneity
ΔG = ΔH − TΔS
General Chemistry 18.3
Lesson Overview
Gibbs free energy, defined by ΔG = ΔH − TΔS, determines whether a chemical reaction proceeds spontaneously by combining enthalpy and entropy changes into a single measurable quantity. The 4 possible sign combinations of ΔH and ΔS produce reactions that are always spontaneous, never spontaneous, or conditionally spontaneous above or below a threshold temperature calculated as T = ΔH/ΔS. Worked examples include predicting the spontaneity of ammonia synthesis and calculating the boiling point of water from thermodynamic data.
Table of Contents
What Is Gibbs Free Energy?
Definition and the Core Equation
Gibbs free energy is represented by the symbol G, and in chemistry what we almost always care about is its change: ΔG, the change in Gibbs free energy. The quantity is defined by one of the most important equations in all of thermodynamics:
The Gibbs Free Energy Equation:
ΔG = ΔH − TΔS
where ΔH is the enthalpy change of the system, T is the absolute temperature in Kelvin, and ΔS is the entropy change of the system.
A common point of confusion: the word "free" in Gibbs free energy does not mean the energy costs nothing. Think of it the way you use the word in everyday speech — "Are you free for lunch on Friday?" In other words, Gibbs free energy is energy that is available, or free, to do work.
Spontaneous vs. nonspontaneous — a concrete example: Spontaneous processes occur naturally without requiring a net input of free energy from the surroundings. For instance, ice at 10°C and 1 atm melts spontaneously, whereas ice at −10°C and 1 atm does not. During a spontaneous process the system "uses up" some of its free energy, so ΔG is negative. During a nonspontaneous process, free energy must be supplied from the surroundings, so ΔG is positive.
What the Sign of ΔG Tells You About Spontaneity
To make sense of ΔG, it helps to think about the relationship between a system (the reaction you are studying) and its surroundings (everything else in the universe). A system cannot create its own energy. If it ends up with more free energy, the surroundings must have supplied it. Conversely, a system can spend its free energy to drive a process forward.
That spending of free energy is precisely what spontaneity means in thermodynamics:
- ΔG < 0 (negative): The reaction is spontaneous — the system releases free energy and the process occurs on its own.
- ΔG > 0 (positive): The reaction is non-spontaneous — the system would need to absorb free energy from the surroundings to proceed.
- ΔG = 0: The system is at equilibrium — no net change occurs.
| ΔG Value | Meaning |
|---|---|
| ΔG < 0 | Spontaneous |
| ΔG > 0 | Nonspontaneous |
| ΔG = 0 | At Equilibrium |
ΔG vs. ΔG° (Delta G Standard)
You will frequently see two related symbols in thermodynamics problems: plain ΔG and ΔG° (Delta G standard). The small circle denotes standard conditions, and the distinction matters for general chemistry exams.
Standard conditions are defined as:
- All aqueous species at a concentration of 1 M
- All gaseous species at a partial pressure of 1 atm
Common misconception: Standard conditions do not specify a temperature. Thermodynamic data in textbooks are most often tabulated at 298 K (25 °C), so many students assume 25 °C is part of the definition — but it is not. You can have standard conditions at any temperature. The standard symbol only constrains concentrations and pressures, not temperature.
In this lesson, all discussion applies to plain ΔG — the change in Gibbs free energy under whatever actual conditions a given reaction occurs. Whether a reaction is spontaneous (ΔG < 0), non-spontaneous (ΔG > 0), or at equilibrium (ΔG = 0) is always evaluated under the conditions being described.
Why ΔH is often the dominant term: The Gibbs equation applies under standard or non-standard conditions alike. For a reaction to have any chance of being spontaneous, at least one of the following must be true: ΔH is negative, or ΔS is positive.
The first term, ΔH, is often the dominant term because ΔS is typically much smaller in magnitude. This is why we commonly expect exothermic reactions (negative ΔH) to be spontaneous and endothermic reactions (positive ΔH) to be non-spontaneous — though this is not always the case.
The second term, −TΔS, gains importance as temperature increases. Even a small ΔS can dominate at very high temperatures, which is why temperature is the deciding factor in the two conditional spontaneity cases.
Derivation of ΔG from the Second Law of Thermodynamics
Where did the Gibbs free energy equation come from? The starting point is the Second Law of Thermodynamics, which requires that for any spontaneous process:
Second Law Requirement for Spontaneity:
ΔSsystem + ΔSsurroundings > 0
The total entropy of the universe must increase for a process to be spontaneous.
This expression is thermodynamically rigorous, but deeply impractical — directly measuring the entropy change of the entire surroundings is not feasible. Gibbs recognized that the surroundings contribution can be rewritten entirely in terms of the system.
Substitution and Rearrangement
Recall that entropy can be expressed as Qreversible / T, and that for a constant-pressure process, heat equals enthalpy: Q = ΔH. Therefore:
ΔSsurroundings = ΔHsurroundings / T
Because the system and surroundings simply exchange enthalpy — what the system loses, the surroundings gain — the enthalpy changes are equal in magnitude but opposite in sign:
ΔHsurroundings = −ΔHsystem
Substituting into the Second Law inequality and dropping the "system" label (unlabeled thermodynamic quantities always refer to the system by convention):
ΔS − ΔH/T > 0
Multiplying through by −T (which flips the inequality sign, since T is always positive in Kelvin):
ΔH − TΔS < 0
This is the spontaneity requirement expressed entirely in terms of the system. Gibbs recognized this combination as a single, elegant quantity and defined it as ΔG:
ΔG = ΔH − TΔS
A spontaneous process requires ΔG < 0. No measurement of the surroundings is needed — only the enthalpy and entropy changes of the system itself.
Unlabeled quantities refer to the system: ΔG, ΔH, and ΔS written without a subscript always mean the system (typically a chemical reaction). If you mean the surroundings or the universe, you must label them explicitly: ΔSsurr, ΔSuniv.
What the Universe "Wants": 2 Driving Forces for Spontaneity
Before working through the mathematics of ΔG, it is useful to build intuition about what drives spontaneity. The universe has two thermodynamic preferences:
- Lower energy (negative ΔH): Exothermic reactions release heat to the surroundings, lowering the system's enthalpy. The universe favors this.
- Greater disorder (positive ΔS): The Second Law tells us that the entropy of the universe tends to increase. A reaction that increases the system's entropy contributes to this trend.
Give the universe both, and the reaction is spontaneous at every temperature. Deny the universe both, and the reaction is never spontaneous. Provide only one of the two, and the reaction is conditionally spontaneous — whether it proceeds depends on the temperature.
Key insight for ΔG = ΔH − TΔS: The second term, −TΔS, always has the opposite sign of ΔS, because temperature is always positive in Kelvin. If ΔS is positive, −TΔS is negative; if ΔS is negative, −TΔS is positive. Keep this in mind as you evaluate each of the 4 cases below.
The 4 Cases for Spontaneity (ΔH and ΔS Sign Combinations)
Case 1: ΔH Negative, ΔS Positive — Spontaneous at All Temperatures
This is the "best case" scenario for spontaneity: the universe gets both things it wants. ΔH is negative, so the first term in ΔG = ΔH − TΔS is negative. ΔS is positive, so −TΔS is also negative (negative T multiplied by positive ΔS, with the leading minus sign). The sum of two negative numbers is always negative, regardless of the temperature.
Result: ΔG is negative at every temperature. The reaction is spontaneous at all temperatures.
Case 2: ΔH Positive, ΔS Negative — Non-Spontaneous at All Temperatures
The universe gets nothing it wants. ΔH is positive, so the first term is positive. ΔS is negative, so −TΔS is positive (negative times negative equals positive). The sum of two positive numbers is always positive, regardless of temperature.
Result: ΔG is positive at every temperature. The forward reaction is non-spontaneous at all temperatures. Equivalently, the reverse reaction is spontaneous at all temperatures.
Case 3: ΔH Positive, ΔS Positive — Spontaneous at High Temperatures
Now for the conditional cases. When ΔH is positive (unfavorable) and ΔS is positive (favorable), the first term is positive and the second term, −TΔS, is negative. The sign of ΔG depends on which term is larger. Since neither ΔH nor ΔS changes much with temperature, the lever is T itself. A higher temperature amplifies the magnitude of the −TΔS term, eventually making it larger than ΔH and driving ΔG negative.
Result: ΔG becomes negative above some threshold temperature. The reaction is spontaneous at high temperatures (above the threshold).
Classic example: Boiling water — H₂O(l) → H₂O(g). Vaporization is endothermic (positive ΔH) and increases entropy (positive ΔS). It becomes spontaneous above 100 °C (373 K) at 1 atm.
What counts as "high"? "High temperature" means above the threshold for that specific reaction — it is not a judgment about whether humans would find the temperature comfortable. The threshold for melting ice is 0 °C; the threshold for boiling water is 100 °C. Both are "high" for their respective processes.
Case 4: ΔH Negative, ΔS Negative — Spontaneous at Low Temperatures
When ΔH is negative (favorable) and ΔS is negative (unfavorable), the first term is negative and −TΔS is positive. Again, which term dominates depends on temperature. Here, a lower temperature keeps the positive −TΔS term small, allowing the negative ΔH term to dominate and produce a negative ΔG.
Result: ΔG becomes negative below some threshold temperature. The reaction is spontaneous at low temperatures (below the threshold).
Classic example: Freezing water — H₂O(l) → H₂O(s). Freezing is exothermic (negative ΔH) but decreases entropy (negative ΔS). It becomes spontaneous below 0 °C. Condensing steam to liquid water follows the same logic: it is spontaneous below 100 °C.
Finding the Threshold Temperature (T = ΔH / ΔS)
For the 2 conditional cases (Cases 3 and 4), you can calculate the exact temperature at which the reaction transitions between spontaneous and non-spontaneous. At the threshold temperature, the system is at equilibrium, so ΔG = 0.
Threshold Temperature Formula:
Set ΔG = 0 in the Gibbs free energy equation:
0 = ΔH − TΔS
Tthreshold = ΔH / ΔS
The result is in Kelvin. At temperatures above Tthreshold, a positive-ΔS reaction becomes spontaneous; at temperatures below Tthreshold, a negative-ΔS reaction becomes spontaneous.
The ΔH–ΔS relationship at phase transitions: At a phase-change temperature (melting point or boiling point), the two phases coexist in equilibrium, so ΔG = 0. Substituting into the Gibbs equation gives a simple relationship used frequently in exam problems:
ΔG = ΔH − TΔS
0 = ΔH − TΔS
TΔS = ΔH
This leads directly to the expressions students are expected to know for phase transitions: ΔSfus = ΔHfus / Tfus (melting) and ΔSvap = ΔHvap / Tbp (vaporization).
It is from this last expression that undergraduate students are presented with equations that relate the freezing temperature to the ΔH and ΔS of fusion and the boiling temperature to the ΔH and ΔS of vaporization:
Solving for a freezing or boiling point: Rearranging TΔS = ΔH to solve for T gives the threshold temperature at which a phase change occurs, or the temperature at which a reaction with both-negative or both-positive ΔH and ΔS crosses between spontaneous and nonspontaneous. Remember that ΔG and ΔH values are often in kJ/mol while ΔS is in J/(K·mol) — always match units before calculating. All temperatures must be in Kelvin.
Unit warning — this is a frequent exam trap: ΔH is typically reported in kJ/mol, while ΔS is typically reported in J/(mol·K). These units do not cancel directly. Before dividing, convert both to the same energy unit — either convert ΔH to joules (multiply by 1000) or convert ΔS to kilojoules (divide by 1000). The resulting threshold temperature will be in Kelvin. Watch out for answer choices expressed in °C — professors often include the Kelvin value converted to Celsius (incorrectly labeled as °C) as a distractor.
Summary Table: All 4 Spontaneity Cases
Commit this table to memory, or be prepared to derive each result quickly from ΔG = ΔH − TΔS.
| ΔH | ΔS | Spontaneity | Reasoning |
|---|---|---|---|
| Negative | Positive | Spontaneous at all temperatures | Both terms in ΔG are negative; sum is always negative. |
| Positive | Negative | Never spontaneous (reverse is spontaneous) | Both terms in ΔG are positive; sum is always positive. |
| Positive | Positive | Spontaneous at high temperatures | −TΔS dominates at large T; need T above threshold. |
| Negative | Negative | Spontaneous at low temperatures | ΔH dominates at small T; need T below threshold. |
The following table also shows the sign of the −TΔS° term for each combination, which can help reinforce the reasoning behind each case:
| ΔH° | ΔS° | −TΔS° | Conditions for Spontaneity |
|---|---|---|---|
| − | + | − | Spontaneous at All Temperatures |
| + | − | + | Nonspontaneous at All Temperatures |
| − | − | + | Spontaneous at Low Temperatures |
| + | + | − | Spontaneous at High Temperatures |
For a review of Enthalpy, see: 5.1 The First Law of Thermodynamics, Enthalpy, and Phase Changes
For a review of Entropy, see: 18.2 Entropy
2 Worked Example Problems
Example 1: Predicting Spontaneity for N₂ + 3H₂ → 2NH₃
Problem: For the reaction N₂(g) + 3H₂(g) → 2NH₃(g), ΔH is negative. Is this reaction spontaneous at all temperatures, no temperatures, high temperatures, or low temperatures?
Step 1 — Identify ΔH: Given. ΔH is negative (exothermic).
Step 2 — Determine the sign of ΔS: ΔS is not provided, but we can predict it from the change in moles of gas. The reactant side has 1 + 3 = 4 moles of gas; the product side has 2 moles of gas. The number of moles of gas decreases, so entropy decreases: ΔS is negative.
Step 3 — Apply the spontaneity table: ΔH negative and ΔS negative → spontaneous at low temperatures (Case 4).
Answer: The reaction is spontaneous at low temperatures only. At high temperatures, the positive −TΔS term becomes too large, and ΔG turns positive.
Example 2: Calculating the Boiling Point of Water from Thermodynamic Data
At a phase transition temperature — boiling point or melting point — the 2 phases coexist in equilibrium. By definition, ΔG = 0 at equilibrium. This gives a direct way to calculate the transition temperature from ΔH and ΔS data using the Gibbs free energy equation.
Problem: Given that the enthalpy of vaporization of water is ΔHvap = 40.68 kJ/mol and the entropy of vaporization is ΔSvap = 109 J/(mol·K), calculate the normal boiling point of water.
Setup: At the boiling point, ΔG = 0, so:
0 = ΔHvap − Tbp ΔSvap
Tbp = ΔHvap / ΔSvap
Unit conversion: ΔH is in kJ/mol; ΔS is in J/(mol·K). Convert ΔH to joules:
40.68 kJ/mol × 1000 J/kJ = 40,680 J/mol
Calculation:
Tbp = 40,680 J/mol ÷ 109 J/(mol·K) = 373 K
Answer: 373 K = 100 °C — exactly the known boiling point of water. This confirms the method works. For any other liquid, the same threshold temperature approach gives the boiling point even if you haven't memorized it.
Exam distractor alert: After correctly computing 373 K, professors will often include "373 °C" as a wrong answer choice. Remember: the threshold temperature formula always yields Kelvin. Convert to Celsius only if explicitly asked, and do so correctly: 373 K − 273 = 100 °C, not 373 °C.
7 Key Takeaways
- ΔG = ΔH − TΔS is the defining Gibbs free energy equation. Memorize it.
- Gibbs free energy represents the energy available to do work. When a system spends that energy to drive a process, ΔG is negative and the reaction is spontaneous.
- ΔG < 0 → spontaneous; ΔG > 0 → non-spontaneous; ΔG = 0 → equilibrium.
- Distinguish ΔG (actual conditions) from ΔG° (standard conditions: 1 M aqueous, 1 atm gas). Standard conditions do not specify a temperature.
- The 4 spontaneity cases are determined by the signs of ΔH and ΔS: negative/positive → always spontaneous; positive/negative → never spontaneous; positive/positive → spontaneous above threshold T; negative/negative → spontaneous below threshold T.
- At a phase transition (boiling point, melting point), the 2 phases are in equilibrium, so ΔG = 0 and the threshold temperature is T = ΔH / ΔS.
- Always check units before calculating the threshold temperature: ΔH in kJ/mol and ΔS in J/(mol·K) cannot be divided directly. Convert to matching units first. The result is always in Kelvin.