Entropy of the Surroundings & Gibbs Free Energy: Spontaneity and Temperature Dependence

Entropy of the Surroundings

Heat Transfer and Entropy Change

The entropy change of the surroundings (ΔSsurr) is a key factor in determining the spontaneity of chemical reactions. It is directly related to the heat exchanged with the surroundings and the temperature at which the process occurs.

• Formula: $\Delta S_{\text{surr}} = -\frac{\Delta H_{\text{sys}}}{T}$, where T is in Kelvins.

• Exothermic reactions (ΔHsys < 0): Surroundings gain heat, so ΔSsurr > 0.

• Endothermic reactions (ΔHsys > 0): Surroundings lose heat, so ΔSsurr < 0.

• Temperature dependence: At low T, ΔSsurr is large; at high T, ΔSsurr is small for the same ΔH.

Example: When water freezes (liquid → solid), ΔSsys < 0 (system becomes more ordered), but the process is exothermic, so ΔSsurr > 0. At low temperatures, the positive ΔSsurr dominates, making the process spontaneous.

Temperature Dependence of Spontaneity

Why Phase Transitions Occur at Specific Temperatures

The spontaneity of a process depends on the balance between the entropy changes of the system and the surroundings. The sum, ΔSuniv = ΔSsys + ΔSsurr, determines whether a process is spontaneous (ΔSuniv > 0) or not.

• At low temperature: ΔSsurr is large and can outweigh a negative ΔSsys, making ΔSuniv positive (spontaneous).

• At high temperature: ΔSsurr is small, so a negative ΔSsys dominates, making ΔSuniv negative (nonspontaneous).

This explains why, for example, water freezes spontaneously below 0°C but not above.

Gibbs Free Energy: The Criterion for Spontaneity

Definition and Equation

Gibbs free energy (G) is a thermodynamic function that combines enthalpy and entropy to predict the spontaneity of a process at constant temperature and pressure.

• Equation: $G = H - TS$

• Change in Gibbs free energy: $\Delta G = \Delta H - T\Delta S$

• Relationship to the universe: $\Delta G = -T\Delta S_{\text{universe}}$

Spontaneity criterion:

• ΔG < 0: Process is spontaneous

• ΔG = 0: System is at equilibrium

• ΔG > 0: Process is nonspontaneous (reverse is spontaneous)

Physical meaning: |ΔG| is the maximum useful work obtainable from a process (if ΔG < 0), or the minimum work required to drive a nonspontaneous process (if ΔG > 0).

The Four Cases: How ΔH, ΔS, and T Determine Spontaneity

Classification of Reaction Spontaneity

The sign of ΔH (enthalpy change) and ΔS (entropy change), along with temperature, determines whether a reaction is spontaneous.

• Case 1: ΔH < 0, ΔS > 0 — Always spontaneous (ΔG < 0 at all T)

• Case 2: ΔH > 0, ΔS < 0 — Never spontaneous (ΔG > 0 at all T)

• Case 3: ΔH < 0, ΔS < 0 — Spontaneous at low T only

• Case 4: ΔH > 0, ΔS > 0 — Spontaneous at high T only

Crossover temperature: The temperature at which ΔG changes sign is $T_{\text{crossover}} = \frac{\Delta H}{\Delta S}$.

Practice Problems and Applications

Sample Calculations

Example 1: For the reaction 2 N2(g) + O2(g) → 2 N2O(g), ΔH°rxn = +163.2 kJ, ΔSsys = −109 J/K at 25°C.

• Calculate ΔSsurr at 25°C (T = 298.15 K): $\Delta S_{\text{surr}} = -\frac{163,200\ \text{J}}{298.15\ \text{K}} = -547.4\ \text{J/K}$

• Calculate ΔSuniv:

• $\Delta S_{\text{univ}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} = (−109) + (−547.4) = −656.4\ \text{J/K}$

• Since ΔSuniv < 0, the reaction is NOT spontaneous at 25°C.

Example 2: For C2H4(g) + H2(g) → C2H6(g), ΔH = −137.5 kJ, ΔS = −120.5 J/K:

• Case 3: Spontaneous at low T only (ΔH−, ΔS−)

• Crossover T: $T = \frac{137,500}{120.5} \approx 1141\ \text{K}$

Example 3: Electrolysis of water (2H2O(l) → 2H2(g) + O2(g)): ΔH > 0, ΔS > 0 — Spontaneous only at high T, but requires electrical input at room temperature.

Summary Table: Effect of ΔH, ΔS, and T on Spontaneity

ΔH	ΔS	Low Temperature	High Temperature	Example
−	+	Spontaneous (ΔG < 0)	Spontaneous (ΔG < 0)	2 N2(g) + O2(g) → 2 N2O(g)
+	−	Nonspontaneous (ΔG > 0)	Nonspontaneous (ΔG > 0)	3 O2(g) → 2 O3(g)
−	−	Spontaneous (ΔG < 0)	Nonspontaneous (ΔG > 0)	H2O(g) → H2O(l)
+	+	Nonspontaneous (ΔG > 0)	Spontaneous (ΔG < 0)	H2O(l) → H2O(g)

Key Takeaways

• ΔSsurr = −ΔHsys/T: Exothermic reactions increase surroundings entropy; endothermic reactions decrease it.

• ΔSsurr is larger at low T.

• Gibbs free energy: ΔG = ΔH − TΔS = −TΔSuniverse; ΔG < 0 → spontaneous.

• Four cases: (1) ΔH−, ΔS+: always spontaneous; (2) ΔH+, ΔS−: never spontaneous; (3) ΔH−, ΔS−: spontaneous at low T only; (4) ΔH+, ΔS+: spontaneous at high T only.

• Crossover temperature: Tcrossover = ΔH/ΔS (where ΔG changes sign).