Thermodynamics: Spontaneity and Equilibrium HL

Why do some reactions go to completion while others reach a standstill? Why can we predict the voltage of a battery from a table of data? In IB Chemistry HL, the bridge between thermodynamics—the study of energy changes—and chemical equilibrium provides the answers. This connection is not just theoretical; it allows chemists to calculate the exact position of equilibrium for industrial processes like Haber-Bosch ammonia synthesis or to design efficient electrochemical cells. Mastering this link is key to predicting and controlling chemical behavior.

Spontaneity and the Gibbs Free Energy Change

A spontaneous process is one that, given the opportunity, will proceed on its own without ongoing external intervention. Crucially, spontaneity says nothing about speed; a spontaneous reaction might be immeasurably slow. The primary criterion for spontaneity at constant temperature and pressure is the Gibbs free energy change ($\Delta G$).

Gibbs free energy ($G$) combines the system's enthalpy ($H$) and entropy ($S$) into a single value. The change in Gibbs free energy for a reaction is given by: $$\Delta G = \Delta H - T\Delta S$$ where $T$ is the absolute temperature in Kelvin. The sign of $\Delta G$ dictates spontaneity:

• $\Delta G < 0$: The reaction is spontaneous in the forward direction.
• $\Delta G > 0$: The reaction is non-spontaneous forward (spontaneous reverse).
• $\Delta G = 0$: The system is at equilibrium; no net change occurs.

The standard Gibbs free energy change ($\Delta G^\circ$) is the value when all reactants and products are in their standard states (e.g., 1 mol dm$^{-3}$ for solutions, 100 kPa for gases) at a specified temperature (usually 298 K). While $\Delta G$ tells you about spontaneity for specific, non-standard conditions, $\Delta G^\circ$ is a fixed value for a given reaction at a given temperature and is crucial for calculating equilibrium constants.

Linking Gibbs Free Energy to the Equilibrium Constant

This is the core conceptual leap. While $\Delta G$ determines the direction of a reaction, the equilibrium constant ($K$) quantifies the position of equilibrium—the ratio of product to reactant concentrations (or partial pressures) at equilibrium. They are directly connected by the fundamental equation: $$\Delta G = \Delta G^\circ + RT \ln Q$$ Here, $R$ is the gas constant (8.314 J mol$^{-1}$ K$^{-1}$), $T$ is temperature, and $Q$ is the reaction quotient, which has the same form as $K$ but uses the current, non-equilibrium concentrations or pressures.

At equilibrium, $\Delta G = 0$ and $Q = K$. Substituting these into the equation gives the pivotal relationship: $$0 = \Delta G^\circ + RT \ln K$$ Rearranging yields the two most useful forms: $$\Delta G^\circ = -RT \ln K \quad \text{or} \quad K = e^{-\Delta G^\circ / RT}$$

Interpretation: A large, negative $\Delta G^\circ$ (highly spontaneous under standard conditions) corresponds to a very large $K$, meaning the equilibrium mixture is rich in products. A large, positive $\Delta G^\circ$ corresponds to a very small $K$, meaning reactants are favored. If $\Delta G^\circ = 0$, then $K = 1$.

Worked Example: Calculate $K$ at 298 K for a reaction where $\Delta G^\circ = -22.5$ kJ mol$^{-1}$.

1. Convert $\Delta G^\circ$ to J mol$^{-1}$: -22500 J mol$^{-1}$.
2. Use the formula: $K = e^{-\Delta G^\circ / RT}$.
3. Substitute: $K = e^{-(-22500) / (8.314 \times 298)}$.
4. Calculate exponent: $22500 / (8.314 \times 298) \approx 9.08$.
5. Therefore, $K = e^{9.08} \approx 8.8 \times 10^3$.

This large $K$ confirms the reaction is product-favored at equilibrium.

Electrochemical Connections: Cell Potential and Gibbs Free Energy

In electrochemistry, the driving force for a redox reaction in a galvanic cell is the cell potential ($E^\circ_{\text{cell}}$). This, too, is directly related to Gibbs free energy. The maximum electrical work a cell can perform is equal to the decrease in Gibbs free energy. The relationship is: $$\Delta G^\circ = -nFE^\circ_{\text{cell}}$$ where $n$ is the number of moles of electrons transferred in the balanced redox equation, and $F$ is the Faraday constant (96500 C mol$^{-1}$). A positive $E^\circ_{\text{cell}}$ (spontaneous cell reaction) yields a negative $\Delta G^\circ$.

This creates a powerful bridge. You can:

1. Use standard electrode potentials to find $E^\circ_{\text{cell}}$, then calculate $\Delta G^\circ$.
2. Use that calculated $\Delta G^\circ$ to find the equilibrium constant $K$ for the redox reaction using $\Delta G^\circ = -RT \ln K$.

This explains why the standard hydrogen electrode is assigned $E^\circ = 0$ V; it's the thermodynamic baseline from which all other tendencies are measured.

The Role of Temperature in Spontaneity and Equilibrium

Temperature is not a passive variable. It directly influences both $\Delta G$ and $K$, sometimes changing the outcome of a reaction. Recall $\Delta G = \Delta H - T\Delta S$. The effect of temperature depends on the signs of $\Delta H$ and $\Delta S$.

• For an exothermic reaction ($\Delta H < 0$) with a positive entropy change ($\Delta S > 0$): Both terms favor spontaneity ($\Delta G < 0$) at all temperatures.
• For an endothermic reaction ($\Delta H > 0$) with a negative entropy change ($\Delta S < 0$): Both terms oppose spontaneity ($\Delta G > 0$) at all temperatures.
• The interesting cases involve opposing signs:
• $\Delta H < 0, \Delta S < 0$ (e.g., bond formation, which is exothermic but creates order): The reaction is spontaneous only at low temperatures where the $-T\Delta S$ term is small. At high temperatures, it becomes non-spontaneous.
• $\Delta H > 0, \Delta S > 0$ (e.g., melting or evaporation): The reaction is spontaneous only at high temperatures where the $T\Delta S$ term can overcome the positive $\Delta H$.

This temperature dependence is encoded in the equation for $K$. From $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ = -RT \ln K$, we can derive the van't Hoff equation, which shows how $K$ varies with $T$: $$\ln \left( \frac{K_2}{K_1} \right) = \frac{\Delta H^\circ}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$$ For an exothermic reaction ($\Delta H^\circ < 0$), increasing temperature ($T_2 > T_1$) decreases $K$ (shifts equilibrium toward reactants, in line with Le Châtelier's principle). For an endothermic reaction, increasing temperature increases $K$.

Common Pitfalls

1. Confusing $\Delta G$ with $\Delta G^\circ$: A common error is using the standard value $\Delta G^\circ$ to judge spontaneity under non-standard conditions. Remember, $\Delta G^\circ$ only tells you about spontaneity when all substances are at 1 M or 100 kPa. You must use $\Delta G = \Delta G^\circ + RT \ln Q$ for real-world, non-equilibrium mixtures.
2. Equating spontaneity with rate: A spontaneous reaction ($\Delta G < 0$) may have a very high activation energy and thus proceed infinitely slowly (e.g., diamond turning into graphite). Thermodynamics tells you if a reaction can happen; kinetics tells you how fast.
3. Misapplying the temperature rules: Students often memorize that "increasing temperature favors the endothermic direction" but fail to connect this to the signs of $\Delta H$ and $\Delta S$ in the $\Delta G$ equation. Always ground Le Châtelier observations in the underlying thermodynamic equations.
4. Sign errors in electrochemical calculations: Ensure $n$ is correct from the balanced half-equations and that you use a consistent set of units (J, C, V). Forgetting to convert kJ to J is a frequent mistake when using $\Delta G^\circ = -nFE^\circ_{\text{cell}}$.

Summary

• The Gibbs free energy change ($\Delta G$) is the ultimate predictor of spontaneity at constant temperature and pressure. A negative $\Delta G$ means a process is spontaneous.
• At equilibrium, $\Delta G = 0$, leading to the critical link: $\Delta G^\circ = -RT \ln K$. A large negative $\Delta G^\circ$ corresponds to a large equilibrium constant $K$, favoring products.
• In electrochemistry, cell potential relates to Gibbs energy via $\Delta G^\circ = -nFE^\circ_{\text{cell}}$, allowing you to connect voltage tables to equilibrium constants.
• Temperature critically affects both spontaneity and $K$. The sign and magnitude of $\Delta H^\circ$ and $\Delta S^\circ$ determine whether a reaction becomes more or less favorable as temperature changes, as described by the van't Hoff equation.
• Always distinguish between standard conditions ($\Delta G^\circ$, $E^\circ$) and actual conditions ($\Delta G$, $E$, $Q$). The reaction quotient $Q$ is essential for determining the direction of a reaction not at equilibrium.