AP Chemistry Unit 9: Entropy, Free Energy, and Favorability

Introduction to Entropy

Thermodynamics is the study of energy and its interconversions. While Enthalpy ($\Delta H$) tells us about heat flow, it isn't enough to predict if a reaction will happen naturally. To do that, we need a new quantity.

Defining Entropy ($S$)

Entropy ($S$) is a measure of the dispersal of matter and energy. Often simplified as "disorder" or "randomness," defined more rigorously in AP Chemistry involves microstates.

• Microstates: The specific configuration of all atom positions and energies at a given instant. High entropy means a system has a vast number of possible microstates.
• Second Law of Thermodynamics: The entropy of the universe increases in any thermodynamically favorable (spontaneous) process. \Delta S{univ} = \Delta S{sys} + \Delta S_{surr} > 0

States of Matter and Molecular Complexity

Entropy depends heavily on the physical state and structure of the substance.

Absolute Entropy and Entropy Change

Unlike Enthalpy ($H$), absolute Entropy ($S$) can be measured.

The Third Law of Thermodynamics

The entropy of a perfect crystal at absolute zero (0 K) is exactly zero. This provides a baseline reference point. Note: Elements in their standard states have non-zero standard entropies (unlike $\Delta H_f^\circ$, which is zero for pure elements).

Calculating $\Delta S^\circ_{rxn}$

To calculate the standard entropy change for a reaction, we use Hess's Law logic:

\Delta S^\circ{rxn} = \Sigma S^\circ{products} - \Sigma S^\circ_{reactants}

• Positive $\Delta S$ ($+S$): Matter is becoming more dispersed (e.g., solid $\rightarrow$ gas, or $2$ gas moles $\rightarrow$ $4$ gas moles).
• Negative $\Delta S$ ($-S$): Matter is becoming more ordered/constrained (e.g., gas $\rightarrow$ liquid, precipitation).

Gibbs Free Energy and Thermodynamic Favorability

Gibbs Free Energy ($G$) combines enthalpy and entropy to predict thermodynamic favorability (formerly called spontaneity). Favorability means a process will occur without continuous external intervention.

The Gibbs-Helmholtz Equation

This is arguably the most critical equation in Unit 9:

\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ

• Units: Be careful! $\Delta H$ is usually in kJ/mol, while $\Delta S$ is in J/mol$\,\cdot\,$K. You must convert $\Delta S$ to kJ or $\Delta H$ to J before subtracting.
• Standard Conditions: $T = 298\,K$, $1\,atm$, $1\,M$.

Interpreting Signs

The sign of $\Delta G$ determines favorability:

• $\Delta G < 0$: Thermodynamically favorable (spontaneous).
• $\Delta G > 0$: Thermodynamically unfavorable (non-spontaneous).
• $\Delta G = 0$: The system is at equilibrium.

Temperature Dependence of $\Delta G$

The signs of $\Delta H$ and $\Delta S$ determine at which temperatures a reaction is favorable.

Mnemonic: To remember High/Low cases, look at $\Delta S$. If $\Delta S$ is positive, you want high $T$ to make the $-T\Delta S$ term large enough to dominate.

Thermodynamic vs. Kinetic Control

Just because a reaction is "thermodynamically favorable" ($\Delta G < 0$) does not mean it will happen quickly.

Kinetic Control

A reaction is under kinetic control if it has a large negative $\Delta G$ (should happen) but occurs at a negligible rate.

• Reason: High Activation Energy ($E_a$).
• Example: Diamond turning into graphite is thermodynamically favorable ($\Delta G < 0$), but the reaction is so slow (millions of years) that diamonds are effectively stable. We say this reaction is kinetically controlled.

Free Energy and Equilibrium

$\Delta G^\circ$ tells us about the favorability from standard states to equilibrium. The Equilibrium Constant ($K$) tells us how far the reaction proceeds.

The Relationship

The bridge between Thermodynamics and Equilibrium is calculated as:

\Delta G^\circ = -RT \ln K

Where:

• $R = 8.314\, J/(mol\cdot K)$ (Gas constant in energy units)
• $T =$ Temperature in Kelvin
• $K =$ Equilibrium constant ($K{eq}$, $Kp$, $K_c$, etc.)

Qualitative Implications

• If $\Delta G^\circ < 0$, then $\ln K > 0$, so $K > 1$. The equilibrium favors products.
• If $\Delta G^\circ > 0$, then $\ln K < 0$, so $K < 1$. The equilibrium favors reactants.
• If $\Delta G^\circ = 0$, then $\ln K = 0$, so $K = 1$.

Coupled Reactions

Biological systems and industrial processes often need to perform reactions that are thermodynamically unfavorable ($\Delta G > 0$). To achieve this, they use coupled reactions.

The Principle

Two reactions that share a common intermediate can be coupled. If you add the reactions together, you add their $\Delta G$ values (Hess's Law).

1. Reaction A (Unfavorable): $\Delta G_1 = +20\,kJ$
2. Reaction B (Highly Favorable): $\Delta G_2 = -50\,kJ$

Net Reaction: $\Delta G_{total} = +20 + (-50) = -30\,kJ$

Because the net $\Delta G$ is negative, the overall process becomes favorable.

• Example: ATP hydrolysis ($
  \Delta G < 0$) is often coupled with protein synthesis ($ \Delta G > 0$) in the body to drive the synthesis forward.

Common Mistakes and Pitfalls

1. The Unit Trap: Students consistently fail to convert units in $\Delta G = \Delta H - T\Delta S$.
   • $\Delta H$ is usually kJ.
   • $\Delta S$ is usually J.
   • Fix: Divide $\Delta S$ by 1000 to get kJ/K before calculating.
2. "Spontaneous" Confusion: Don't assume "spontaneous" means "fast." Rusting is spontaneous, but slow. Always distinguish between Thermodynamics (Direction) and Kinetics (Rate).
3. Standard Conditions vs. Equilibrium: $\Delta G^\circ$ is the free energy change at standard conditions (all pure, 1M, 1atm). At equilibrium, calculate $\Delta G$ (without the degree symbol), which equals 0. Do not confuse $\Delta G^\circ$ with $\Delta G$.
4. Temperature Units: Using Celsius instead of Kelvin. Always use $K = ^\circ C + 273.15$.