Free Energy and Thermodynamics: Study Notes for General Chemistry

Free Energy and Thermodynamics

Introductory Concepts

Thermodynamics is the study of energy transformations, particularly the relationships between heat (q), work (w), and internal energy (ΔE). It provides the foundation for understanding spontaneity, equilibrium, and the direction of chemical processes.

• Internal Energy (ΔE): The total energy contained within a system. It is a state function, meaning its value depends only on the state of the system, not the path taken to reach that state.

• First Law of Thermodynamics: Energy cannot be created or destroyed, only transferred or converted. Expressed as: $\Delta E = q + w$

• Heat Capacity: $q = m C_s \Delta T$ or $q = C \Delta T$

• Relationship between Enthalpy and Internal Energy: $\Delta H = \Delta E + P\Delta V$

• Work at Constant Pressure: $w = -P\Delta V$

Thermodynamics and Spontaneity

Spontaneity refers to the natural tendency of a process to occur without external intervention. Thermodynamics helps predict whether a reaction or process is spontaneous by comparing the potential energy of the system before and after the process.

Comparing Potential Energy

The direction of spontaneity can be determined by comparing the potential energy of the system at the start and the end. Systems tend to move toward lower potential energy, which is often associated with increased stability.

Thermodynamics vs. Kinetics

Thermodynamics determines whether a reaction is possible and the direction it will proceed, while kinetics describes the rate at which the reaction occurs. A process can be thermodynamically favorable but kinetically slow.

Thermodynamic Favorability

Thermodynamic favorability is influenced by two main factors: enthalpy (ΔH) and entropy (ΔS). Generally, the enthalpy factor is more significant, but entropy becomes increasingly important at higher temperatures.

Enthalpy

Standard Enthalpies of Formation

The standard enthalpy of formation (ΔHf°) is the enthalpy change when one mole of a compound is formed from its elements in their standard states. These values are used to calculate the enthalpy changes of reactions.

Entropy (S)

Definition and Properties

Entropy (S) is a measure of the dispersal of energy and the degree of disorder in a system. Systems with higher entropy have energy more widely spread among their particles.

• Boltzmann Equation: $S = k \ln W$ where k is the Boltzmann constant ($1.38 \times 10^{-23}$ J/K) and W is the number of microstates (ways to arrange the system).

• Units: J/K

Natural Tendencies in Spontaneous Processes

There are two driving forces for spontaneous processes:

• Tendency to achieve a lower energy state

• Tendency toward a more disordered state (higher entropy)

Changes in Entropy (ΔS)

Entropy increases when a system becomes more disordered. Several types of changes increase entropy:

• Phase changes: solid < liquid < gas

• Dissolution of solids into ions

• Reactions producing more product molecules than reactant molecules

• Increase in temperature

Entropy Change and State Change

Entropy always increases with temperature and increases dramatically during a phase change, such as melting or vaporization.

The Second Law of Thermodynamics

Statement and Mathematical Formulation

The Second Law of Thermodynamics states that the entropy of the universe increases for any spontaneous process:

• $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings}$

• For a spontaneous process: $\Delta S_{universe} > 0$

• Heat exchange with surroundings: $q_{surr} = -q_{sys} = -\Delta H_{sys}$

• $\Delta S_{surr} = \frac{-\Delta H_{sys}}{T}$

• $\Delta S_{univ} = \Delta S_{sys} - \frac{\Delta H_{sys}}{T}$

Gibbs Free Energy (ΔG)

Definition and Criteria for Spontaneity

Gibbs free energy (ΔG) combines enthalpy and entropy to predict spontaneity at constant temperature and pressure:

• $\Delta G_{sys} = \Delta H_{sys} - T\Delta S_{sys}$

• If $\Delta G < 0$, the process is spontaneous.

• If $\Delta G > 0$, the process is nonspontaneous.

• If $\Delta G = 0$, the system is at equilibrium.

Effects of ΔH, ΔS, and T on Spontaneity

The sign and magnitude of ΔG depend on the values of ΔH, ΔS, and temperature (T). The table below summarizes the effect of these variables:

ΔH	ΔS	Low Temperature	High Temperature	Example
-	+	Spontaneous (ΔG < 0)	Spontaneous (ΔG < 0)	2 NH3 → N2 + 3 H2
+	-	Nonspontaneous (ΔG > 0)	Nonspontaneous (ΔG > 0)	2 H2O → 2 H2 + O2
-	-	Spontaneous (ΔG < 0)	Nonspontaneous (ΔG > 0)	H2O(l) → H2O(s)
+	+	Nonspontaneous (ΔG > 0)	Spontaneous (ΔG < 0)	H2O(s) → H2O(l)

The Third Law of Thermodynamics and Absolute Entropy

Absolute Entropy

The Third Law of Thermodynamics states that the entropy of a perfect crystal at absolute zero (0 K) is zero. All substances have positive entropy values at temperatures above 0 K.

Standard Molar Entropies

The standard molar entropy (S°) is the entropy of one mole of a substance in its standard state (1 atm, 25°C). Entropy values are always positive and increase with temperature.

Factors Affecting Entropy

• Physical State: S(solid) < S(liquid) < S(gas)

• Molar Mass: Larger molar mass generally means higher entropy.

• Allotropes: Less constrained structures have higher entropy.

• Molecular Complexity: More complex molecules have higher entropy.

• Dissolution: Dissolved solids have higher entropy than their undissolved forms.

Calculating ΔS°, ΔH°, and ΔG° for Reactions

Standard State Calculations

• $\Delta S_{rxn}^\circ = \sum n_p S^\circ (\text{products}) - \sum n_r S^\circ (\text{reactants})$

• $\Delta H_{rxn}^\circ = \sum n_p \Delta H_f^\circ (\text{products}) - \sum n_r \Delta H_f^\circ (\text{reactants})$

• $\Delta G_{rxn}^\circ = \sum n_p \Delta G_f^\circ (\text{products}) - \sum n_r \Delta G_f^\circ (\text{reactants})$

Free Energy Change and Equilibrium

Relationship between ΔG, ΔG°, and Q

• $\Delta G = \Delta G^\circ + RT \ln Q$

• At equilibrium, $\Delta G = 0$ and $Q = K_{eq}$, so $\Delta G^\circ = -RT \ln K_{eq}$

Temperature Dependence of Keq (van't Hoff Equation)

The van't Hoff equation relates the equilibrium constant to temperature:

• $\ln K = -\frac{\Delta H^\circ}{R} \left( \frac{1}{T} \right) + \frac{\Delta S^\circ}{R}$

Standard Thermodynamic Quantities for Selected Elements

Summary Table: Key Equations

• First Law: $\Delta E = q + w$

• Enthalpy: $\Delta H = \Delta E + P\Delta V$

• Entropy (Boltzmann): $S = k \ln W$

• Second Law: $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}$

• Gibbs Free Energy: $\Delta G = \Delta H - T\Delta S$

• Relationship to Equilibrium: $\Delta G = \Delta G^\circ + RT \ln Q$

• At Equilibrium: $\Delta G^\circ = -RT \ln K_{eq}$