Thermochemistry and Spontaneity: Energy, Enthalpy, Entropy, and Free Energy

Thermochemistry and Spontaneity

Energy, Work, and Heat

Thermochemistry studies the energy changes that occur during chemical reactions and physical processes. Energy can be transferred as heat (q) or work (w), and the system's energy change is the sum of these transfers.

• Energy (E): The capacity to do work or produce heat. It exists as potential or kinetic energy.

• Work (w): Energy transfer resulting from a force acting over a distance. Formula: $w = -P\Delta V$ (for pressure-volume work)

• Heat (q): Energy transfer due to temperature difference between system and surroundings.

• System and Surroundings: The system is the part of the universe under study; everything else is the surroundings.

• First Law of Thermodynamics: Energy is conserved; it can be transferred or transformed but not created or destroyed. Formula: $\Delta E = q + w$

Example: If a system loses 10 kJ of energy, the surroundings gain 10 kJ.

Exothermic and Endothermic Processes

Processes are classified by the direction of heat flow:

• Exothermic: Heat is released by the system to the surroundings ($q < 0$).

• Endothermic: Heat is absorbed by the system from the surroundings ($q > 0$).

Example: Sublimation (solid to gas) is typically endothermic because energy is required to break intermolecular forces.

State Functions

State functions depend only on the current state of the system, not the path taken to reach it.

• Examples: Internal energy (E), enthalpy (H), pressure (P), volume (V), temperature (T), entropy (S).

• Enthalpy (H): $H = E + PV$

• Change in Enthalpy: $\Delta H = \Delta E + P\Delta V$ (at constant pressure)

Calorimetry and Heat Calculations

Calorimetry measures heat changes in physical and chemical processes.

• Specific Heat Capacity (c): Amount of heat required to raise the temperature of 1 g of a substance by 1°C. Formula: $q = mc\Delta T$

• Heat of Fusion (melting) and Vaporization (boiling): Energy required for phase changes at constant temperature. Formula: $q = n\Delta H_{fus}$ or $q = n\Delta H_{vap}$

Example: To heat 1.00 kg of water from 0°C to 100°C: $q = (1000\,g)(4.184\,J/g\cdot K)(100\,K) = 418,400\,J = 418.4\,kJ$

Thermochemical Equations and Hess's Law

Thermochemical equations show the enthalpy change for reactions. Hess's Law allows calculation of enthalpy changes by combining equations.

• Hess's Law: The total enthalpy change for a reaction is the sum of enthalpy changes for individual steps.

• Reverse a reaction: Change the sign of $\Delta H$.

• Multiply a reaction: Multiply $\Delta H$ by the same factor.

Example: If you need 3 moles of a reactant, multiply the reaction and its $\Delta H$ by 3.

Standard Enthalpy of Formation

The standard enthalpy of formation ($\Delta H_f^\circ$) is the enthalpy change when 1 mole of a compound forms from its elements in their standard states at 298 K.

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

Example: For the formation of water: $2H_2(g) + O_2(g) \rightarrow 2H_2O(l)$, $\Delta H_f^\circ$ for $H_2O(l)$ is -285.8 kJ/mol.

Bond Enthalpies

Bond dissociation enthalpy is the energy required to break one mole of a bond in the gas phase. Breaking bonds is endothermic; forming bonds is exothermic.

• Bond Enthalpy Table (kJ/mol):

Bond	Bond Enthalpy
H-H	436
C-H	413
C-C	358
C=O	463
O=O	498
N≡N	941
O-H	463
Additional info: Values inferred from standard tables.

• Bond Enthalpy Calculation: $\Delta H_{rxn} = \sum D_{bonds\,broken} - \sum D_{bonds\,formed}$

Entropy and Spontaneity

Entropy (S)

Entropy is a measure of disorder or randomness in a system. Spontaneous processes tend to increase the entropy of the universe.

• Change in Entropy: $\Delta S = S_{final} - S_{initial}$

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

• Units: J/(mol·K)

Example: Melting (fusion) is endothermic and increases entropy; freezing decreases entropy.

Second Law of Thermodynamics

The total entropy of the universe increases in a spontaneous process.

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

• If $\Delta S_{univ} > 0$, the process is spontaneous.

Entropy Changes in Phase Transitions

• Fusion (melting): $\Delta S$ is positive (more disorder).

• Solidification (freezing): $\Delta S$ is negative (less disorder).

• Vaporization: $\Delta S$ is positive (much more disorder).

Predicting Entropy Changes

• More particles or more complex molecules usually mean higher entropy.

• Gases have higher entropy than liquids, which have higher entropy than solids.

• Mixing increases entropy.

Gibbs Free Energy and Spontaneity

Gibbs Free Energy (G)

Gibbs free energy predicts the spontaneity of a process at constant temperature and pressure.

• Formula: $\Delta G = \Delta H - T\Delta S$

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

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

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

Standard Free Energy Change and Equilibrium

• Relationship to Equilibrium Constant (K): $\Delta G^\circ = -RT \ln K$

• Relationship to Reaction Quotient (Q): $\Delta G = \Delta G^\circ + RT \ln Q$

• At equilibrium, $\Delta G = 0$ and $Q = K$.

Example: For a reaction with $K = 5.3 \times 10^{13}$ at 298 K, $\Delta G^\circ = -RT \ln K$.

Temperature Dependence of Spontaneity

• Spontaneity can depend on temperature, especially when $\Delta H$ and $\Delta S$ have the same sign.

• Endothermic reactions ($\Delta H > 0$) can be spontaneous at high T if $\Delta S$ is positive.

• Exothermic reactions ($\Delta H < 0$) are more likely spontaneous at low T if $\Delta S$ is negative.

Solution Formation and Entropy

Enthalpy and Entropy in Solution Formation

Forming a solution involves changes in enthalpy and entropy:

• Enthalpy of Solution ($\Delta H_{soln}$): $\Delta H_{soln} = \Delta H_{solute} + \Delta H_{solvent} + \Delta H_{mix}$

• Solution formation can be endothermic or exothermic.

• Entropy usually increases when a solution forms due to increased disorder.

Example: Dissolving ammonium nitrate is endothermic and feels cold; dissolving magnesium sulfate is exothermic and feels warm.

Spontaneity of Solution Formation

• Even if $\Delta H_{soln}$ is positive (endothermic), solution formation can be spontaneous if $\Delta S$ is sufficiently positive.

• For ionic salts with highly charged ions, $\Delta S_{soln}$ can be negative, making dissolution nonspontaneous unless temperature increases.

Summary Table: Key Thermodynamic Quantities

Quantity	Symbol	Definition	Units
Internal Energy	E	Total energy (kinetic + potential) of a system	J
Enthalpy	H	Heat content at constant pressure	J
Entropy	S	Disorder/randomness of a system	J/(mol·K)
Gibbs Free Energy	G	Maximum useful work obtainable from a process at constant T, P	J
Additional info: Table entries inferred from standard definitions.