Chapter 19: Thermodynamics and Equilibrium – Study Notes

Thermodynamics and Equilibrium

Introduction and SI Units

Thermodynamics is the study of energy changes, particularly heat and work, in chemical reactions and physical processes. Equilibrium refers to the state where the forward and reverse reactions occur at equal rates, resulting in no net change in the system.

• Gas Constant (R): $R = 8.31 \text{ J/mol·K}$

• SI Units:

  • Pressure (P): measured in pascals (Pa), $1.013 \times 10^5 \text{ Pa} = 1 \text{ atm}$

  • Volume (V): measured in cubic meters (m3), $1000 \text{ L} = 1 \text{ m}^3$

Internal Energy (U)

Internal energy is the total energy contained within a system, including both kinetic and potential energies of its particles.

• Definition: $U = PE + KE$

• State Function: Internal energy is a state function, meaning its value depends only on the current state of the system, not the path taken to reach that state.

• Change in Internal Energy: $\Delta U = U_f - U_i$

State Functions

State functions are properties that depend only on the present state of the system, not on how the system arrived there.

• Examples: $U$, $P$, $T$, $H$

Work (w) and Heat (q)

Work and heat are two ways energy can be transferred between a system and its surroundings.

• Work (w): Energy exchange resulting from a force moving an object through a distance. $w = F \cdot d$

• Heat (q): Energy that flows into or out of a system due to temperature difference.

• Sign Conventions:

  • $q > 0$: Endothermic, heat gained by the system

  • $q < 0$: Exothermic, heat lost by the system

  • $w > 0$: Work done on the system by surroundings

  • $w < 0$: Work done by the system on surroundings

First Law of Thermodynamics

The first law states that energy cannot be created or destroyed, only transferred or transformed.

• Equation: $\Delta U = q + w$

• At constant pressure: $\Delta U = q_p - P\Delta V$

• Work at constant pressure: $w = -P\Delta V$

• Expansion: $\Delta V > 0$, $w < 0$

• Contraction: $\Delta V < 0$, $w > 0$

Enthalpy (H)

Enthalpy is a state function that represents the heat content of a system at constant pressure.

• Definition: $H = U + PV$

• Change in Enthalpy: $\Delta H = H_f - H_i$

• At constant pressure: $\Delta H = q_p$

• Standard Enthalpy Change: $\Delta H^\circ = \sum n \Delta H^\circ_f (\text{products}) - \sum n \Delta H^\circ_f (\text{reactants})$

Spontaneous Processes and Entropy (S)

A spontaneous process occurs without outside intervention. Entropy is a measure of the randomness or disorder in a system.

• Entropy (S): State function quantifying disorder.

• Change in Entropy: $\Delta S = S_f - S_i$

• Example: Melting of ice: $H_2O(s) \rightarrow H_2O(l)$, $\Delta S = 22 \text{ J/K}$

• $\Delta S > 0$: System becomes more disordered

• $\Delta S < 0$: System becomes more ordered

• Spontaneous processes result in an overall increase in disorder (system + surroundings).

Second Law of Thermodynamics

The total entropy of a system and its surroundings always increases for a spontaneous process.

• No process is 100% efficient; some energy is always lost as heat.

• Equation: $\Delta S > \frac{q}{T}$ (spontaneous process)

• At equilibrium: $\Delta S = \frac{q}{T}$

• Entropy Change for Phase Transition: $\Delta S = \frac{\Delta H_{fus}}{T}$

Entropy and Disorder: Boltzmann Equation

The Boltzmann equation relates entropy to the number of possible arrangements (microstates) of particles.

• Equation: $S = k \ln W$

• $k$ = Boltzmann constant = $1.38 \times 10^{-23} \text{ J/K}$

• $W$ = Number of ways particles can be arranged

• Change in Entropy: $\Delta S = k \ln \left( \frac{W_f}{W_i} \right)$

• Example: Shuffling a deck of cards increases disorder

Third Law of Thermodynamics

A perfectly crystalline substance at 0 K has zero entropy.

• Equation: $S = k \ln W$; at $W = 1$, $S = 0$

Standard Entropy ($S^\circ$)

Standard entropy is the entropy value for 1 mole of a substance at standard conditions (1 atm, 1 M, 25°C).

• All elements have nonzero $S^\circ$ values, unlike $\Delta H^\circ_f$ values.

• Equation: $\Delta S^\circ_{rxn} = \sum S^\circ_{products} - \sum S^\circ_{reactants}$

• Entropy usually increases when:

  1. A molecule is broken into smaller molecules

  2. There is a net increase in moles of gas

  3. A solid changes to a liquid or gas, or a liquid changes to a gas

Free Energy (G) and Gibbs Equation

Free energy is a thermodynamic quantity that predicts the spontaneity of a process.

• Definition: $G = H - TS$

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

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

• Criteria for Spontaneity:

  • $\Delta G < 0$: Reaction is spontaneous

  • $\Delta G = 0$: Reaction is at equilibrium

  • $\Delta G > 0$: Reaction is nonspontaneous

Derivation of the Gibbs Equation

The Gibbs equation is derived from the second law of thermodynamics and relates entropy and enthalpy changes to spontaneity.

• At constant pressure: $q_p = \Delta H$

• For spontaneous processes: $T \Delta S > \Delta H$

• At equilibrium: $T \Delta S = \Delta H$

• For nonspontaneous processes: $T \Delta S < \Delta H$

Standard Free Energy ($\Delta G^\circ$)

Standard free energy change is the change in free energy under standard conditions (1 atm, 1 M, 25°C).

• Equation: $\Delta G^\circ = \Delta H^\circ - T \Delta S^\circ$

• Standard Free Energy of Formation ($\Delta G^\circ_f$): Free energy change when 1 mole of a substance is formed from its elements in their standard states.

• Equation: $\Delta G^\circ_f = \sum n \Delta G^\circ_f (\text{products}) - \sum n \Delta G^\circ_f (\text{reactants})$

Criterion for Spontaneity Using $\Delta G^\circ$

The sign and magnitude of $\Delta G^\circ$ indicate whether a reaction is spontaneous, nonspontaneous, or at equilibrium.

• Large negative $\Delta G^\circ$: Reaction goes to completion

• Large positive $\Delta G^\circ$: Reaction does not proceed as written

• $\Delta G^\circ = 0$: Reaction is at equilibrium

• Small negative or positive $\Delta G^\circ$: Reaction is close to equilibrium

Maximum Work ($w_{max}$)

The maximum work obtainable from a chemical or physical process is equal to the free energy change.

• Equation: $w_{max} = \Delta G$

• Also: $w_{max} = \Delta H - T \Delta S$

• Coupling of Reactions: Nonspontaneous reactions can be driven by coupling with spontaneous ones.

Effect of Temperature on Spontaneity

The spontaneity of a reaction depends on the signs of $\Delta H$ and $\Delta S$ and the temperature.

Math	Description	Spontaneous at...
$\Delta G = \Delta H - T \Delta S$	$\Delta H < 0$, $\Delta S > 0$	All T
$\Delta H > 0$, $\Delta S < 0$	Nonspontaneous at any T	None
$\Delta H < 0$, $\Delta S < 0$	Spontaneous at low T	Low T
$\Delta H > 0$, $\Delta S > 0$	Spontaneous at high T	High T

Relating $\Delta G^\circ$ to the Equilibrium Constant (K)

Free energy change is related to the equilibrium constant, allowing prediction of reaction direction and extent.

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

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

• Equation: $\Delta G^\circ = -RT \ln K$

• Where $Q$ is the reaction quotient, $K$ is the equilibrium constant, $R$ is the gas constant, and $T$ is temperature in Kelvin.

Worked Example: Temperature and Spontaneity

For the reaction $\text{CaCO}_3(s) \rightarrow \text{CaO}(s) + \text{CO}_2(g)$:

• $\Delta H^\circ = 178.3 \text{ kJ/mol}$

• $\Delta S^\circ = 159.0 \text{ J/mol·K}$

• At equilibrium: $\Delta G^\circ = 0$

• Find $T$ where reaction becomes spontaneous:

$T = \frac{\Delta H^\circ}{\Delta S^\circ} = \frac{178.3 \times 10^3 \text{ J/mol}}{159.0 \text{ J/mol·K}} = 1121 \text{ K}$

Summary of Key Equations

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

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

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

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

• $\Delta G^\circ = -RT \ln K$

• $w = -P \Delta V$ (at constant P)

• $\Delta S > \frac{q}{T}$ (spontaneous), $\Delta S = \frac{q}{T}$ (at equilibrium)

• $\Delta H = q$ (at constant P)

Table: Thermodynamic Data at 25°C

Substance	$\Delta H_f^\circ$ (kJ/mol)	$\Delta G_f^\circ$ (kJ/mol)	$S^\circ$ (J/mol·K)
$N_2(g)$	0	0	191.5
$H_2(g)$	0	0	130.6
$NH_3(g)$	-45.9	-16.5	192.8

Example Calculation

For the reaction: $N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$

• $\Delta H^\circ_{rxn} = 2(-45.9) - [0 + 3(0)] = -91.8 \text{ kJ/mol}$

• $\Delta G^\circ_{rxn} = 2(-16.5) - [0 + 3(0)] = -33.0 \text{ kJ/mol}$

• $\Delta S^\circ_{rxn} = 2(192.8) - [191.5 + 3(130.6)] = -197.3 \text{ J/mol·K}$

• At what temperature does the reaction become nonspontaneous?

• Set $\Delta G^\circ = 0$:

$T = \frac{\Delta H^\circ_{rxn}}{\Delta S^\circ_{rxn}} = \frac{-91.8 \text{ kJ/mol}}{-0.1973 \text{ kJ/mol·K}} = 465 \text{ K} (192°C)$

Above 192°C, the reaction is nonspontaneous due to the negative entropy factor.