Chapter 19: Free Energy and Thermodynamics – Study Notes

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Free Energy and Thermodynamics

Introduction

This chapter explores the principles of thermodynamics as they apply to chemical systems, focusing on energy, spontaneity, entropy, and free energy. Understanding these concepts is essential for predicting whether chemical reactions will occur and how much energy is available to do work.

Review of Energy Concepts

Thermodynamics and Thermochemistry

Thermodynamics is the study of energy and its interconversions.
Thermochemistry examines the relationships between chemistry and energy, particularly heat exchange in chemical reactions.
Energy is the ability to do work or transfer heat.
Work is a force acting over a distance: $\text{work} = \text{force} \times \text{distance}$
Heat is the flow of energy caused by a temperature difference.

Classification of Energy

Kinetic energy: Energy of motion or energy being transferred.
Thermal energy: Energy associated with temperature; a form of kinetic energy.
Potential energy: Energy stored in an object or associated with its composition and position.
Chemical energy: A type of potential energy due to atomic structure, bonding, and molecular arrangement.

Electrostatic Potential Energy

Arises from interactions between charged particles.
Formula: $E_{el} = \kappa \frac{Q_1 Q_2}{d}$ where $\kappa$ is a constant ($8.99 \times 10^9$ J·m/C2), $Q_1$ and $Q_2$ are charges, $d$ is distance.
Important in atomic and molecular interactions.

First Law of Thermodynamics

Energy Conservation

Energy can be converted from one form to another but cannot be created or destroyed.
Total energy of the universe is constant: energy lost by a system is gained by the surroundings, and vice versa.

System and Surroundings

System: The part of the universe being studied (e.g., reactants in a reaction).
Surroundings: Everything else outside the system.
Closed system: Exchanges energy but not matter with surroundings.

Internal Energy (E)

Sum of kinetic and potential energies of all particles in a system.
Depends only on the initial and final states (state function).
Change in internal energy: $\Delta E = E_{\text{final}} - E_{\text{initial}}$ $\Delta E_{\text{rxn}} = E_{\text{products}} - E_{\text{reactants}}$

State Functions

A property that depends only on the current state, not the path taken to reach it (e.g., internal energy, enthalpy, entropy).
Example: Elevation change from base to peak of a mountain is a state function.

Energy Diagrams

Graphical representations showing energy changes during a process.
Initial state: reactants; Final state: products.

Energy Exchange: Heat and Work

Energy is exchanged as heat ($q$) and work ($w$): $\Delta E = q + w$
Neither $q$ nor $w$ are state functions; they depend on the process.

Quantity	System Gains	System Loses
q (heat)	thermal energy	thermal energy
w (work)	work done on system	work done by system
$\Delta E$	energy flows into system	energy flows out of system

Quantifying Heat Energy

Heat capacity depends on mass and specific heat ($C_s$) of the material.
Formula: $q = m \times C_s \times \Delta T$ where $q$ = heat (J), $m$ = mass (g), $C_s$ = specific heat (J/g·°C), $\Delta T$ = temperature change (°C).

Pressure–Volume Work

Work done by a system during volume change against external pressure: $w = -P_{\text{ext}} \Delta V$
To convert L·atm to J: $1$ L·atm $= 101.3$ J.

Enthalpy and Reaction Energetics

Enthalpy (H)

Enthalpy is the sum of internal energy and the product of pressure and volume: $H = E + PV$
Change in enthalpy: $\Delta H = \Delta E + P \Delta V$
At constant pressure, $\Delta H = q$ (heat at constant pressure).

Endothermic and Exothermic Reactions

Endothermic reaction: $\Delta H > 0$; heat absorbed by the system.
Exothermic reaction: $\Delta H < 0$; heat released by the system.

Standard Conditions and Enthalpy of Formation

Standard state: Pure substance at 1 atm, 25°C, 1 M concentration for solutions.
Standard enthalpy change ($\Delta H^\circ$): Enthalpy change when all reactants and products are in their standard states.
Standard enthalpy of formation ($\Delta H_f^\circ$): Enthalpy change for forming 1 mol of a compound from its elements in their standard states.
$\Delta H_f^\circ$ for a pure element in its standard state is zero.

Substance	Formula	$\Delta H_f^\circ$ (kJ/mol)
Water (g)	H2O(g)	-241.8
Hydrogen (g)	H2(g)	0
Oxygen (g)	O2(g)	0
Carbon dioxide (g)	CO2(g)	-393.5
Methane (g)	CH4(g)	-74.8

Calculating Standard Enthalpy Change for a Reaction

Any reaction can be written as the sum of formation reactions for reactants and products.
Formula: $\Delta H^\circ_{\text{rxn}} = \sum n \Delta H_f^\circ (\text{products}) - \sum n \Delta H_f^\circ (\text{reactants})$
$n$ is the stoichiometric coefficient from the balanced equation.

Thermodynamic Laws and Spontaneity

Thermodynamic Laws

First Law: Energy is conserved.
Second Law: For any spontaneous process, the entropy of the universe increases.
Third Law: The entropy of a perfect crystal at absolute zero (0 K) is zero.

Spontaneity

Spontaneous process: Occurs without ongoing outside intervention.
Nonspontaneous process: Requires energy input to occur.
Spontaneity is not related to the speed of the process (kinetics vs. thermodynamics).

Comparing Potential Energy

The direction of spontaneity can be determined by comparing the potential energy of the system before and after the reaction.
If the system has less potential energy after the reaction, the process is thermodynamically favorable.

Spontaneous Processes and Enthalpy

Most spontaneous processes are exothermic ($\Delta H < 0$), but some endothermic processes ($\Delta H > 0$) are also spontaneous.
Therefore, $\Delta H$ alone cannot predict spontaneity.

Spontaneous Endothermic Processes

Examples: Ice melting, water evaporating, salt dissolving.
These processes involve increased freedom of movement for particles, leading to increased entropy.

Entropy (S)

Definition and Calculation

Entropy (S): A thermodynamic function that increases as the number of energetically equivalent ways of arranging the components increases.
Boltzmann equation: $S = k \ln W$ $\Delta S = k \ln W_{\text{final}} - k \ln W_{\text{initial}}$
$k$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K); $W$ = number of microstates.
Units: J/K or J/(mol·K).
Entropy is a state function.

The Second Law of Thermodynamics

For any spontaneous process, the entropy of the universe increases: $\Delta S_{\text{univ}} > 0$
For a reversible process: $\Delta S_{\text{univ}} = 0$
$\Delta S_{\text{univ}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}}$

Macrostates and Microstates

Macrostate: Defined by a set of conditions (e.g., P, V, T); energy is constant if conditions remain the same.
Microstate: Exact internal energy distribution among particles at an instant; many microstates per macrostate.
More microstates correspond to higher entropy.

Expansion of a Gas into a Vacuum

Spontaneous process with no enthalpy change and no work performed.
Spontaneity is due to increased number of microstates (higher entropy).

Probability and Entropy

Macrostates with more microstates are more probable and have higher entropy.
Example: For two gas molecules, there are six ways (microstates) to distribute them evenly, but only one way to have both in one flask.

Macrostate	Number of Microstates	Relative Probability
Both molecules in left flask (A)	1	1/8
One in each flask (C)	6	6/8
Both in right flask (B)	1	1/8

Molecular Interpretation of Entropy

Molecules contribute to entropy through translational, vibrational, and rotational motion.
More types of motion and higher temperature/volume increase entropy.

Qualitative Predictions of Entropy

Entropy increases with temperature, volume, and number of independently moving particles (especially gases).
Formation of liquids or solutions from solids, and gases from solids or liquids, increases entropy.

The Third Law of Thermodynamics and Absolute Entropy

Absolute entropy ($S^\circ$) is the energy due to dispersal at the particle level.
At 0 K, a perfect crystal has $S = 0$ J/mol·K.
Absolute entropy values are always positive.

Standard Molar Entropy

Standard molar entropy ($S^\circ$): Entropy of 1 mol of substance in its standard state (J/mol·K).
Values depend on molecular complexity, mass, and physical state.

Substance	Physical State	$S^\circ$ (J/mol·K)
O2	gas	205
H2O	liquid	70
NaCl	solid	72

Factors Affecting Standard Entropy

Higher molar mass → higher entropy (for same physical state).
Greater molecular complexity → higher entropy.
Dissolved solids have higher entropy than pure solids.

Calculating Standard Entropy Change

Formula: $\Delta S^\circ_{\text{rxn}} = \sum n S^\circ (\text{products}) - \sum n S^\circ (\text{reactants})$
$n$ is the stoichiometric coefficient.

Entropy of System, Surroundings, and Universe

$\Delta S_{\text{univ}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}}$
Spontaneity depends on $\Delta S_{\text{univ}}$.

Quantifying Entropy Changes in Surroundings

Entropy change in surroundings is proportional to heat exchanged and inversely proportional to temperature:
$\Delta S_{\text{surr}} = -\frac{\Delta H_{\text{sys}}}{T}$ (at constant pressure and temperature)

Gibbs Free Energy (G)

Definition and Calculation

Gibbs free energy ($G$) is the maximum energy available to do work at constant temperature and pressure.
Formula: $\Delta G = \Delta H - T \Delta S$
If $\Delta G < 0$, the process is spontaneous; if $\Delta G > 0$, it is nonspontaneous.

Methods for Calculating $\Delta G^\circ$

Method 1: Use $\Delta H^\circ$ and $\Delta S^\circ$ values: $\Delta G^\circ = \Delta H^\circ - T \Delta S^\circ$
Method 2: Use standard free energies of formation: $\Delta G^\circ_{\text{rxn}} = \sum n \Delta G_f^\circ (\text{products}) - \sum n \Delta G_f^\circ (\text{reactants})$
Method 3: For stepwise reactions, sum the $\Delta G^\circ$ values for each step, reversing sign if the reaction is reversed and multiplying by coefficients as needed.

Free Energy under Nonstandard Conditions

For nonstandard conditions: $\Delta G = \Delta G^\circ + RT \ln Q$
$Q$ is the reaction quotient; $R$ is the gas constant (8.314 J/mol·K); $T$ is temperature in K.

Relationship between $\Delta G^\circ$ and Equilibrium Constant (K)

At equilibrium, $\Delta G = 0$ and $Q = K$:
$0 = \Delta G^\circ + RT \ln K$
$\Delta G^\circ = -RT \ln K$
If $K > 1$, $\Delta G^\circ < 0$ (spontaneous forward reaction); if $K < 1$, $\Delta G^\circ > 0$ (spontaneous reverse reaction).

Temperature Dependence of K

Combining equations: $\Delta G^\circ = \Delta H^\circ - T \Delta S^\circ$ $\Delta G^\circ = -RT \ln K$
Therefore: $\ln K = -\frac{\Delta H^\circ}{R} \left( \frac{1}{T} \right) + \frac{\Delta S^\circ}{R}$
Plotting $\ln K$ vs. $1/T$ yields a straight line; slope = $-\Delta H^\circ/R$, intercept = $\Delta S^\circ/R$.

Summary Table: Key Thermodynamic Quantities

Quantity	Symbol	Definition	Units
Internal Energy	E	Total kinetic and potential energy of system	J
Enthalpy	H	E + PV	J
Entropy	S	Measure of disorder	J/K
Gibbs Free Energy	G	H - TS	J

Example: Calculate $\Delta G^\circ$ for the reaction at 25°C using tabulated $\Delta H^\circ$ and $\Delta S^\circ$ values, then determine spontaneity and equilibrium constant.

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