01
The Core Laws of Thermodynamics
Thermodynamics is the branch of physical chemistry that studies energy transformations — how heat, work, and matter flow between systems. The four fundamental laws form the bedrock of all thermodynamic reasoning, from simple calorimetry to predicting reaction spontaneity.
Zeroth Law of Thermodynamics
⚡ Zeroth Law — Thermal Equilibrium
Statement
If system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then A and C are in thermal equilibrium. This defines the concept of temperature as a fundamental property.
The zeroth law lets us use thermometers. When a mercury thermometer touches a hot body and the reading stabilises, it means the thermometer (A) has reached thermal equilibrium with the object (C) via the scale (B). Temperature is the property that is equal between systems at thermal equilibrium.
First Law — Conservation of Energy
⚡ First Law — Energy is conserved
Statement
The total energy of an isolated system is constant. Energy can be converted between forms but never created or destroyed. ΔU = q + w
First Law
ΔU = q + w
ΔU = Change in internal energy | q = heat absorbed by system | w = work done on system
Sign Convention (IUPAC)
q > 0 : system absorbs heat (endothermic) |
q < 0 : system releases heat (exothermic)w > 0 : work done ON system | w < 0 : work done BY system
Second Law — Entropy and Directionality
⚡ Second Law — Entropy of universe always increases
Statement
The total entropy of an isolated system always increases in any spontaneous process. The entropy of the universe (Suniv) can never decrease.
Second Law — Entropy Universe
ΔSuniverse = ΔSsystem + ΔSsurroundings ≥ 0
The second law gives thermodynamics its arrow of time. It explains why ice melts in warm water but hot water doesn't freeze spontaneously, why gases expand into vacuums, and why every natural process disperses energy.
Third Law — Absolute Zero Entropy
⚡ Third Law — Perfect crystal at 0 K has zero entropy
Statement (Nernst–Simon)
The entropy of a perfectly ordered crystal at absolute zero (0 K, −273.15 °C) is exactly zero. This sets the absolute reference point for entropy.
Third Law
S0 K, perfect crystal = 0
📊 Visual Summary — Four Laws Overview
02
Systems, Boundaries & States
Before applying any thermodynamic law, we must precisely define what we are studying. A system is the portion of the universe under study; everything else is the surroundings. The system and surroundings are separated by a boundary.
Types of Thermodynamic Systems
Isolated System
No exchange of matter or energy with surroundings. Energy and mass are both constant. Example: thermos flask (ideal).
Closed System
Exchanges energy (heat & work) but NOT matter with surroundings. Example: sealed piston-cylinder arrangement.
Open System
Exchanges both energy AND matter with surroundings. Example: boiling water in an open pot, living organisms.
State Functions vs Path Functions
State Functions
Properties that depend ONLY on the current state of the system, not on how that state was reached. Examples: Internal Energy (U), Enthalpy (H), Entropy (S), Gibbs Free Energy (G), Temperature (T), Pressure (P), Volume (V).
Path Functions
Properties that depend on the specific path taken. Examples: Heat (q) and Work (w). You cannot say a system "contains" a certain amount of heat or work.
This distinction is crucial. Because ΔH, ΔU, and ΔS are state functions, we can apply Hess's Law — adding thermochemical equations to calculate changes indirectly without performing the experiment.
Thermodynamic State Variables
| Variable | Symbol | Units | Type |
|---|---|---|---|
| Temperature | T | K (Kelvin) | Intensive / State |
| Pressure | P | Pa, atm, bar | Intensive / State |
| Volume | V | m³, L | Extensive / State |
| Internal Energy | U | J, kJ | Extensive / State |
| Enthalpy | H | J, kJ | Extensive / State |
| Entropy | S | J/K, J mol⁻¹K⁻¹ | Extensive / State |
| Gibbs Free Energy | G | J, kJ | Extensive / State |
| Heat | q | J, kJ | Path Function |
| Work | w | J, kJ | Path Function |
03
Thermodynamic Processes
A thermodynamic process describes how a system transitions from one state to another. Different processes hold specific variables constant, leading to distinct mathematical relationships between heat, work, and internal energy.
The Four Key Processes
Isothermal
Constant Temperature
ΔT = 0
ΔT = 0
ΔU = 0
q = −w
q = −w
Isobaric
Constant Pressure
ΔP = 0
ΔP = 0
q = ΔH
w = −PΔV
w = −PΔV
Isochoric
Constant Volume
ΔV = 0
ΔV = 0
w = 0
q = ΔU
q = ΔU
Adiabatic
No Heat Exchange
q = 0
q = 0
ΔU = w
dS = 0 (rev)
dS = 0 (rev)
Reversible vs Irreversible Processes
Reversible Process
A process carried out infinitely slowly through a series of equilibrium states. The system can be returned to its initial state with no net change in the universe. It represents the maximum work extractable. ΔSuniverse = 0.
Irreversible Process
A spontaneous, real-world process that cannot be reversed without leaving a permanent change in the surroundings. All natural processes are irreversible. ΔSuniverse > 0.
Work Done in Expansion
Pressure-Volume Work
w = −Pext · ΔV (irreversible, constant P)
Reversible Work (isothermal, ideal gas)
w = −nRT · ln(Vf / Vi)
Key insight
For the same initial and final states, reversible work is always GREATER in magnitude than irreversible work for an expansion. The difference is energy "wasted" to entropy production.
Carnot Cycle
The Carnot cycle is an idealised reversible heat engine operating between two reservoirs (Thot and Tcold). It defines the maximum theoretical efficiency of any heat engine and is foundational to the second law.
Carnot Efficiency
η = 1 − Tcold / Thot (temperatures in Kelvin)
04
Energy, Work & Potentials
Thermodynamic potentials are state functions that serve as "bookkeeping" tools for energy under different constraints. Each is appropriate for a specific set of natural variables and tells us the maximum useful work available.
Forms of Energy in Chemistry
Kinetic Energy
Energy of motion. At the molecular level, this is translational, rotational, and vibrational motion of atoms and molecules.
Potential Energy
Stored energy. In chemistry: bond energies, intermolecular forces, coulombic interactions between charged species.
Thermal Energy
Total kinetic energy of all particles. Temperature is a measure of average thermal energy per molecule (proportional to kT).
Chemical Energy
Energy stored in chemical bonds. Released or absorbed during reactions. Tracked via enthalpy (ΔH) at constant pressure.
Thermodynamic Potentials Summary
| Potential | Symbol | Definition | Natural Variables | Minimum at Equilibrium |
|---|---|---|---|---|
| Internal Energy | U | Total energy of system | S, V | Yes (isolated) |
| Enthalpy | H | U + PV | S, P | Yes (const. S,P) |
| Helmholtz Free Energy | A (or F) | U − TS | T, V | Yes (const. T,V) |
| Gibbs Free Energy | G | H − TS = U + PV − TS | T, P | Yes (const. T,P) |
Why Gibbs dominates chemistry
Most chemical reactions occur at constant temperature and pressure (open beakers, lab conditions). Therefore, Gibbs free energy G is the most practically useful thermodynamic potential in chemistry.
Heat Capacity
Heat Capacity at Constant Pressure (Cp)
qp = m · Cp · ΔT Cp = (∂H/∂T)P
Heat Capacity at Constant Volume (Cv)
qv = m · Cv · ΔT Cv = (∂U/∂T)V
Relationship (Ideal Gas)
Cp − Cv = nR γ = Cp/Cv
05
Enthalpy Changes & Hess's Law
Enthalpy (H) is the "heat content" of a system at constant pressure. The enthalpy change ΔH of a reaction equals the heat exchanged with the surroundings at constant pressure. It is one of the most measured quantities in chemistry.
Enthalpy and the First Law
Definition of Enthalpy
H = U + PV
Enthalpy Change at Constant Pressure
ΔH = ΔU + PΔV = ΔU + ΔngasRT
Δngas = moles of gaseous products − moles of gaseous reactants
Endothermic vs Exothermic
Exothermic (ΔH < 0): Heat flows from system to surroundings. Products lower energy than reactants. E.g., combustion.Endothermic (ΔH > 0): Heat flows from surroundings into system. Products higher energy than reactants. E.g., dissolution of ammonium nitrate.
Types of Enthalpy Changes
The enthalpy change when 1 mole of a compound is formed from its constituent elements in their standard states at 298 K and 1 bar. ΔH°f of any pure element in its standard state = 0 kJ/mol. Used with Hess's Law to calculate ΔH°rxn.
Enthalpy change when 1 mole of a substance undergoes complete combustion in excess oxygen under standard conditions. Always negative (exothermic). Measured in bomb calorimeters. C₆H₁₂O₆(s) + 6O₂(g) → 6CO₂(g) + 6H₂O(l) ΔH = −2803 kJ/mol.
Heat required to convert 1 mole of liquid into vapour at constant temperature and pressure. Always positive (endothermic). ΔH°vap of water = +44 kJ/mol at 25°C. Related to intermolecular force strength.
Heat required to melt 1 mole of a solid at its melting point. Always endothermic. ΔH°fus of ice = +6.01 kJ/mol. Always less than ΔH°vap since melting disrupts fewer intermolecular interactions.
Enthalpy change when 1 mole of solute dissolves in a large excess of solvent. Can be positive (endothermic — lattice energy dominates, e.g., NH₄NO₃) or negative (exothermic — hydration enthalpy dominates, e.g., NaOH in water).
Average energy required to break 1 mole of a specific bond in gaseous molecules. All BDEs are positive (endothermic). ΔH°rxn ≈ Σ(bonds broken) − Σ(bonds formed). This gives an approximate ΔH from structural information alone.
Hess's Law of Constant Heat Summation
⚡ Hess's Law — Enthalpy is a state function
Statement
The total enthalpy change for a reaction is the same regardless of whether the reaction is carried out in one step or several steps. Since H is a state function, ΔH depends only on initial and final states.
Hess's Law Application
ΔH°rxn = Σ ΔH°f(products) − Σ ΔH°f(reactants)
🔷 Hess's Law Energy Diagram — Formation of CO₂ via Two Paths
Rules for Manipulating Thermochemical Equations
1. Multiplying an equation by a factor multiplies ΔH by the same factor.2. Reversing an equation changes the sign of ΔH.
3. Adding equations adds their ΔH values.
4. Substances that appear on both sides cancel out.
Born-Haber Cycle
An application of Hess's Law to ionic compounds. The Born-Haber cycle constructs the lattice enthalpy (energy of ionic crystal formation) indirectly from measurable enthalpies: sublimation, ionisation energy, electron affinity, and bond dissociation.
Born-Haber for NaCl
ΔH°f = ΔHsub(Na) + IE1(Na) + ½D0(Cl₂) + EA(Cl) + U(NaCl)
06
Internal Energy
Internal energy (U) is the total energy stored within a thermodynamic system — the sum of all kinetic energies (translational, rotational, vibrational) and potential energies (intermolecular interactions, bond energies) of all particles in the system.
Nature of Internal Energy
Translational KE
Motion of molecules through space. Major contributor in gases. For monatomic ideal gas: U = 3/2 nRT.
Rotational KE
Rotation of polyatomic molecules about axes. Contributes 1/2 RT per degree of rotational freedom.
Vibrational KE
Stretching and bending of bonds. Significant at higher temperatures. Contributes RT per vibrational mode (kinetic + potential).
Electronic Energy
Electronic state of atoms and molecules. Normally fixed (ground state) unless photoexcitation or ionisation occurs.
Internal Energy and Temperature
Equipartition Theorem (Ideal Gas)
U = f/2 · nRT where f = degrees of freedom
Monatomic gas: f=3 → U=3/2 nRT | Diatomic (rigid): f=5 → U=5/2 nRT | Diatomic (non-rigid): f=7 → U=7/2 nRT
📊 Internal Energy vs Temperature — Monatomic vs Diatomic Ideal Gas
ΔU vs ΔH at Constant Pressure
Connecting ΔH and ΔU
ΔH = ΔU + Δng·RT
Δng = Σn(gaseous products) − Σn(gaseous reactants) at temperature T in Kelvin
Practical Note
Bomb calorimeters measure ΔU (constant volume). To convert to ΔH for standard reporting, use ΔH = ΔU + ΔngRT. The difference is small for reactions with Δng = 0 (no net change in gas moles) and significant when gases are produced or consumed.
07
Gibbs Free Energy
Gibbs free energy (G) is the single most important quantity in chemical thermodynamics. It combines enthalpy and entropy to determine whether a process is spontaneous at constant temperature and pressure. Named after Josiah Willard Gibbs (1839–1903).
Definition and Spontaneity Criterion
Gibbs Free Energy
G = H − TS
Gibbs Free Energy Change
ΔG = ΔH − TΔS (at constant T, P)
Spontaneity Criterion
ΔG < 0: Spontaneous (forward reaction) |
ΔG = 0: At equilibrium |
ΔG > 0: Non-spontaneous (reverse reaction spontaneous)
Temperature Dependence of ΔG
| ΔH | ΔS | ΔG = ΔH − TΔS | Spontaneity |
|---|---|---|---|
| − (exothermic) | + (increase) | Always − | Always spontaneous ✅ |
| + (endothermic) | − (decrease) | Always + | Never spontaneous ❌ |
| − (exothermic) | − (decrease) | − at low T | Spontaneous only at low T |
| + (endothermic) | + (increase) | − at high T | Spontaneous only at high T |
📈 ΔG vs Temperature — All Four Cases
Gibbs Free Energy and Reaction Quotient
ΔG at Non-Standard Conditions
ΔG = ΔG° + RT · ln(Q)
At Equilibrium (ΔG = 0)
ΔG° = −RT · ln(Keq)
Keq = Equilibrium constant | R = 8.314 J mol⁻¹ K⁻¹ | T in Kelvin
Physical Interpretation
ΔG° is the standard Gibbs free energy change. Its sign and magnitude directly determine Keq: a large negative ΔG° gives K ≫ 1 (products favoured). A large positive ΔG° gives K ≪ 1 (reactants favoured).
Maximum Non-Expansion Work
Useful Work
wmax = ΔG (at constant T, P, reversible process)
The Gibbs free energy equals the maximum non-PV work obtainable from a process (e.g., electrical work in a galvanic cell). This connects thermodynamics directly to electrochemistry: ΔG° = −nFE°cell.
Electrochemistry Link
ΔG° = −n·F·E°cell
n = number of electrons transferred | F = Faraday constant (96485 C/mol) | E° = standard cell potential (V)
🎯 Quick Check: For a reaction with ΔH = −120 kJ/mol and ΔS = −200 J/mol·K, the reaction is spontaneous when...
ΔG = ΔH − TΔS = −120,000 − T(−200) = −120,000 + 200T. Setting ΔG < 0: 200T < 120,000 → T < 600 K. Correct!
08
Entropy Changes
Entropy (S) is a measure of the dispersal of energy or the number of microstates accessible to a system. Ludwig Boltzmann gave entropy its microscopic interpretation, while Clausius defined it macroscopically through heat flow.
Boltzmann's Statistical Definition
Boltzmann Entropy (Microscopic)
S = kB · ln(Ω)
kB = 1.38 × 10⁻²³ J/K (Boltzmann constant) | Ω = number of microstates (ways energy can be distributed)
More microstates = higher entropy. When you mix two gases, the number of ways molecules can be distributed increases enormously — hence ΔSmix > 0 always. This is why mixing is spontaneous.
Clausius Definition
Entropy Change (Reversible Process)
dS = δqrev / T → ΔS = ∫ δqrev / T
Clausius Inequality (any process)
dS ≥ δq / T
Equality holds for reversible; strict inequality for irreversible (spontaneous) processes
Entropy Change Calculations
Phase Transition at Constant T
ΔStransition = ΔHtransition / Ttransition
ΔSvap(water at 100°C) = 40,700 J/mol ÷ 373 K = +109 J mol⁻¹ K⁻¹. Always positive for vaporisation; negative for condensation.
Trouton's Rule: ΔSvap ≈ 88 J mol⁻¹ K⁻¹ for most non-associating liquids (a useful approximation).
Heating at Constant Pressure
ΔS = n·Cp · ln(Tf / Ti)
Heating at Constant Volume
ΔS = n·Cv · ln(Tf / Ti)
Isothermal Ideal Gas Expansion
ΔS = nR · ln(Vf / Vi) = nR · ln(Pi / Pf)
For free expansion into vacuum (irreversible), heat exchange = 0 but ΔS > 0 — illustrating that entropy can increase without heat flow.
Entropy of Mixing (Ideal Solutions)
ΔSmix = −nR Σ xi ln xi (always ≥ 0)
xi = mole fraction of component i. Since all xi < 1, ln xi < 0, so ΔSmix > 0 always. Mixing is always entropy-favoured.
Qualitative Entropy Trends
Factors that Increase Entropy
• Solid → Liquid → Gas (phase transitions) | More complex molecules (more vibrational modes)• Larger volume | Higher temperature | Mixing of substances
• Increase in number of moles of gas in a reaction (Δng > 0)
• Dissolution of ionic solids in water (usually, if hydration is significant)
📊 Entropy vs Temperature — Phase Transitions of Water
09
Thermochemical Equations
A thermochemical equation is a balanced chemical equation that includes the enthalpy change (ΔH) for the reaction. The physical states of all species must be specified because ΔH depends on whether substances are solid, liquid, or gaseous.
Rules for Thermochemical Equations
Essential Rules
1. Physical states (s, l, g, aq) must be specified for every substance.2. The ΔH value corresponds to the stoichiometry as written (1 mol coefficients).
3. Allotropic forms matter: C(graphite) ≠ C(diamond) — different ΔH°f.
4. Fractional coefficients are allowed in thermochemical equations.
Example Thermochemical Equations
| Reaction | ΔH (kJ/mol) | Type |
|---|---|---|
| C(graphite) + O₂(g) → CO₂(g) | −393.5 | Standard Combustion |
| H₂(g) + ½O₂(g) → H₂O(l) | −285.8 | Standard Formation |
| H₂O(l) → H₂O(g) | +44.0 | Vaporisation |
| N₂(g) + 3H₂(g) → 2NH₃(g) | −92.4 | Haber Process |
| CaCO₃(s) → CaO(s) + CO₂(g) | +178.3 | Thermal Decomposition |
| CH₄(g) + 2O₂(g) → CO₂(g) + 2H₂O(l) | −890.3 | Combustion of Methane |
Worked Hess's Law Problem
Problem: Find ΔH for C(s) + ½O₂(g) → CO(g)
Given:(i) C(s) + O₂(g) → CO₂(g) ΔH₁ = −393.5 kJ/mol
(ii) CO(g) + ½O₂(g) → CO₂(g) ΔH₂ = −283.0 kJ/mol
Strategy: Write equation (i). Reverse equation (ii) to get CO on products side.
(i): C + O₂ → CO₂ ΔH₁ = −393.5 kJ
(ii reversed): CO₂ → CO + ½O₂ ΔH = +283.0 kJ
Add: C + O₂ + CO₂ → CO₂ + CO + ½O₂ (cancel CO₂)
C + ½O₂ → CO ΔH = −393.5 + 283.0 = −110.5 kJ/mol ✓
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Standard Enthalpy & Entropy Changes
Standard conditions provide a universal reference point for comparing thermodynamic data. The standard state is defined as 298.15 K (25°C) and 1 bar (≈ 1 atm) pressure, with all solutes at 1 mol/L. Standard values are denoted with the ° superscript.
Standard Enthalpy Change
ΔH°rxn from Formation Enthalpies
ΔH°rxn = Σ [n · ΔH°f(products)] − Σ [n · ΔH°f(reactants)]
| Substance | State | ΔH°f (kJ/mol) |
|---|---|---|
| H₂O | l | −285.8 |
| H₂O | g | −241.8 |
| CO₂ | g | −393.5 |
| NH₃ | g | −46.2 |
| NaCl | s | −411.2 |
| CH₄ | g | −74.8 |
| C₆H₆ | l | +49.1 |
Standard Entropy Change
ΔS°rxn from Standard Entropies
ΔS°rxn = Σ [n · S°(products)] − Σ [n · S°(reactants)]
Key Distinction: Entropy ≠ Enthalpy of Formation
Unlike ΔH°f, the standard entropy S° of elements in their standard states is NOT zero (only S° at 0 K is zero by the third law). Every element and compound has a positive absolute entropy at 298 K.
| Substance | State | S° (J mol⁻¹ K⁻¹) |
|---|---|---|
| H₂ | g | 130.7 |
| O₂ | g | 205.1 |
| H₂O | l | 69.9 |
| H₂O | g | 188.7 |
| CO₂ | g | 213.8 |
| C (graphite) | s | 5.7 |
| C (diamond) | s | 2.4 |
Kirchhoff's Law — Temperature Dependence of ΔH
Kirchhoff's Law
ΔH°(T₂) = ΔH°(T₁) + ΔCp · (T₂ − T₁)
ΔCp = Σ Cp(products) − Σ Cp(reactants) (assumed constant over temperature range)
11
Spontaneity & Equilibrium
Spontaneity — the tendency of a process to occur without external intervention — is governed by the interplay between enthalpy (energy), entropy (dispersal), and temperature. Thermodynamics predicts whether a reaction will proceed, and where it will stop (equilibrium).
The Driving Forces of Spontaneity
Enthalpy Drive
Exothermic reactions (ΔH < 0) release energy to surroundings, increasing Ssurr. This favours spontaneity but is not the sole criterion.
Entropy Drive
Processes that increase disorder (ΔS > 0) increase the microstates. More dispersal = more probable = more spontaneous.
Temperature Mediation
Temperature determines which force wins when ΔH and ΔS oppose each other. High T amplifies the entropy term (TΔS).
Gibbs Free Energy and Equilibrium Constant
Van't Hoff Equation
ln(K) = −ΔG° / RT = −ΔH°/RT + ΔS°/R
Van't Hoff (temperature dependence of K)
ln(K₂/K₁) = −ΔH°/R · (1/T₂ − 1/T₁)
Plotting ln K vs 1/T gives a straight line: slope = −ΔH°/R, intercept = ΔS°/R
📊 Van't Hoff Plot — ln K vs 1/T (Endothermic vs Exothermic)
Le Chatelier's Principle — Thermodynamic Perspective
Thermodynamic Basis of Le Chatelier
When a system at equilibrium is disturbed, ΔG of the reaction shifts away from zero. The system responds by shifting in the direction that returns ΔG → 0 (equilibrium). Le Chatelier's principle is a qualitative statement of the thermodynamic minimisation of G.
| Disturbance | Effect on K | Equilibrium Shift |
|---|---|---|
| Add reactant | No change | Forward (→) until new equilibrium |
| Remove product | No change | Forward (→) |
| Increase T (exothermic rxn) | K decreases | Reverse (←) |
| Increase T (endothermic rxn) | K increases | Forward (→) |
| Increase pressure | No change | Toward fewer moles of gas |
| Add inert gas (const. V) | No change | No shift |
| Add catalyst | No change | No shift (reaches eq. faster) |
Reaction Quotient (Q) and Spontaneity
Comparing Q to K
Q < K → ΔG < 0 → Forward reaction spontaneous
Q = K → ΔG = 0 → At equilibrium
Q > K → ΔG > 0 → Reverse reaction spontaneous
Q = K → ΔG = 0 → At equilibrium
Q > K → ΔG > 0 → Reverse reaction spontaneous
📊 ΔG vs Reaction Progress — The Free Energy Minimum at Equilibrium
Coupled Reactions — Thermodynamic Driving
A non-spontaneous reaction (ΔG > 0) can be driven by coupling it to a spontaneous one (ΔG < 0) such that the total ΔG < 0. Nature uses this constantly: ATP hydrolysis (ΔG ≈ −30 kJ/mol) drives biosynthetic reactions that are thermodynamically unfavourable.
Coupled Reaction Principle
A(non-spontaneous): ΔG₁ > 0B(spontaneous): ΔG₂ < 0
Combined: ΔGtotal = ΔG₁ + ΔG₂ < 0 → Spontaneous overall ✅
🎯 Final Challenge: A reaction has ΔH° = +50 kJ/mol and ΔS° = +100 J/mol·K. The equilibrium constant K at 800 K is approximately...
ΔG° = ΔH° − TΔS° = 50,000 − 800(100) = 50,000 − 80,000 = −30,000 J/mol. ln K = −ΔG°/RT = 30,000/(8.314×800) ≈ 4.5, so K ≈ e^4.5 ≈ 90. Product-favoured but not enormously so. Best described as K ≈ moderately high. Closest correct interpretation is "near equal" at this temperature transition.