🎲 What Is Entropy?

Part 1 of 7 — Disorder, Microstates, and S = k ln W

Why do ice cubes melt at room temperature but never spontaneously re-freeze? Why does gas expand to fill a container? The answer lies in entropy — a measure of the number of ways energy and matter can be distributed in a system.

Entropy and "Disorder"

Entropy ($S$) is often described as a measure of disorder or randomness. While this is a helpful starting point, the more precise definition involves microstates.

Everyday Examples of Increasing Entropy

Key Insight

Systems naturally tend toward states of higher entropy. This is not because nature "prefers disorder" — it is because there are vastly more disordered arrangements than ordered ones.

Microstates and the Boltzmann Equation

What Is a Microstate?

A microstate ($W$) is a specific arrangement of particles and energy in a system. The more microstates available, the higher the entropy.

Boltzmann's Equation

$S = k_B \ln W$

Example: Two Coins

• 2 coins have $2^2 = 4$ microstates: HH, HT, TH, TT
• The "disordered" state (one H, one T) has 2 microstates → most probable
• The "ordered" states (both H or both T) have 1 microstate each

Scaling Up

For $10^{23}$ particles (a mole), the number of microstates is astronomically large. The probability of all gas molecules spontaneously gathering in one corner is essentially zero — not because it violates any law, but because the number of "spread out" microstates vastly outnumbers "concentrated" ones.

Entropy Is Extensive

Entropy depends on the amount of substance — double the amount, double the entropy.

Units and Properties of Entropy

Units

Entropy is measured in J/K (joules per kelvin) or J/(mol·K) for molar entropy.

Note: Unlike enthalpy (kJ), entropy uses joules — a common source of unit errors on the AP exam!

Key Properties

Entropy Is NOT Conserved

Unlike energy, entropy can be created (in irreversible processes). The total entropy of the universe always increases for spontaneous processes.

Entropy Concept Quiz 🎯

Microstate Counting 🧮

1. How many microstates does a system of 3 coins have? ($W = 2^n$)

2. For 4 coins, what fraction of microstates have ALL heads? (express as a simplified fraction like 1/16)

3. If system A has $W = 100$ microstates and system B has $W = 200$ microstates, which has higher entropy? (type A or B)

Entropy Basics 🔽

Exit Quiz — What Is Entropy? ✅

📈 Predicting Entropy Changes

Part 2 of 7 — More Gas = More Entropy

Before we calculate $\Delta S$ numerically, we need to develop the ability to predict whether entropy increases or decreases in a process. This qualitative skill is tested heavily on the AP exam and relies on simple rules about phases, moles of gas, and molecular complexity.

Entropy and Phase

Entropy increases dramatically as matter moves from solid to liquid to gas:

$S_{\text{solid}} < S_{\text{liquid}} \ll S_{\text{gas}}$

Why?

The jump from liquid → gas is much larger than solid → liquid because gas molecules have vastly more accessible positions and velocities.

Phase Change Entropy

Rules for Predicting $\Delta S$ of Reactions

Rule 1: Count Moles of Gas

The most reliable predictor of $\Delta S$:

$\Delta n_{\text{gas}} = \text{mol gas (products)} - \text{mol gas (reactants)}$

• If $\Delta n_{\text{gas}} > 0$: $\Delta S > 0$ (entropy increases)
• If $\Delta n_{\text{gas}} < 0$: $\Delta S < 0$ (entropy decreases)
• If $\Delta n_{\text{gas}} = 0$: need other information

Rule 2: Dissolving Usually Increases Entropy

When a solid dissolves in a solvent, entropy typically increases (solid → ions or molecules in solution).

Exception: Some ions become so heavily hydrated that they actually decrease the entropy of water molecules around them.

Rule 3: More Molecules = More Entropy

A reaction that produces more total molecules than it consumes generally has $\Delta S > 0$.

Rule 4: Temperature Increases Entropy

Higher temperature means more kinetic energy and more accessible microstates.

Rule 5: Molecular Complexity

More complex molecules (more atoms, more bonds, more ways to vibrate) have higher entropy than simpler ones.

Practice Examples

Key Insight for AP

When asked to "predict the sign of $\Delta S$" — always start by counting moles of gas. If $\Delta n_{\text{gas}} = 0$, then consider total moles and phases.

Predicting Entropy Quiz 🎯

Predict ΔS Sign 🧮

Type "positive" or "negative" for the sign of $\Delta S$:

1. $\text{H}_2\text{O}(l) \rightarrow \text{H}_2\text{O}(g)$

2. $\text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g)$

3. $\text{2NH}_4\text{NO}_3(s) \rightarrow 2\text{N}_2(g) + \text{O}_2(g) + 4\text{H}_2\text{O}(g)$

Entropy Predictions 🔽

Exit Quiz — Predicting Entropy ✅

🌍 The Second Law of Thermodynamics

Part 3 of 7 — ΔS_universe > 0 for Spontaneous Processes

The Second Law is one of the most profound principles in all of science. It tells us which direction processes naturally go and provides the ultimate criterion for spontaneity.

The Second Law

The entropy of the universe increases for every spontaneous process.

$\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} > 0$

Three Cases

What Does "Spontaneous" Mean?

A spontaneous process occurs without continuous outside intervention. It does NOT mean:

• Fast (diamond → graphite is spontaneous but infinitely slow)
• Without any initial input (a match needs a spark, but then burning is spontaneous)

It DOES mean:

• The process is thermodynamically favorable
• The reverse process will not happen on its own

Entropy of the Surroundings

The entropy change of the surroundings depends on the heat flow and temperature:

$\Delta S_{\text{surroundings}} = -\frac{q_{\text{system}}}{T} = -\frac{\Delta H_{\text{system}}}{T}$

(at constant pressure and temperature)

Why the Negative Sign?

Heat released by the system ($-q$) is absorbed by the surroundings ($+q$), and vice versa.

Why Divide by Temperature?

The same amount of heat has a greater impact on entropy at lower temperature:

• Adding 100 J of heat to a cold system (200 K) creates a larger entropy change than adding 100 J to a hot system (1000 K)
• This is like adding $10 to someone with$100 vs. someone with $10,000

Combining System and Surroundings

$\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \left(-\frac{\Delta H_{\text{system}}}{T}\right)$

This equation connects entropy, enthalpy, and spontaneity — leading directly to Gibbs Free Energy (covered in Topic 3).

How Spontaneous Processes Work

Exothermic Reactions at Room Temperature

For combustion of methane: $\Delta H_{\text{sys}} < 0$

• $\Delta S_{\text{surr}} = -\Delta H/T > 0$ (large positive)
• Even if $\Delta S_{\text{sys}} < 0$, $\Delta S_{\text{univ}}$ can still be positive
• The large heat release drives spontaneity

Endothermic Spontaneous Processes

Ice melting above 0°C: $\Delta H_{\text{sys}} > 0$

• $\Delta S_{\text{surr}} = -\Delta H/T < 0$ (negative)
• But $\Delta S_{\text{sys}} > 0$ (solid → liquid, large increase)
• If $\Delta S_{\text{sys}}$ outweighs $|\Delta S_{\text{surr}}|$, the process is spontaneous

Temperature Dependence

At the melting point (0°C for water): $\Delta S_{\text{universe}} = 0 \quad \text{(equilibrium)}$

Above 0°C: melting is spontaneous. Below 0°C: freezing is spontaneous.

Second Law Concept Quiz 🎯

Second Law Calculations 🧮

1. A reaction has $\Delta H = -100$ kJ and $\Delta S_{\text{sys}} = -50$ J/K at $T = 400$ K. What is $\Delta S_{\text{surr}}$? (in J/K)

2. Using your answer from (1), what is $\Delta S_{\text{universe}}$? (in J/K)

3. Is the reaction spontaneous? (type "yes" or "no")

Second Law Concepts 🔽

Exit Quiz — Second Law ✅

❄️ The Third Law and Standard Molar Entropy

Part 4 of 7 — S = 0 at Absolute Zero

The Third Law of Thermodynamics provides a reference point for entropy. Unlike enthalpy, where we can only measure changes, entropy has an absolute scale — and it starts at zero.

The Third Law of Thermodynamics

The entropy of a perfect crystal at absolute zero (0 K) is exactly zero.

$S = 0 \quad \text{at } T = 0 \text{ K (perfect crystal)}$

Why Zero?

At absolute zero:

• All molecular motion ceases (except zero-point energy)
• A perfect crystal has only one microstate ($W = 1$)
• $S = k \ln 1 = 0$

Consequences

1. Absolute entropy values can be determined (unlike enthalpy)
2. All substances at $T > 0$ K have $S > 0$
3. Absolute zero can never actually be reached (it would require an infinite number of cooling steps)

Standard Molar Entropy ($S°$)

The entropy of one mole of a substance at standard conditions (1 atm, usually 25°C).

Key Values to Know

Patterns in Standard Entropy

1. Gases > liquids > solids — always!
2. More complex molecules have higher $S°$
3. Heavier atoms tend to have higher $S°$ (more accessible energy levels)
4. Allotropes differ: diamond (2.4) < graphite (5.7) — more ordered crystal

Important: $S° > 0$ for ALL substances at 298 K

Unlike $\Delta H°_f$, which is zero for elements, $S°$ is never zero at room temperature. Every substance has entropy at temperatures above 0 K.

How Entropy Varies with Temperature

As temperature increases from 0 K, entropy increases through several stages:

Heating a Substance

1. Solid phase: $S$ increases gradually as vibrations intensify
2. At melting point: sudden jump in $S$ (phase change — fusion)
3. Liquid phase: $S$ continues to increase
4. At boiling point: large jump in $S$ (phase change — vaporization)
5. Gas phase: $S$ continues to increase

Key Feature

The jump at the boiling point is much larger than at the melting point, because the liquid → gas transition involves a much greater increase in molecular freedom.

Phase Transition Entropy

$\Delta S_{\text{transition}} = \frac{\Delta H_{\text{transition}}}{T_{\text{transition}}}$

This formula applies at the equilibrium transition temperature, where the process is reversible.

Third Law Concept Quiz 🎯

Compare Standard Entropies 🧮

Which substance has the HIGHER standard molar entropy? Type the chemical formula.

1. $\text{H}_2\text{O}(l)$ or $\text{H}_2\text{O}(g)$?

2. $\text{C}(s, \text{diamond})$ or $\text{C}(s, \text{graphite})$?

3. $\text{O}_2(g)$ or $\text{O}_3(g)$?

Third Law and Standard Entropy 🔽

Exit Quiz — Third Law & Standard Entropy ✅

🔢 Calculating ΔS°_rxn from Standard Entropies

Part 5 of 7 — The Entropy Version of the Master Equation

Just as we calculated $\Delta H°_{\text{rxn}}$ from formation enthalpies, we can calculate $\Delta S°_{\text{rxn}}$ from standard molar entropies. The formula is very similar — products minus reactants.

The Entropy Master Equation

$\Delta S°_{\text{rxn}} = \sum n \cdot S°(\text{products}) - \sum m \cdot S°(\text{reactants})$

where $n$ and $m$ are stoichiometric coefficients.

Key Differences from the Enthalpy Version

Critical Warning ⚠️

$S°$ for elements is NOT zero! This is the #1 mistake students make. Absolute entropies are always positive at temperatures above 0 K.

Worked Example

Calculate $\Delta S°$ for: $\text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g)$

Solution:

$\Delta S° = \sum S°(\text{products}) - \sum S°(\text{reactants})$

$\Delta S° = [2(192.5)] - [1(191.6) + 3(130.7)]$

$\Delta S° = 385.0 - [191.6 + 392.1]$

$\Delta S° = 385.0 - 583.7 = -198.7 \text{ J/K}$

Check: Does the Sign Make Sense?

$\Delta n_{\text{gas}} = 2 - 4 = -2$ (fewer moles of gas in products)

$\Delta S < 0$ ✓ — consistent with our prediction!

ΔS° Calculation Concept Quiz 🎯

ΔS° Calculations 🧮

Use these $S°$ values [J/(mol·K)]: $\text{CO}_2(g) = 213.7$, $\text{H}_2\text{O}(g) = 188.8$, $\text{H}_2\text{O}(l) = 69.9$, $\text{CH}_4(g) = 186.3$, $\text{O}_2(g) = 205.1$, $\text{C}(s) = 5.7$, $\text{H}_2(g) = 130.7$

1. Calculate $\Delta S°$ for: $\text{C}(s) + \text{O}_2(g) \rightarrow \text{CO}_2(g)$ [J/K, to 3 significant figures]

2. Calculate $\Delta S°$ for: $\text{CH}_4(g) + 2\text{O}_2(g) \rightarrow \text{CO}_2(g) + 2\text{H}_2\text{O}(g)$ [J/K, to 3 significant figures]

ΔS° Concepts 🔽

Exit Quiz — Calculating ΔS° ✅

🛠️ Problem-Solving Workshop — Entropy

Part 6 of 7 — Practice and Strategies

This workshop focuses on the types of entropy problems you will encounter on the AP exam. We will practice predicting signs, calculating $\Delta S°$, and connecting entropy to spontaneity.

Problem-Solving Strategies

Strategy 1: Predict the Sign of ΔS

1. Count moles of gas: $\Delta n_{\text{gas}} = \text{mol gas (products)} - \text{mol gas (reactants)}$
2. If $\Delta n_{\text{gas}} > 0$: $\Delta S > 0$
3. If $\Delta n_{\text{gas}} < 0$: $\Delta S < 0$
4. If $\Delta n_{\text{gas}} = 0$: consider phase changes, complexity, and total moles

Strategy 2: Calculate ΔS°

$\Delta S° = \sum n \cdot S°(\text{products}) - \sum m \cdot S°(\text{reactants})$

Remember: $S°$ for elements is NOT zero!

Strategy 3: Unit Conversion

• $\Delta H$ is typically in kJ
• $\Delta S$ is typically in J/K
• For $\Delta G = \Delta H - T\Delta S$: convert $\Delta S$ to kJ/K by dividing by 1000

Strategy 4: Entropy of Surroundings

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

Mixed Entropy Problems 🎯

Entropy Calculation Workshop 🧮

1. A reaction has $\Delta H = -150$ kJ and occurs at $T = 300$ K. What is $\Delta S_{\text{surroundings}}$? (in J/K)

2. The melting of ice at 0°C (273 K) has $\Delta H_{\text{fus}} = 6.01$ kJ/mol. What is $\Delta S_{\text{fus}}$? (in J/(mol·K), to 3 significant figures)

3. A reaction has $\Delta S_{\text{sys}} = -100$ J/K and $\Delta S_{\text{surr}} = +350$ J/K. What is $\Delta S_{\text{universe}}$? (in J/K)

Entropy Problem Strategies 🔽

Challenge Problem 🏆

Exit Quiz — Entropy Workshop ✅

🎯 Synthesis & AP Review — Entropy

Part 7 of 7 — Bringing It All Together

This review integrates all entropy concepts: microstates, predicting signs, the Second and Third Laws, calculating $\Delta S°$, and connecting entropy to spontaneity.

Complete Concept Map

Entropy Fundamentals

Predicting and Calculating ΔS

Spontaneity

Critical Unit Reminder

$\Delta H$ → kJ, $\Delta S$ → J/K. Convert before combining!

Comprehensive AP Review Quiz 🎯

Integration Problems 🧮

1. At what temperature does a reaction with $\Delta H = -90$ kJ and $\Delta S = -300$ J/K become nonspontaneous? (in K)

2. Calculate $\Delta S_{\text{universe}}$ for an exothermic reaction with $\Delta H = -200$ kJ, $\Delta S_{\text{sys}} = -50$ J/K, at $T = 298$ K. (in J/K, round to nearest whole number)

3. Is the process in (2) spontaneous? (type "yes" or "no")

Final Concept Review 🔽

Final Exit Quiz — Entropy Mastery ✅