# Entropy and the Second Law

Chemistry I · Energy, Equilibrium and Electrochemistry · https://tryals.app/learn/chemistry-i/entropy-and-the-second-law

## The Arrow of Time

The first law permits any process that conserves energy, including a cup of coffee spontaneously reheating itself from the cool air around it. That never happens, so energy conservation is not the whole story.

**Entropy** supplies the missing direction. Thermodynamically it is defined by heat transferred along a reversible route:

$$\Delta S = \frac{q_{rev}}{T}$$

Statistically, Boltzmann defined it by counting:

$$S = k \ln W$$

where $W$ is the number of microscopic arrangements consistent with the observed macroscopic state and $k = 1.381 \times 10^{-23}$ J/K. These two definitions were developed independently and agree exactly, one of the deepest results in physical science.

The **second law** states that the entropy of the universe increases in any spontaneous process:

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

Note the scope. A *system* may lose entropy — water freezing, a crystal forming, a cell building a protein — provided the surroundings gain more. Life does not violate the second law; it exports entropy.

The **third law** gives the scale an absolute origin: a perfect crystal at 0 K has $S = 0$, because there is exactly one way to arrange it, and $\ln 1 = 0$. Unlike enthalpy, entropy therefore has genuine absolute values, and tables list $S^\circ$ rather than $\Delta S^\circ$.

Predicting the sign of $\Delta S$ is usually a matter of counting freedom. Entropy rises going solid to liquid to gas, on dissolving a solid, on heating, and when a reaction produces more gas molecules than it consumes.

> **Common pitfall:** thinking a spontaneous process must increase the entropy of the *system*. Water freezing at $-10$ °C is spontaneous and its entropy falls sharply, but it dumps enough heat into the surroundings that the universe still gains. Always ask which entropy you are being asked about.

## Practice questions

8 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. A spontaneous process must always increase the entropy of the system itself.

**Answer:** False

**Why:** False, only the entropy of the *universe* must rise. Freezing, crystallisation and biological assembly all lower the system’s entropy while releasing enough heat to raise the surroundings’ entropy by more.

Page: https://tryals.app/practice/chemistry-i/entropy-and-the-second-law/a-spontaneous-process-must-always-increase-the-entropy-of-the-system

### 2. A system has 4 accessible microstates. Using $S = k \ln W$ with $k = 1.381 \times 10^{-23}$ J/K, compute its entropy in units of $10^{-23}$ J/K, to two decimal places.

**Answer:** 1.91 (within ±0.04)

**Why:** $S = k\ln W = 1.381 \times 10^{-23} \times \ln 4 = 1.381 \times 1.386 = 1.91 \times 10^{-23}$ J/K. Entropy grows only logarithmically, so doubling the microstate count adds a fixed amount rather than doubling $S$.

Page: https://tryals.app/practice/chemistry-i/entropy-and-the-second-law/a-system-has-4-accessible-microstates-using-s-k-ln-w-with-k

### 3. Living cells continuously assemble ordered macromolecules, yet this does not violate the second law of thermodynamics. Why is this local decrease in entropy permissible?

A. The cell exports sufficient entropy to the environment
B. The second law applies to mechanical not living systems
C. Cellular reactions conserve total free energy throughout
D. Ordered structures generate zero net entropy in solution

**Answer:** A. The cell exports sufficient entropy to the environment

**Why:** Confusing the system with the universe leads to the mistaken belief that biological order defies physics; cells remain non-isolated systems whose metabolic heat dissipation ensures a net increase in total universal entropy.

Page: https://tryals.app/practice/chemistry-i/entropy-and-the-second-law/living-cells-continuously-assemble-ordered-macromolecules-yet-this

### 4. Why does a perfect crystal at absolute zero have exactly zero entropy?

A. The number of accessible microstates becomes zero at absolute zero temperature
B. There is only one way to arrange it, and the logarithm of one is zero
C. The scale is defined such that pure substances have zero entropy when freezing
D. Thermal energy vanishes, so heat can no longer be transferred reversibly

**Answer:** B. There is only one way to arrange it, and the logarithm of one is zero

**Why:** With every atom fixed in a unique arrangement, $W = 1$ and $S = k\ln 1 = 0$. This is the third law, and it is what gives entropy genuine absolute values where enthalpy has only differences.

Page: https://tryals.app/practice/chemistry-i/entropy-and-the-second-law/why-does-a-perfect-crystal-at-absolute-zero-have-exactly-zero-entropy

### 5. Sort each change by what it does to the entropy of the system.

**Answer:**

- Entropy increases: Ice melting to liquid water, A solid dissolving in water, A gas expanding into a vacuum
- Entropy decreases: Water vapour condensing to liquid, A protein folding into one compact shape, Two gas molecules combining into one

**Why:** Melting, dissolving and expanding all increase the number of accessible arrangements. Condensing, folding into one shape, and reducing the count of gas molecules all restrict them, so entropy falls.

Page: https://tryals.app/practice/chemistry-i/entropy-and-the-second-law/sort-each-change-by-what-it-does-to-the-entropy-of-the-system

### 6. Which changes generally increase the entropy of a chemical system?

A. Raising the temperature of a liquid
B. Crystallising a salt from solution
C. A reaction producing more moles of gas than it consumes
D. Converting a solid directly to a gas

**Answer:** A. Raising the temperature of a liquid; C. A reaction producing more moles of gas than it consumes; D. Converting a solid directly to a gas

**Why:** Sublimation, heating and net gas production all widen the range of accessible arrangements. Crystallisation does the opposite, pinning mobile ions into a regular lattice, so it lowers the system’s entropy.

Page: https://tryals.app/practice/chemistry-i/entropy-and-the-second-law/which-changes-generally-increase-the-entropy-of-a-chemical-system

### 7. Match each entropy expression to what it describes.

**Answer:**

- $\Delta S = q_{rev}/T$ → The thermodynamic definition, via reversible heat
- $S = k \ln W$ → The statistical definition, via microstate counting
- $\Delta S_{universe} > 0$ → The condition for a process to be spontaneous
- $S = 0$ at 0 K → The absolute origin of the entropy scale

**Why:** The first two are independent definitions that agree exactly; the third is the spontaneity criterion; the fourth is the third law, which makes absolute entropies tabulatable.

Page: https://tryals.app/practice/chemistry-i/entropy-and-the-second-law/match-each-entropy-expression-to-what-it-describes

### 8. A process releases 4500 J of heat reversibly to surroundings held at 250 K. Compute the entropy change of the SURROUNDINGS in J/K.

**Answer:** 18 (within ±0.2)

**Why:** $\Delta S_{surr} = q_{rev}/T = 4500/250 = +18$ J/K. The surroundings gain entropy even though the system lost heat, and it is this gain that can license a spontaneous process in which the system’s own entropy falls.

Page: https://tryals.app/practice/chemistry-i/entropy-and-the-second-law/a-process-releases-4500-j-of-heat-reversibly-to-surroundings-held-at