# Second Law of Thermodynamics and Entropy

Physics I · Principles of Waves, Fluids and Thermodynamics · https://tryals.app/learn/physics-i/second-law-of-thermodynamics-and-entropy

## The Arrow of Time

Break an egg, stir cream into coffee, let perfume fill a room. You never see the reverse, though mechanics allows it. The **second law** governs this, driven by sheer probability.

Disordered arrangements so vastly outnumber ordered ones that with $10^{23}$ particles, the unlikely becomes impossible.

### Core Statements

| Statement | Meaning |
| --- | --- |
| **Clausius** | Heat cannot spontaneously flow from cold to hot. |
| **Kelvin-Planck** | No engine can convert heat entirely into work. |

Entropy is time's compass. **Entropy** measures disorder, defined for reversible paths as $\Delta S = \int \frac{dQ_{\text{rev}}}{T}$. For isolated systems, $\Delta S \geq 0$.

## Engines and Microstates

The **Carnot engine** sets the absolute efficiency limit for engines operating between $T_H$ and $T_C$:

$$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$$

No real engine can exceed this value. Boltzmann linked entropy to statistics via **Boltzmann's entropy formula**:

$$S = k_B\ln\Omega$$

Here, $\Omega$ is the number of **microstates**. Systems naturally evolve toward the macrostate with the largest $\Omega$.

### Common Pitfall

Entropy can decrease locally, like inside your freezer, provided more entropy is exported to the surroundings. The **second law** binds only the total system plus surroundings, never a subsystem alone.

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Which of the following are irreversible processes?

A. Friction converting kinetic energy to heat
B. A perfectly quasi-static compression of an ideal gas
C. A gas expanding freely into a vacuum
D. Heat flowing from hot to cold spontaneously

**Answer:** A. Friction converting kinetic energy to heat; C. A gas expanding freely into a vacuum; D. Heat flowing from hot to cold spontaneously

**Why:** Free expansion, heat flow, and friction are irreversible (entropy increases). Quasi-static processes are reversible idealizations.

Page: https://tryals.app/practice/physics-i/second-law-of-thermodynamics-and-entropy/which-of-the-following-are-irreversible-processes

### 2. Order these milestones in the history of entropy and the second law.

**Answer:**

1. Carnot introduces concept of maximum engine efficiency
2. Clausius defines entropy: $dS = \delta Q_{rev}/T$
3. Boltzmann gives statistical definition: $S = k_B \ln \Omega$
4. Planck formulates the third law of thermodynamics
5. Shannon connects entropy to information theory

**Why:** Carnot (1824), Clausius (1865), Boltzmann (1877), Planck (1911), Shannon (1948).

Page: https://tryals.app/practice/physics-i/second-law-of-thermodynamics-and-entropy/order-these-milestones-in-the-history-of-entropy-and-the-second-law

### 3. A drop of ink disperses through a glass of water and never spontaneously regathers. Yet each molecular collision is perfectly **reversible**. What actually forbids the ink from re-forming?

A. Collisions destroy the information needed to run backwards
B. The ink molecules chemically bond to the water
C. Nothing forbids it; it is overwhelmingly improbable
D. Friction between ink and water molecules dissipates the motion

**Answer:** C. Nothing forbids it; it is overwhelmingly improbable

**Why:** Every microscopic motion could legally reverse; there are simply astronomically more ways to be mixed than gathered, so chance never finds the gathered ones again. Entropy increase is probability at work, with $10^{23}$ molecules, "overwhelmingly likely" is indistinguishable from "always".

Page: https://tryals.app/practice/physics-i/second-law-of-thermodynamics-and-entropy/a-drop-of-ink-disperses-through-a-glass-of-water-and-never

### 4. A heat engine runs between a 600 K flame and a 300 K environment. Estimate the **maximum possible efficiency**, in percent.

**Answer:** 50 (within ±5)

**Why:** $\eta_{max} = 1 - 300/600 = 50\%$, and no engineering cleverness can beat it, because it follows from the second law itself. Real engines between these temperatures manage perhaps 30%. Waste heat is not a design flaw; it is physics rent.

Page: https://tryals.app/practice/physics-i/second-law-of-thermodynamics-and-entropy/a-heat-engine-runs-between-a-600-k-flame-and-a-300-k-environment

### 5. The entropy of an isolated system can spontaneously decrease.

**Answer:** False

**Why:** The second law: $\Delta S_{universe} \geq 0$. Entropy of an isolated system never decreases spontaneously; it can only stay the same (reversible) or increase (irreversible).

Page: https://tryals.app/practice/physics-i/second-law-of-thermodynamics-and-entropy/the-entropy-of-an-isolated-system-can-spontaneously-decrease

### 6. A Carnot engine takes in $900$ J of heat per cycle from a reservoir at $T_h = 450$ K and rejects heat at $T_c = 150$ K. The work it delivers per cycle is $\_\_\_$ J.

**Answer:** 600 (within ±0.5)

**Why:** $\eta = 1 - T_c/T_h = 1 - 150/450 = 2/3$, so $W = \eta Q_h = \tfrac{2}{3} \times 900 = 600$ J. Efficiency fixes only the *fraction* of the intake that can become work; the remaining $300$ J must be dumped to the cold reservoir, and no refinement of the engine removes that debt.

Page: https://tryals.app/practice/physics-i/second-law-of-thermodynamics-and-entropy/a-carnot-engine-takes-in-900-j-of-heat-per-cycle-from-a-reservoir-at

### 7. The second law forbids processes the first law permits. What does it add?

A. A correction to the first law for irreversible processes
B. The requirement that energy be conserved in every subsystem separately
C. A direction, which energy conservation alone does not give
D. A limit on the total amount of energy in the universe

**Answer:** C. A direction, which energy conservation alone does not give

**Why:** The first law is silent about direction. A cup un-smashing conserves energy perfectly; the second law is what makes it never happen, and it is the only fundamental law that distinguishes past from future.

Page: https://tryals.app/practice/physics-i/second-law-of-thermodynamics-and-entropy/the-second-law-forbids-processes-the-first-law-permits-what-does-it