The Timpson Lyceum · University Physics I (2025)

Second Law of Thermodynamics - Lecture 20 (University Physics I)

Overview

This lecture introduces the second law of thermodynamics and the physics of converting heat into useful work. While turning work into heat is trivially easy, the reverse — extracting work from heat — is fundamentally limited, and the second law quantifies that limit. We build up the idea through heat engines (including the Otto cycle of a gasoline engine), refrigerators and heat pumps, and the idealized Carnot cycle, which sets the maximum possible efficiency. We then define entropy three complementary ways — as a state variable computed from $dQ/T$, as a measure of disorder, and as the count of microstates underlying a macrostate — and show why entropy in an isolated system always tends to increase. The chapter closes by connecting entropy to information theory.

Key Concepts

The second law, qualitatively

• Energy spontaneously flows from hotter to colder, never the reverse, on its own. Ordered motion (a sliding book) decays into random thermal motion (heat), but random heat never spontaneously reassembles into ordered motion.
• Entropy measures disorder/randomness in a system, and equivalently the tendency of energy toward less-useful forms. In any isolated system, total entropy stays constant (only for reversible processes) or — far more typically — increases.

Reversible processes (an idealization)

• A reversible process is one perpetually on the verge of equilibrium: an infinitesimal change $\pm dT$ reverses its direction. It keeps a well-defined temperature throughout and changes continuously (no jumps).
• Real processes are irreversible, but we model them with fictitious reversible paths because entropy is a state variable — its change depends only on start and end states, not the path taken (analogous to gravitational potential energy).

Heat engines

• A heat engine takes in heat $Q_H$ from a hot reservoir, exhausts $Q_C$ to a cold reservoir, and delivers work $W = Q_H - Q_C$.
• Thermal efficiency is the fraction of input heat turned into work: $e = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}$

The Otto cycle (gasoline engine)

• Four strokes: $A\to B$ adiabatic compression, $B\to C$ isochoric (constant-volume) ignition adding $Q_H$, $C\to D$ adiabatic power stroke, $D\to A$ isochoric exhaust releasing $Q_C$.
• Using the adiabatic relation $TV^{\gamma-1}=\text{const}$ across the two adiabatic strokes, the efficiency reduces to depend only on the compression ratio $r$: $e = 1 - \frac{1}{r^{\gamma - 1}}$ where $\gamma \approx 1.4$ for a diatomic gas. Higher $r$ means higher efficiency, but too high a ratio overheats the fuel-air mix and causes pre-ignition (knock) — which is why premium fuel (higher autoignition temperature) and diesel engines (inject fuel only after full compression, no spark plug) exist.

Refrigerators and heat pumps

• A refrigerator does work $W$ to pump heat $Q_C$ out of a cold space and dump $Q_H = Q_C + W$ into the hot surroundings. A heat pump runs the same machine to heat a house, and beats a resistive space heater because it moves existing outside energy inward in addition to converting work to heat.
• Performance is the coefficient of performance, not efficiency: $K = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C}$ $K$ can exceed 1 and has no upper bound of 1, unlike engine efficiency.

Two equivalent statements of the second law

• No engine can convert heat entirely into work with zero exhaust ($e = 1$ is impossible).
• No refrigerator can move heat from cold to hot with zero work input. These two statements are logically equivalent.

The Carnot cycle (maximum efficiency)

• The most efficient possible engine uses only reversible processes: two isothermal steps (heat exchanged with reservoirs at essentially equal temperature) and two adiabatic steps. Its efficiency depends only on the absolute (Kelvin) reservoir temperatures: $e_{\text{Carnot}} = 1 - \frac{T_C}{T_H}, \qquad \frac{Q_C}{Q_H} = \frac{T_C}{T_H}$
• No real engine can beat a Carnot engine operating between the same two temperatures; assuming otherwise lets you build a device that violates the second law. Efficiency reaches 1 only if $T_C = 0$ K — impossible — which is one way Kelvin's absolute-zero scale arises.

Entropy — the thermodynamic definition

• For a reversible process at temperature $T$, the infinitesimal entropy change is $dS = dQ/T$, so $\Delta S = \int_{T_1}^{T_2} \frac{dQ}{T}$
• The same heat added to a cold system disorders it far more than when added to a hot system, hence the $1/T$ weighting. Units are $\text{J/K}$.
• For an isothermal process, $\Delta S = Q/T$. For an adiabatic reversible process, $dQ = 0$, so $\Delta S = 0$. Over any complete cycle, $\Delta S_{\text{system}} = 0$ (a Carnot cycle changes neither the system's nor the universe's entropy).

Entropy — the microscopic (statistical) definition

• A macrostate (measured $P$, $V$, $T$) corresponds to many microstates (specific positions/velocities of every molecule). Let $W$ be the number of microstates; then $S = k_B \ln W, \qquad \Delta S = k_B \ln\!\frac{W_2}{W_1}$ where $k_B = R/N_A \approx 1.38\times10^{-23}\,\text{J/K}$.
• More microstates means higher entropy and higher probability. Systems evolve toward the most probable (highest-$W$) macrostate. The second law is therefore statistical: entropy decreasing is not impossible, merely astronomically unlikely (e.g. all air in a room crowding into half has probability $\sim (1/2)^{N}$ with $N\sim10^{26}$).

Entropy and information

• Shannon information defines the surprise of an outcome as $\log_2(1/p)$ and entropy as the expected surprise, $H = \sum_i p_i \log_2(1/p_i)$, measured in bits.
• A certain outcome ($p=1$) carries zero surprise/entropy; a fair coin carries 1 bit; $n$ equally likely outcomes carry $\log_2 n$ bits. High-entropy systems take more information (more bits) to describe, mirroring the thermodynamic notion of disorder.

Worked Example

Carnot engine. A Carnot engine takes in $Q_H = 2000\ \text{J}$ per cycle from a hot reservoir at $T_H = 500\ \text{K}$ and rejects heat to a cold reservoir at $T_C = 300\ \text{K}$. Find the exhaust heat $Q_C$, the work $W$, and the efficiency $e$.

Because a Carnot engine's heat ratio equals its temperature ratio, $Q_C = Q_H\,(T_C/T_H)$; the work is $W = Q_H - Q_C$; and the efficiency is $e = 1 - T_C/T_H$:

As a check, $e = W/Q_H = 800/2000 = 0.40$, consistent.

Entropy of mixing. Mix $1\ \text{kg}$ of water at $100^\circ\text{C}$ ($373\ \text{K}$) with $1\ \text{kg}$ at $0^\circ\text{C}$ ($273\ \text{K}$); both settle to $50^\circ\text{C}$ ($323\ \text{K}$). With $dQ = mc\,dT$ and $c = 4190\ \text{J/(kg}\cdot\text{K)}$, integrate $\Delta S = \int mc\,dT/T = mc\ln(T_f/T_i)$ for each:

Total entropy increases, exactly as the second law requires — the mixed state is more disordered than the separated one.

Summary

• Converting work to heat is effortless; the second law sets a hard ceiling on converting heat back to work — no engine reaches 100% efficiency and no refrigerator runs on zero work.
• Heat-engine efficiency is $e = 1 - Q_C/Q_H$; for the Otto cycle it becomes $e = 1 - r^{-(\gamma-1)}$, favoring high compression ratios but limited by fuel pre-ignition.
• The reversible Carnot cycle is the most efficient engine between two temperatures, with $e = 1 - T_C/T_H$ using absolute (Kelvin) temperatures; reaching $e=1$ would require $T_C = 0$ K.
• Refrigerators and heat pumps are characterized by a coefficient of performance $K = Q_C/W$, which can exceed 1.
• Entropy is a state variable with $\Delta S = \int dQ/T$ (reversible path); equivalently $S = k_B \ln W$ counting microstates. Entropy of an isolated system never decreases and almost always increases.
• The second law is ultimately statistical — disorder grows because high-entropy macrostates overwhelmingly outnumber low-entropy ones — and the same entropy concept reappears in information theory as bits needed to describe a system.