Second Law of Thermodynamics
The Second Law of Thermodynamics states that in any spontaneous process, the total entropy of an isolated system can only increase or remain constant, never decrease. This gives thermodynamics a preferred direction of time, explaining why heat flows from hot to cold, why mechanical energy converts irreversibly to heat, and why perpetual motion machines of the second kind are impossible. It is the thermodynamic basis for the arrow of time and sets fundamental efficiency limits on all heat engines.
Key Formula
dS ≥ δQ / T
LaTeX: dS \geq \frac{\delta Q}{T}
| Symbol | Meaning | Unit |
|---|---|---|
| dS | Differential change in entropy of the system | J/K |
| δQ | Differential heat added to the system | J |
| T | Absolute temperature of the system | K |
Worked Example
Problem
A heat engine operates between a hot reservoir at 600 K and a cold reservoir at 300 K. Calculate the maximum possible (Carnot) efficiency of this engine.
Solution
Step 1: Identify temperatures. T_H = 600 K, T_C = 300 K. Step 2: Use the Carnot efficiency formula derived from the Second Law: η_Carnot = 1 − (T_C / T_H). Step 3: Substitute: η_Carnot = 1 − (300 / 600) = 1 − 0.5 = 0.5.
Answer
Maximum efficiency = 50%. No real engine operating between these temperatures can exceed this.
Statements of the Second Law of Thermodynamics
| Formulation | Author | Statement Summary | Implication |
|---|---|---|---|
| Kelvin–Planck | Lord Kelvin & Planck | No engine converts heat entirely to work in a cycle | Limits heat engine efficiency |
| Clausius | Rudolf Clausius | Heat cannot spontaneously flow from cold to hot | Explains direction of heat flow |
| Entropy | Clausius | Total entropy of isolated system never decreases | Arrow of time |
| Statistical | Boltzmann | Systems evolve toward macrostates of higher probability | Disorder tends to increase |
| Caratheodory | Caratheodory | Adiabatically inaccessible states exist near any state | Mathematical formulation |
Interactive Tools
PhET Reversible Reactions Simulation
Visualise entropy changes and equilibrium in thermodynamic processes
Open ToolBrilliant – Thermodynamics Course
Interactive problem sets on entropy, efficiency, and the Second Law
Open ToolKhan Academy – Second Law of Thermodynamics
AP-level video lectures and exercises on entropy and heat engines
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Entropy
Entropy is a thermodynamic state function that quantifies the degree of disorder, randomness, or the number of microstates available to a system at a given macrostate. Macroscopically, it is defined via the Clausius inequality as the ratio of reversible heat exchange to absolute temperature; microscopically, Boltzmann's formula connects it to the number of microscopic configurations. Entropy always increases in irreversible processes in isolated systems, driving systems toward equilibrium and explaining the thermodynamic arrow of time.
Carnot Cycle
The Carnot Cycle is an idealized, reversible thermodynamic cycle consisting of two isothermal and two adiabatic processes, first described by Sadi Carnot in 1824 as the most efficient possible heat engine operating between two fixed temperatures. No real engine can exceed the efficiency of a Carnot engine operating between the same hot and cold reservoirs, making it the theoretical upper bound for heat engine performance. The Carnot efficiency depends only on the absolute temperatures of the reservoirs and sets the fundamental limit imposed by the Second Law of Thermodynamics.
Heat Engine
A heat engine is a device that converts thermal energy into mechanical work by exploiting the temperature difference between a high-temperature heat source (hot reservoir) and a low-temperature heat sink (cold reservoir). The engine absorbs heat Q_H from the hot reservoir, converts part of it to useful work W, and rejects the remainder Q_C to the cold reservoir, operating in a cyclic process. The thermal efficiency of a heat engine is always less than 100% due to the Second Law of Thermodynamics, and the maximum theoretical efficiency is set by the Carnot efficiency.
Formulated independently by Rudolf Clausius and Lord Kelvin around 1850–1851. Clausius introduced the concept of entropy in 1865 from the Greek "en" (in) + "trope" (transformation). The law grew out of Sadi Carnot's 1824 work on heat engine efficiency.