The second law of thermodynamics is a fundamental principle of thermodynamics governing the direction of physical processes and the limits of energy conversion. In its entropy formulation, the entropy of an isolated thermodynamic system cannot decrease: it remains constant in an ideal reversible process and increases in an irreversible process. Unlike the first law of thermodynamics, which expresses conservation of energy, the second law distinguishes transformations that conserve energy but cannot occur spontaneously. It explains why heat flows naturally from hotter bodies to colder ones and why a cyclic engine cannot convert heat from a single reservoir entirely into work. (ocw.mit.edu)
Classical formulations
Two equivalent classical formulations express the second law without first defining entropy:
- Clausius statement: No process is possible whose sole net result is the transfer of heat from a colder body to a hotter body.
- Kelvin–Planck statement: No device operating in a cycle can have as its sole net result the absorption of heat from a single thermal reservoir and its complete conversion into work.
The qualifications “sole net result” and “operating in a cycle” are essential. Refrigerators transfer heat from colder to hotter regions, but require work input. A gas can convert absorbed heat completely into work during an individual expansion, but restoring the gas to its original state requires additional changes. The two statements are equivalent because a hypothetical device violating either could be combined with an ordinary engine or refrigerator to violate the other. (ocw.mit.edu)
Entropy and its mathematical expression
Entropy, conventionally denoted (S), is a state function: its change depends on the initial and final equilibrium states, not on the path connecting them. For a reversible process,
[ dS=\frac{\delta Q_{\mathrm{rev}}}{T}, ]
where (\delta Q_{\mathrm{rev}}) is heat transferred into the system and (T) is its absolute temperature. Entropy has units of joules per kelvin. The subscript emphasizes that entropy changes are calculated using a reversible path, even when the actual transformation is irreversible. (feynmanlectures.caltech.edu)
For a closed system exchanging heat through boundary regions at temperatures (T_j), an entropy balance can be written
[ \Delta S=\sum_j\int\frac{\delta Q_j}{T_j}+S_{\mathrm{gen}}, \qquad S_{\mathrm{gen}}\geq0. ]
Here (S_{\mathrm{gen}}) is entropy generated by irreversibility. An isolated system exchanges neither energy nor matter, so its entropy balance reduces to (\Delta S=S_{\mathrm{gen}}\geq0). A nonisolated system can lose entropy by transferring it to its surroundings; the second law does not require every individual object's entropy to increase. (ocw.mit.edu)
A related cyclic expression is the Clausius inequality,
[ \oint\frac{\delta Q}{T_{\mathrm{b}}}\leq0, ]
where (T_{\mathrm{b}}) is the boundary temperature at the location of heat transfer. Equality holds for a reversible cycle. (ocw.mit.edu)
Reversibility and physical examples
A reversible process is an ideal limiting transformation that can be undone without leaving any net change in either the system or its surroundings. Real processes involving friction, mixing, or heat transfer across a finite temperature difference generate entropy. Merely restoring a system to its initial state does not make the preceding process reversible: the surroundings must also be restored. (ocw.mit.edu)
Consider an ideal gas expanding freely into a vacuum inside an insulated, rigid container. No work is performed and no heat enters, so its internal energy remains constant. Its temperature also remains constant, but its entropy increases:
[ \Delta S=Nk_{\mathrm B}\ln!\left(\frac{V_2}{V_1}\right)>0. ]
Here (N) is the number of particles and (k_{\mathrm B}) the Boltzmann constant. Thus an adiabatic process—one without heat transfer—is not necessarily isentropic, or constant in entropy. (ocw.mit.edu)
Heat engines and efficiency
A heat engine absorbs heat (Q_{\mathrm H}) from a hot reservoir, delivers work (W), and rejects heat to a colder reservoir. The second law establishes the upper bound
[ \eta=\frac{W}{Q_{\mathrm H}} \leq1-\frac{T_{\mathrm C}}{T_{\mathrm H}}, ]
with reservoir temperatures expressed in kelvin. Equality is attained by a reversible engine, exemplified by the Carnot cycle. The bound depends on reservoir temperatures rather than the engine's working substance. Irreversibilities reduce actual efficiency below this limit. The law therefore restricts the usability of energy, not its conservation. (feynmanlectures.caltech.edu)
Statistical interpretation
Statistical mechanics relates entropy to the number of microscopic configurations compatible with a macroscopic state. For equally probable accessible configurations, Boltzmann's expression is
[ S=k_{\mathrm B}\ln\Omega, ]
where (\Omega) counts those configurations. High-entropy macrostates generally correspond to overwhelmingly more microscopic arrangements than low-entropy ones. Evolution toward equilibrium is consequently understood through probability, rather than a prohibition imposed on every microscopic motion. “Disorder” is a qualitative analogy, not a universal mathematical definition of entropy. (feynmanlectures.caltech.edu)
For small systems, fluctuations can produce negative entropy production over individual trajectories. Fluctuation theorems quantify relationships between positive and negative fluctuations, while the appropriate ensemble-average entropy production remains nonnegative. These results refine the macroscopic second law rather than establish a cyclic source of work from a single equilibrium reservoir. (arxiv.org)
Equilibrium and chemical transformations
For an isolated system, stable thermodynamic equilibrium corresponds to maximum entropy under the applicable constraints. Different environmental constraints lead to equivalent criteria involving thermodynamic potentials. At fixed temperature and pressure, spontaneous changes in a closed system performing only pressure–volume work decrease its Gibbs free energy, (G=H-TS), where (H) is enthalpy. Equilibrium corresponds to a minimum of (G). These criteria determine the thermodynamic direction of a chemical reaction, but not its speed; a favorable transformation can remain slow because of kinetic barriers. (live.ocw.mit.edu)