The Second Law of Thermodynamics and Entropy
The second law of thermodynamics states that heat does not flow spontaneously from a colder body to a hotter body and that no cyclic engine can convert heat completely into work; equivalently, the entropy of an isolated system never decreases, and for a reversible transfer of heat Q at temperature T, ΔS = Q ÷ T. In brief: entropy is a measure of the dispersal of energy, or of the number of microscopic arrangements W corresponding to a macroscopic state, S = k ln W. Natural processes (mixing, free expansion, heat flow, friction) increase the total entropy. This gives the “arrow of time”. Local entropy can decrease (a refrigerator, a living organism) only if the entropy of the surroundings increases by at least as much.
Hot coffee cools down, a dropped glass breaks but never reassembles itself, and gas spreads through a room but does not gather in one corner. The first law allows all of these in either direction; the second law tells which direction actually happens. This guide states the second law in its equivalent forms, defines entropy macroscopically and statistically, compares reversible and irreversible processes in a table, works through entropy calculations and corrects common mistakes.
What are the statements of the second law?
- Clausius statement: it is impossible for a cyclic process to have as its only result the transfer of heat from a colder to a hotter body. (Refrigerators need work.)
- Kelvin-Planck statement: it is impossible for a cyclic engine to take heat from a single reservoir and convert it entirely into work. (Engines must reject heat to a colder reservoir.)
- Entropy statement: the total entropy of an isolated system can never decrease: ΔS ≥ 0, with equality only for reversible processes.
The statements are equivalent: a violation of one would allow a violation of the others. A machine that violates the second law is called a perpetual motion machine of the second kind (for example, a ship driven by heat extracted from the ocean).
What is entropy?
Entropy S is a state function. When a heat Q is added reversibly to a system at a constant absolute temperature T, the entropy change is ΔS = Q ÷ T, in J/K (in general, dS = dQrev ÷ T). For a spontaneous transfer of heat Q from a hot body at Th to a cold body at Tc, the hot body loses entropy Q ÷ Th, and the cold body gains Q ÷ Tc, which is larger since Tc < Th. The total ΔS = Q(1/Tc − 1/Th) > 0. The greater the temperature difference, the larger the entropy generated. In an ideal reversible process, ΔStotal = 0. The entropy changes for common processes: melting, ΔS = mLf ÷ Tm; heating with specific heat c from T1 to T2, ΔS = mc ln(T2/T1); isothermal expansion of an ideal gas, ΔS = nR ln(V2/V1).
What is the statistical meaning of entropy?
Ludwig Boltzmann showed that entropy measures the number W of microstates (arrangements of molecules) that correspond to a given macrostate: S = k ln W. A state with a large number of microstates is much more probable. For example, when a gas is allowed to expand into twice the volume, each molecule has twice as many positions available, so W increases by a factor 2N and S by Nk ln 2 = nR ln 2. The probability that all the N molecules go back to one half of the container spontaneously is (1/2)N: for N = 1023, this is so small that it never happens. The second law is thus statistical. Entropy is often described as “disorder”, but the better description is the dispersal of energy among the available microstates. The second law has consequences for the “heat death” of the universe, and for the arrow of time: the laws of mechanics are the same in both time directions, but the entropy of the universe increases. Information theory (Shannon, 1948) has a similar entropy formula, and Maxwell’s demon and Landauer’s principle (erasing a bit dissipates at least kT ln 2 of heat) connect information with thermodynamics.
Reversible and irreversible processes compared
| Feature | Reversible | Irreversible |
|---|---|---|
| Total entropy change | Zero | Positive |
| Conditions | Quasi-static, no friction, infinitesimal temperature differences | Finite differences, friction, mixing, free expansion |
| Examples | Ideal Carnot cycle | Heat flow across a finite temperature difference, a falling stone coming to rest |
| Real processes | Idealization | All real processes |
Worked example: entropy changes
- Heat flow. 1,000 J of heat flows from a hot body at 500 K to a cold body at 300 K. ΔShot = −1,000 ÷ 500 = −2.00 J/K. ΔScold = +1,000 ÷ 300 = +3.33 J/K. ΔStotal = +1.33 J/K > 0. The reverse flow would give −1.33 J/K and is forbidden.
- Melting ice. 0.100 kg of ice at 273.15 K melts: Q = mLf = 0.100 × 334,000 = 33,400 J. ΔS = 33,400 ÷ 273.15 = 122 J/K.
- Free expansion. 1.0 mol of an ideal gas expands into a vacuum to double its volume. Q = 0 and W = 0, but ΔS = nR ln 2 = 8.314 × 0.693 = 5.76 J/K (calculated along a reversible isothermal path between the same states).
- Heating water. 1.0 kg of water is heated from 20 °C (293.15 K) to 80 °C (353.15 K): ΔS = mc ln(T2/T1) = 1.0 × 4,180 × ln(353.15/293.15) = 4,180 × 0.1861 = 778 J/K.
- Heat engine limit. An engine that takes 1,000 J from a reservoir at 400 K and converts it all to work would decrease the entropy of the universe by 1,000 ÷ 400 = 2.5 J/K, which the second law prohibits. The engine has to reject at least Qc = QhTc ÷ Th to a cold reservoir at Tc.
- Boltzmann entropy. A system with 106 microstates has S = k ln 106 = 1.381 × 10−23 × 13.8 = 1.9 × 10−22 J/K, a tiny value for a small number of states; macroscopic systems have ln W of order 1024.
How was the law formulated?
Sadi Carnot’s work on the limits of heat engines (1824) led to the second law. Rudolf Clausius stated it in 1850 and introduced the concept of entropy in 1865 (from the Greek tropē, transformation). William Thomson (Kelvin) gave an equivalent statement in 1851. Ludwig Boltzmann’s statistical interpretation (1877) is inscribed on his tombstone in Vienna as S = k log W. Max Planck introduced the constant k in the form we use today in 1900, and the third law (Nernst, 1906) states that the entropy of a perfect crystal tends to zero at absolute zero.
Which mistakes are common?
Mistake 1: Saying that entropy always increases for every system
It applies to isolated systems (or to the system plus its surroundings). A refrigerator or a growing crystal decreases its own entropy while increasing that of the surroundings.
Mistake 2: Using Celsius temperatures
ΔS = Q ÷ T uses absolute temperature.
Mistake 3: Thinking that the second law forbids the conversion of heat into work
It forbids complete conversion in a cycle from one reservoir. Heat engines do convert part of the heat into work.
Mistake 4: Confusing “disorder” with chaos in a visual sense
Entropy counts microstates and the spread of energy, not just visual messiness.