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Clausius showed that entropy always tends to increase, but not why. Ludwig Boltzmann gave the reason: there are vastly more ways for atoms to be spread out than bunched up, so a system drifts towards the arrangements that are most numerous. This post explains that idea, works through the numbers, and looks at the cost of defending atoms before most physicists accepted them.
Who was Ludwig Boltzmann?
Boltzmann was born in Vienna on 20 February 1844 and died at Duino, near Trieste, on 5 September 1906 (MacTutor, University of St Andrews). He earned his doctorate at the University of Vienna in 1866 under Josef Stefan, and held chairs in Graz, Vienna, Munich and Leipzig before returning to Vienna in 1902.
MacTutor lists his main results: the Maxwell-Boltzmann distribution of molecular speeds (1871), a derivation in 1884 of Stefan's T⁴ law for radiation from thermodynamics, and the statistical explanation of the second law. The Neue Deutsche Biographie adds that he was an excellent pianist.
What does S = k log W mean?
The Neue Deutsche Biographie records that Boltzmann's honorary grave in Vienna's Central Cemetery carries the inscription S = k log W as the essence of his work. S is entropy. W is the number of microscopic arrangements (microstates) that give the same large-scale state, which Boltzmann showed how to calculate by counting. The "log" is the natural logarithm.
His key 1877 paper, on the relation between the second law and probability, is available in an English translation in the journal Entropy.
Worked example: counting arrangements
Put 4 labelled molecules in a box with a left half and a right half. Each molecule can be on either side, so there are 2⁴ = 16 microstates in total.
All 4 on the left: 1 way.
3 left, 1 right: 4 ways.
2 left, 2 right: C(4,2) = 6 ways.
1 left, 3 right: 4 ways.
All 4 on the right: 1 way.
The even split is six times as likely as "all on the left". With 4 molecules that is a mild preference. With 10²³ molecules, the chance of finding them all on one side is so small that you will never see it. That is the second law as Boltzmann understood it: entropy increases not because it must, but because the alternative is overwhelmingly improbable. MacTutor puts it the same way: he held that entropy increases "almost always, rather than always".
Does this agree with the gas laws?
Let one mole of an ideal gas double its volume at constant temperature. Each of the N molecules now has twice as much room, so the number of microstates is multiplied by 2^N.
ΔS = k ln(2^N) = N k ln 2.
For one mole, N k = N_A k = R. With k = 1.380649 × 10⁻²³ J/K (NIST CODATA value) and N_A = 6.02214076 × 10²³ mol⁻¹, R ≈ 8.314 J/(mol·K).
ΔS = 8.314 × ln 2 = 8.314 × 0.693 ≈ 5.76 J/K.
The classical thermodynamics result for isothermal expansion is ΔS = nR ln(V₂/V₁) = 1 × 8.314 × ln 2, the same 5.76 J/K. Counting atoms and measuring heat give one answer. For the kinetic theory side of this, see our guide to rms speed, degrees of freedom and specific heats.
Common mistakes:
Using log base 10. Boltzmann's log is the natural log. ln 2 is 0.693; log₁₀ 2 is 0.301.
Counting macrostates instead of microstates. In the 4-molecule box there are 5 macrostates but 16 microstates.
Treating "disorder" as the definition. It is a rough picture. The definition is the count W.
Who named the constant k?
Max Planck. In his 1920 Nobel lecture he said the constant "is often referred to as Boltzmann's constant, although, to my knowledge, Boltzmann himself never introduced it". Planck needed it for his radiation law, the work that introduced the quantum of action; read our post on Max Planck and the quantum. In today's SI, k has an exact value with no uncertainty, as the NIST listing shows.
Why did his ideas meet such resistance?
Many leading scientists of the time, including Wilhelm Ostwald and Ernst Mach, doubted that atoms were real, or thought physics should not depend on them. MacTutor describes a famous 1895 debate at Lübeck between Ostwald and Boltzmann, which Arnold Sommerfeld, who was there, said Boltzmann won.
Boltzmann also lived with severe swings of mood, which MacTutor describes as alternating depressed and elevated states. He took his own life in 1906 while on holiday. MacTutor cautions that the cause may have been wrongly attributed to the rejection of his ideas, and that mental illness may have been the real cause. If you or someone you know is struggling, India's national mental health helpline, Tele MANAS, is free on 14416 or 1-800-891-4416.
What can you learn from Boltzmann?
Explain the big from the small. He started with molecules and derived what thermometers and pressure gauges show.
Check a new idea against an old result. The 5.76 J/K match above is the kind of test that made his theory convincing.
Accept probability as an answer. "Almost always" was a hard idea in his time. It is now central to physics and chemistry.
If microstates and entropy still feel abstract, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can build from probability to entropy step by step.
Your next step: count the microstates for 6 molecules in two halves and find how much more likely a 3-3 split is than all on one side.
