On this page
Entropy is in the JEE Main chemistry syllabus, next to Gibbs energy and the second law, and it is in physics too. Students often learn it as "disorder" and then struggle with the numbers. This post goes back to the man who defined it, shows what his definition means, and works through two calculations.
Who was Rudolf Clausius?
Clausius was born in Köslin, Prussia (now Koszalin, Poland) on 2 January 1822 and died in Bonn on 24 August 1888 (MacTutor, University of St Andrews). He studied mathematics and physics in Berlin and received his doctorate from Halle in 1848. He later held chairs in Zürich, Würzburg and Bonn.
MacTutor records that he was elected a Fellow of the Royal Society of London in 1868 and received its Copley Medal in 1879.
What did his 1850 paper change?
Before Clausius, most theories of heat engines, including Sadi Carnot's, treated heat as a weightless fluid called caloric that is never created or destroyed. Joule's experiments had shown that work and heat can be converted into each other, which did not fit.
In *Über die bewegende Kraft der Wärme* ("On the motive power of heat"), read to the Berlin Academy on 18 February 1850 and published in Annalen der Physik, Clausius dropped the caloric assumption. MacTutor summarises the result: he kept Carnot's idea that the work from an ideal engine depends only on the temperatures of the hot and cold bodies, but rebuilt it on the equivalence of heat and work. He wrote the balance we now call the first law, and stated an early form of the second: heat tends to flow from hot bodies to cold ones.
The same line of work produced the Clausius-Clapeyron equation, which relates the pressure and temperature at which two phases of a substance are in equilibrium.
How did Clausius define entropy?
Over the next fifteen years he looked for a quantity that captured the second law in mathematical form. In his 1865 paper in Annalen der Physik, he defined S for a body through the heat it absorbs reversibly, divided by the absolute temperature at which it absorbs it:
dS = dQ_rev / T
In an English translation of the key passages published by Le Moyne College, he proposed to name S after "the Greek word ἡ τροπή = the transformation", giving "the entropy of the body". He then stated the two laws for the universe as a whole: "The energy of the universe is constant" and "The entropy of the universe strives toward a maximum."
Worked example 1: entropy change when heat flows
A hot block at 400 K passes 1,000 J of heat to a cold block at 300 K. Both are large enough that their temperatures barely change. What is the total entropy change?
Hot block loses heat: ΔS_hot = −1,000 / 400 = −2.50 J/K.
Cold block gains heat: ΔS_cold = +1,000 / 300 = +3.33 J/K.
Total: ΔS = −2.50 + 3.33 = +0.83 J/K.
The total is positive, so the process can happen on its own. Run it backwards, heat flowing from cold to hot with nothing else changing, and the total would be −0.83 J/K. That is the second law in one line of arithmetic.
Worked example 2: melting ice
A sample of ice absorbs 6,000 J of heat while melting at 0 °C. Melting at the melting point happens at constant temperature and can be treated as reversible.
ΔS = Q / T = 6,000 / 273.15 ≈ 22.0 J/K.
Common mistakes:
Using °C. T must be in kelvin. 6,000 / 0 is meaningless.
Dropping signs. Heat lost gives negative ΔS for that body.
Forgetting the surroundings. The second law is about the total, system plus surroundings. A system's entropy can fall if the surroundings gain more.
If the signs of heat and work still trip you up, our post on sign conventions in thermodynamics sorts out the physics and chemistry versions.
Was Clausius right about everything?
His achievements are secure, but MacTutor also notes sharp priority disputes with British scientists such as Tait, and a historian's observation that Clausius showed little interest in the work of Gibbs or Boltzmann, who built directly on his idea of entropy. Boltzmann later explained entropy in terms of counting molecular arrangements; read our post on Ludwig Boltzmann and statistical mechanics for that next step.
What can you learn from how Clausius worked?
Keep the result, change the foundation. He kept Carnot's conclusion and rebuilt the reasoning when the caloric idea failed.
Name a quantity once you can define it. He worked with S for years and named it only when he could state exactly how to calculate it.
Return to a hard problem. MacTutor counts eight more papers on the second law in the fifteen years after 1850.
If you want to rebuild entropy from the ground up, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, from heat and temperature to the second law.
Now try this: find the total entropy change if the same 1,000 J flows from 500 K to 250 K, and compare it with the first example.
