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Every time you write Kc = [C][D]/[A][B] in a Class 11 equilibrium problem, you are using an idea that two friends in Christiania (now Oslo) worked out in the 1860s. One was a mathematician, the other a chemist. This post covers who they were, what they actually wrote, a worked example built on their own illustration, and what you can learn from how they worked.
Who were Guldberg and Waage?
Cato Maximilian Guldberg was born in Christiania on 11 August 1836 and died there on 14 January 1902, according to Norsk biografisk leksikon. He trained in mathematics and physics, taught at the military college, and in 1869 became professor of applied mathematics at the university.
Peter Waage was born on 29 June 1833 on the island of Hidra and died on 13 January 1900, according to Store norske leksikon. He studied chemistry and mineralogy, worked with Robert Bunsen in Heidelberg, and became professor of chemistry in Christiania.
The two met as students. In 1858 they and four friends founded a small physics and chemistry club that met on Saturdays to give talks to each other. They also became brothers-in-law twice over. Guldberg and Waage married two sisters, and after Waage's first wife died he married Guldberg's sister.
What did they propose in 1864?
NCERT's Class 11 Equilibrium chapter says it plainly: Guldberg and Waage "proposed in 1864" that the concentrations in an equilibrium mixture are related by the equilibrium equation, and the equation is called the law of mass action "because in the early days of chemistry, concentration was called 'active mass'". Their first account was given as three lectures to the Academy of Science in Christiania in 1864. An English translation, Studies concerning affinity, appeared in the Journal of Chemical Education in 1986.
In 1867 they published a fuller French version, Études sur les affinités chimiques. Its key sentence, in our translation, is: "We have deduced from our experiments that the force is proportional to the product of the active masses of the two substances A and B." For a reaction A + B ⇌ A' + B', with active masses p and q, and an "affinity coefficient" k, the forward force is kpq. The backward force is k'p'q'. At balance they wrote
kpq = k'p'q'
Rearranged, p'q'/pq = k/k', a constant. That is the shape of the modern Kc.
What experiment did they use?
Their typical reaction was barium sulfate with potassium carbonate, which forms barium carbonate and potassium sulfate. They stressed that the reverse reaction also runs under the same conditions, so neither direction ever finishes. They observed that adding more potassium carbonate pushed more barium sulfate to react, and that a large enough excess could convert it almost completely.
Worked example: the numbers from their own illustration
In the 1867 paper they illustrate the idea with round numbers: start with 100 molecules each of A and B, and suppose the reaction stops when 20 have been converted. Treat this as an idealised sample (one fixed volume, all four substances dissolved).
At balance. A = B = 100 − 20 = 80 and A' = B' = 20.
The constant. K = (20 × 20)/(80 × 80) = 400/6400 = 1/16 = 0.0625.
Run it backwards. Start with 100 each of A' and B' and let y convert back. Then (100 − y)²/y² = 1/16, so (100 − y)/y = 1/4 and y = 80. The mixture ends at the same 80 : 20 point from either side, which is exactly what Guldberg and Waage expected.
Add excess B. Start with 100 A and 300 B, and let x convert. Then x²/[(100 − x)(300 − x)] = 1/16. Multiplying out, 16x² = 30000 − 400x + x², so 15x² + 400x − 30000 = 0. The positive root is x = (−400 + 1400)/30 = 33.3.
Tripling B raises the share of A that reacts from 20 per cent to about 33 per cent. That is the effect they saw with extra potassium carbonate, and it is the same reasoning as Le Chatelier's principle, which came later. Our posts on equilibrium constant calculations and Henry Le Chatelier take this further.
What mistakes do students make with mass action?
Using moles instead of concentrations. Active mass means amount per volume. In step 4 the volume cancels only because both sides have the same number of particles.
Thinking equilibrium means equal amounts. At balance above, A is 80 and A' is 20. Equal rates, not equal amounts.
Forgetting temperature. K changes with temperature. Guldberg and Waage themselves noted that heating favoured the breakdown of barium sulfate.
Including solids. Later work showed that pure solids do not appear in the Kc expression, which is why the barium sulfate system is harder than the idealised example.
If the algebra in step 4 is where you stall, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
How did their idea grow?
They kept refining it. In 1879 they published Ueber die chemische Affinität in the Journal für praktische Chemie, a German account of the law. Norsk biografisk leksikon notes that in this work they showed how the law can be derived from a kinetic picture of reacting particles. A 1965 article in the Journal of Chemical Education, Guldberg and Waage and the law of mass action, looks back at the pair and their law.
What can you take from how they worked?
Pair different skills. A mathematician and an experimental chemist each saw half the problem.
Look at both directions. Their insight came from taking the reverse reaction seriously.
Write the idea as an equation. One line, kpq = k'p'q', made a vague word, "affinity", something you could measure.
Keep improving it. They published in 1864, 1867 and 1879, each time sharper.
Next step: redo step 4 with 100 A and 200 B and check that more B still gives a bigger share of A reacted.
