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Fertiliser bags list N, P and K for a reason: a crop needs each one, and too little of any one holds it back. The idea has a name, the law of the minimum, and it is usually credited to the German chemist Justus von Liebig. This post covers what Liebig argued, what the law actually says, how to use it in a simple calculation, and the expensive mistake he made when he tried to sell it.
Who was Justus von Liebig?
The Neue Deutsche Biographie entry on Liebig records that he was born in Darmstadt on 12 May 1803 and died in Munich on 18 April 1873. He left school to become a pharmacist's apprentice, studied chemistry at Bonn and Erlangen, and then worked in Paris, where Gay-Lussac presented his work on fulminates to the Academy. On Alexander von Humboldt's suggestion he was made professor at Giessen in 1824, at 21. There he built a teaching laboratory with set practical courses and exams, a model that spread across Germany. He moved to Munich in 1852.
The same entry names three areas where his influence was greatest: organic elemental analysis, the radical theory and agricultural chemistry. Our post on estimating carbon, hydrogen, halogens and sulphur covers the first.
What did Liebig argue about plants?
In 1840 he published *Die organische Chemie in ihrer Anwendung auf Agricultur und Physiologie*, organic chemistry applied to agriculture and physiology. The NDB summarises the argument:
Plants do not get their carbon from humus in the soil. Some grow well in soils poor in humus, and humus in fields tends to increase, not fall, even though harvests carry off large amounts of carbon.
So carbon must come from carbon dioxide in the air. Carbon, hydrogen and oxygen come from air and water.
The soil supplies the inorganic substances. A farmer who removes them harvest after harvest without replacing them exhausts the soil.
What does the law of the minimum say?
In the NDB's words, Liebig's law of the minimum says the fertility of a soil for a given crop depends on its content of the nutrient present in the lowest amount, and a shortage of one substance cannot be made good by a surplus of one or more others.
Was it his idea? A careful study in the Soil Science Society of America Journal, On the origin of the theory of mineral nutrition of plants and the law of the minimum (1999), found that Carl Sprengel had refuted the humus theory in 1826 and published, in 1828, a journal article that "contained in essence the Law of the Minimum". Sprengel's ideas reappear in Liebig's books of 1840 and 1855. German agricultural research stations now award a Sprengel-Liebig Medal to honour both.
Worked example: which nutrient limits the crop?
These are sample numbers for a classroom exercise, not real agronomic data. Suppose each tonne of grain needs 20 kg of nitrogen, 4 kg of phosphorus and 5 kg of potassium. One hectare of soil can supply 100 kg N, 12 kg P and 40 kg K this season.
Yield allowed by nitrogen: 100 ÷ 20 = 5 tonnes.
Yield allowed by phosphorus: 12 ÷ 4 = 3 tonnes.
Yield allowed by potassium: 40 ÷ 5 = 8 tonnes.
The smallest is 3 tonnes, so phosphorus is limiting and the field yields about 3 tonnes.
Now add 50 kg of extra nitrogen. The nitrogen limit becomes 150 ÷ 20 = 7.5 tonnes, but the yield stays at 3 tonnes, because phosphorus has not changed. Add 8 kg of phosphorus instead: the phosphorus limit becomes 20 ÷ 4 = 5 tonnes, and now nitrogen and phosphorus both limit at 5 tonnes.
Common mistakes to avoid:
Adding up the nutrients. The law is about the smallest ratio, not the total.
Fixing the wrong nutrient. Extra N on a P-limited field is wasted money and can wash into water.
Forgetting the limit moves. Once you fix one shortage, another nutrient becomes the new minimum.
If the nitrogen and phosphorus cycles behind these numbers feel hazy, our post on nutrient cycles explains them, and the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, one idea at a time.
Why did Liebig's own fertiliser fail?
The NDB and the older Allgemeine Deutsche Biographie on the same Deutsche Biographie page tell the story. Liebig designed special mineral fertilisers for wheat, potatoes and grass based on the ash composition of each crop. Worried that rain would wash soluble salts away, he deliberately made them poorly soluble, by fusing the ingredients with lime. The patent fertiliser, made with the Muspratt company in Liverpool, gave little or no increase in yield. Meanwhile Lawes and Gilbert in England had excellent results with soluble fertilisers made from ammonium salts and superphosphate.
The failure turned many farmers against his ideas, and he spent years searching for the flaw. According to the ADB, around 1860 he recognised that he had neglected a known property of soil: it can hold on to dissolved nutrients so that water cannot wash them out. That made insoluble fertiliser unnecessary. His principle was sound; his product ignored how soil behaves.
What can a student learn from Liebig?
A correct principle can still give a wrong product. Test the practical step as carefully as the theory.
Read what came before. Sprengel's earlier work shows that the most famous name is not always the first.
Admit the failure and look for the cause. Liebig did not drop the law. He found the missing piece about soil.
Try it now: using the sample numbers above, work out how much extra N, P and K the field would need to reach 6 tonnes.
