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Measure how much heat it takes to warm a gram of silver, multiply by silver's atomic weight, and you get about the same number as for copper or lead. Two young Paris scientists spotted this pattern in 1819, and for decades it helped chemists work out atomic weights. This post covers who they were, why the rule works, worked examples using NCERT's own table, and where the rule breaks.
Who were Dulong and Petit?
Pierre-Louis Dulong was born in Rouen on 12 February 1785 and died in Paris on 18 July 1838, according to Britannica. He worked as an assistant to the chemist Claude-Louis Berthollet. While studying the explosive nitrogen trichloride, which he discovered in 1813, he lost the sight of one eye. He became professor of physics at the École Polytechnique in Paris in 1820 and its director in 1830.
Alexis-Thérèse Petit was born in Vesoul on 2 October 1791 and died in Paris on 21 June 1820, according to his MacTutor biography from the University of St Andrews. He reached the entry standard for the École Polytechnique at ten and a half but had to wait until sixteen to enrol, and then entered first in the 1807 exams. He earned a doctorate on capillary action in 1811 and took the physics chair at the École Polytechnique in 1815. Dulong was a witness at his wedding in 1814. Petit died of tuberculosis at 28.
What did they find together?
MacTutor describes how the pair began working together around 1815 on a prize problem set by the Paris Academy about thermometers and cooling. Britannica notes that in 1817 they showed Newton's law of cooling holds only for small temperature differences, and that their 1818 work on temperature measurement and heat transfer was honoured by the Academy.
In 1819 they published the rule now named after them. Britannica's entry on the Dulong–Petit law states it as: the gram-atomic heat capacity (specific heat times atomic weight) of a solid element is about the same for all, roughly six calories per gram atom. MacTutor explains why chemists cared: at the time they struggled to tell atomic weights apart from equivalent weights, and the law gave an estimate of an element's atomic weight from one heat measurement.
Why does C = 3R?
NCERT's Class 11 Physics chapter on kinetic theory (section 12.6.4) derives it. Each atom in a solid vibrates about its mean position in three directions. Each direction has average energy k_BT, so a mole has U = 3RT. For a solid, the volume barely changes, so the molar heat capacity is C = 3R ≈ 24.9 J/mol K. Six calories is 6 × 4.186 ≈ 25.1 J, so the 1819 number and the textbook number agree.
Worked example: atomic mass from specific heat
NCERT's chapter on thermal properties of matter (Table 10.3) lists specific heats in J/kg K. Convert to J/g K and use M ≈ 3R ÷ c. Standard atomic weights below are from CIAAW.
Aluminium: c = 0.900 J/g K, so M ≈ 24.94 ÷ 0.900 = 27.7 (true value 26.98)
Copper: c = 0.3864, so M ≈ 64.5 (true 63.55)
Silver: c = 0.2361, so M ≈ 105.6 (true 107.87)
Lead: c = 0.1277, so M ≈ 195 (true 207.2)
Carbon: c = 0.5065, so M ≈ 49 (true 12.01)
The first four land within about 6%. Carbon is out by a factor of four, which is why NCERT calls it the exception.
Here is how such an estimate settles a doubt. Suppose chemical analysis tells you copper combines in the ratio of 31.8 g of copper per 8 g of oxygen. The atomic weight could be 31.8, 63.6 or 95.4, depending on the formula of the oxide. The heat measurement says about 64.5, so you pick 63.6 and the oxide is CuO. This is an illustration of the logic, not a record of a particular historical calculation.
Other common slips: using R = 8.314 when your heat is in calories (use about 2 cal/mol K, so 3R ≈ 6), and applying the law to gases, whose heat capacities follow different rules.
Where does the law break, and who explained it?
Britannica notes that low-temperature measurements later showed the heat capacity of every solid falls towards zero when it is cold enough. The law is now treated as an approximation that holds at ordinary and higher temperatures. Carbon falls well short even at room temperature, as NCERT's table shows.
The explanation needed quantum ideas. In 1907 Albert Einstein published a paper applying Planck's quantum theory of radiation to the specific heat of solids, treating atomic vibrations as quantised. That is the starting point for the modern theory.
What can a student take from Dulong and Petit's way of working?
Look for a product that stays constant. Specific heats alone looked random. Multiplying by atomic weight revealed the pattern.
Take exceptions seriously. Carbon refused to fit, and that misfit pointed to physics nobody yet understood.
Measure carefully before you theorise. Their earlier prize work on thermometers and cooling is what made their heat data trustworthy.
When you are unsure which formula a heat problem needs, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. For more practice, see our ideal gas problems guide.
Next step: take tungsten's specific heat from NCERT Table 10.3, estimate its atomic mass with 3R ÷ c, and compare it with the periodic table.
