Newton's Law of Cooling

  In the March-April 1701 issue of Philosophical Transactions a temperature scale and law of cooling was published anonymously which is now attributed to Isaac Newton. Here is an excerpt and translation of the relevant portion.

"Constructa fuit hæc Tabula ope Thermometri & ferri candentis. Per Thermometrum inveni mensuram caloruni omnium usq; ad calorem quo stannum funditur & per ferrum calefaƈtum corporibus frigidis sibi contiguis dato tempore communicat, hoc est calor quem ferrum dato tempore amittit est ut calor totus ferri. Ideoq; si tempora refrigerii sumantur æqualia calores erunt in ratione geometrica & propterea per tabulam logarithmorum facile inveniri possunt."

"This table was constructed by the help of a thermometer and of heated iron. With the thermometer I found the measure of all the heats up to that at which lead melts and by the hot iron I found the measure of the other heats. For the heat which the hot iron communicates in a given time to cold bodies which are near it, that is, the heat which the iron loses in a given time, is proportional to the whole heat of the iron. And so, if the times of cooling are taken equal, the heats will be in a geometrical progression and consequently can easily be found with a table of logarithms."

At this time Newton became occupied with his new duties as Master of the Mint after resigning from his professorship at Cambridge. This appears to be the background for Newton's law of cooling:

  1694 Newton becomes Warden of the Mint
  1699 Newton becomes Master of the Mint
  1701 Newton retires professorship at Cambridge
  1701 Scala graduum Caloris appears in Philosophical Transactions
  1703 Newton becomes President of the Royal Society of London
  1705 Newton knighted

How might one deduce Newton's the law of cooling? If one had access to a thermometer, as Newton did, measuring the temperature of a cooling object at given intervals of time one would reveal that the rate of cooling decreases monotonically with time. Initially the rate of cooling is highest but slows down as one approached the temperature of the surroundings. One gets a crude approximation of the curve if one assumes in each interval of time the object loses the same fraction of its heat content. The result is a geometrical series similar to that in the race between Achilles and the Tortoise found in one of Zeno's paradoxes. In successive intervals the object loses fractions q, q², q³,...,qⁿ,... of its heat. The total heat lost is q+q²+q³+…+qⁿ+…=q/(1-q). Note that if the sum is 1 corresponding to all the heat in excess of thermal equilibrium being lost then q=1/2 which is what one finds in the paradox. Taking the interval, Δt, to be one second we have ΔQ=-q₁Q=-λ₁ΔtQ or ΔQ/Δt=-λ₁Q which goes to dQ/dt=-λQ as Δt goes to 0. Taking Q=CT where C is the heat capacity of the body and T its temperature we get  d(CT)/dt=-λ(CT) or dT/dt=-λT. The cooling is offset by heating from the environment at temperature Tₑ so there is an additional term, λTₑ, and so we set dT/dt=λ(T₀-Tₑ). Integrating this gives,

T=Tₑ+(T₀-Tₑ)exp(-λt)

This is Newton's law of cooling. In actuality it is more qualitative than quantitative but it is needed to understand some content in Fourier's theory of heat.


Bibliography

Zeno's Paradox

Newton temperature scale

Newton's law of cooling

Bolton - Evolution of the thermometer, 1592-1743

Brewster - The Life of Sir Isaac Newton