How Newton's Law of Cooling Works
Newton's Law of Cooling says that the rate at which an object loses or gains heat is proportional to the difference between its own temperature and the temperature of its surroundings. A hot object in a cool room loses heat fastest right at the start, when the temperature gap is largest, and cools more and more slowly as it approaches room temperature — it never overshoots, only approaches the ambient value asymptotically. This calculator finds the object's temperature at a chosen elapsed time, how much of the total temperature change has happened, and the process's characteristic time constant.
Deriving the exponential decay formula
The law starts as a differential equation: dT/dt = −k(T − T∞), where T is the object's temperature at time t, T∞ is the (assumed constant) ambient temperature, and k is a positive cooling constant with units of inverse time. Separating variables and integrating from the initial temperature T₀ at t = 0 gives the closed-form solution used by this calculator: T(t) = T∞ + (T₀ − T∞)e^(−kt). The temperature gap (T₀ − T∞) shrinks exponentially, so the object gets arbitrarily close to T∞ but reaches it only as t → ∞.
Finding the cooling constant k
k is not a fixed physical constant like the speed of light — it depends on the object's surface area and shape, its material (specifically the ratio of the convective heat-transfer coefficient to the mass times specific heat), and the surrounding medium (still air, moving air, water, and so on cool objects at very different rates). In practice, k is measured: record the temperature T₁ at a known elapsed time t₁, then solve k = −ln[(T₁ − T∞)/(T₀ − T∞)] / t₁. Once k is known for a given object and environment, it can be reused to predict the temperature at any other time under the same conditions.
Assumptions and limits
Newton's Law of Cooling assumes the object's internal temperature is nearly uniform at every instant (a low Biot number — the object conducts heat internally much faster than it loses heat at its surface) and that heat leaves mainly by convection into surroundings held at a constant temperature. It is a good approximation for everyday objects like a cup of coffee or a warm meal cooling in a kitchen. It becomes less accurate for large temperature differences or when radiative heat transfer dominates, since radiation follows the Stefan–Boltzmann T⁴ law rather than this linear approximation — for something like a metal bar cooling from red heat, expect noticeable deviation from the simple exponential model.