Newton's Law of Cooling Calculator — Temperature vs Time

Formula

How this is calculated

Newton's law of cooling states that the rate at which an object's temperature changes is proportional to the difference between the object's current temperature and the ambient temperature: dT/dt = −k(T − T∞). Solving this first-order linear ODE gives the exponential equation T(t) = T∞ + (T₀ − T∞) × e^(−k·t), where T₀ is the initial temperature, T∞ is the ambient (surrounding) temperature, k is the positive cooling constant, and t is time.

The cooling constant k depends on the object's geometry, surface area, heat capacity (specific heat × mass), and the heat transfer coefficient of the surrounding medium. A larger k means faster cooling — metal cools much faster than foam, and an object cooling in moving water cools faster than one in still air. The temperature difference halves every ln(2)/k time units (the thermal half-life).

The same formula applies to heating (when T₀ < T∞): the object warms exponentially toward the ambient temperature. Key limitations: the law assumes a uniform body temperature (the Biot number must be small, Bi < 0.1), constant ambient temperature, and constant k. For thin metal objects cooling in still air these assumptions hold well; for liquids, complex geometries, or large objects, more detailed heat-transfer models are needed.