Newton's Law of Cooling Calculator

Newton's law of cooling uses the formula T(t) = T_amb + (T₀ − T_amb) × e^(−kt), where T₀ is the initial temperature, T_amb is the ambient temperature, k is the cooling constant, and t is time. Plug in your known values to solve for the unknown. This calculator supports all five possible unknowns.

The cooling constant k (in min⁻¹ or s⁻¹) characterises how quickly an object exchanges heat with its surroundings. It depends on the object's surface area (A), heat transfer coefficient (h), and thermal capacity (C) via k = hA/C. If you have two temperature readings at known times, you can rearrange the formula to solve for k directly using this calculator's 'Calculate Cooling Constant k' mode.

Yes — Newton's law of cooling works with any consistent temperature scale, including Fahrenheit. The key requirement is that the temperature differences (not absolute values) are used consistently. Simply enter all temperatures in °F and the formula remains mathematically valid.

The instantaneous cooling rate is the rate of change of temperature with respect to time: dT/dt = −k(T − T_amb). At the start, when the temperature difference is greatest, the cooling rate is highest. As the object approaches ambient temperature, the rate decreases exponentially.

Using a typical cooling constant of k ≈ 0.05 min⁻¹, you can rearrange Newton's law to get t = −(1/k) × ln((T(t) − T_amb) / (T₀ − T_amb)). For T₀=90°C, T_amb=22°C, T(t)=55°C, this gives approximately t ≈ 15.5 minutes. Use the 'Calculate Time Required' mode in this calculator for any scenario.

Yes — the same formula applies when an object is warming up toward the ambient temperature (T₀ < T_amb). In that case, T(t) increases exponentially toward T_amb rather than decreasing. The equation is mathematically identical; only the direction of heat flow changes.

Newton's law of cooling assumes: (1) the temperature difference between the object and surroundings is relatively small, (2) the ambient temperature remains constant, (3) heat transfer is dominated by convection and conduction at the surface, and (4) the cooling constant k remains uniform over time. For large temperature differences or radiative heat loss, more complex models may be needed.

Newton's law of cooling is used in forensic science (estimating time of death), food safety (predicting how quickly cooked food cools to safe storage temperature), electronics thermal management (CPU heat dissipation), HVAC design, industrial process control, and materials science for heat treatment scheduling.