How do I calculate Newton's law of cooling?
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. See also our Sensible Heat Calculator.
What is the cooling constant k, and how do I find it?
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.
Can I use Fahrenheit for Newton's law of cooling?
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.
How do I calculate the cooling rate?
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. You might also find our Specific Heat Calculator useful.
How long does it take for a 90°C cup of coffee to cool to 55°C in a 22°C room?
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.
Does Newton's law of cooling apply to heating as well?
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.
What assumptions does Newton's law of cooling make?
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.
What are practical applications of Newton's law of cooling?
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.