Calculate temperature over time, time to reach a target temperature, or find the cooling constant using Newton's law of cooling with exponential decay.
Newton's Law of Cooling states that the rate of temperature change of an object is proportional to the difference between its temperature and the ambient temperature: dT/dt = −k(T − Ta). The solution is the exponential decay T(t) = Ta + (T₀ − Ta)·e^(−kt).
This calculator operates in three modes. Forward mode computes temperature at any time. Reverse mode finds how long it takes to reach a target temperature. The third mode determines the cooling constant k from a measured temperature at a known time.
Applications range from everyday scenarios (when is my coffee cool enough to drink?) to forensic science (estimating time of death from body temperature) and engineering (heatsink cooling, metal quenching, food safety). The cooling curve table shows temperature at regular intervals, and the half-life tells you how quickly the temperature difference halves.
Preset buttons load common scenarios including coffee cooling, forensic body temperature, metal quenching, and electronics thermal management.
Newton's cooling law appears in physics, engineering, food safety, biology, and forensic science. This calculator handles all three standard problems (find T, find t, find k) in one tool.
The cooling curve table and half-life output give a complete picture of the thermal process without needing to solve the differential equation manually.
T(t) = Ta + (T₀ − Ta) · e^(−kt). t = −ln((Tt − Ta)/(T₀ − Ta)) / k. k = −ln((Tm − Ta)/(T₀ − Ta)) / t. Half-life: t½ = ln(2)/k.
Result: T = 45.7°C, 65.1% cooled
T(30) = 22 + (90−22)·e^(−0.035×30) = 22 + 68·e^(−1.05) = 22 + 23.7 = 45.7°C. Half-life = ln(2)/0.035 = 19.8 min.
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Measure the temperature at two different times. Use this calculator's "Find Cooling Constant" mode with the initial temperature, ambient temperature, and one measurement point.
Newton's law of cooling is most accurate when the temperature difference is modest. For very hot objects (radiation-dominated cooling), Stefan-Boltzmann law is more appropriate.
k depends on the heat transfer coefficient, surface area, mass, and specific heat capacity: k = hA/(mc). Better insulation, larger mass, or higher specific heat all reduce k.
Forensic pathologists measure body temperature and ambient temperature, then use the cooling law to estimate time since death. The Henssge nomogram is a refined version of this approach.
The time for the temperature difference (T − Ta) to decrease by half. Like radioactive half-life, it is constant: t½ = ln(2)/k ≈ 0.693/k.
Yes! The same formula works for heating (object cooler than surroundings). T₀ < Ta, and the temperature exponentially approaches ambient from below.