How to use this calculator

Pick what you want to find. To predict a temperature, enter the starting temperature, the surroundings, the cooling constant k and the time. To find how long cooling takes, enter the target temperature instead of the time. If you don’t know k, choose Cooling constant and enter one measured temperature and the time it was taken; you can then reuse that k to predict anything else. The graph shows the whole cooling curve with your point marked.

Formulas

The differential equationdTdt = −k(T − T_a)

Its solutionT(t) = T_a + (T₀ − T_a) e^−kt

Time and cooling constantt = −1k lnT − T_aT₀ − T_a τ = 1k t_½ = ln 2k

The equation is separable: divide by (T − Tₐ), integrate both sides to get ln|T − Tₐ| = −kt + C, and apply T(0) = T₀. It is the same first-order linear ODE that describes an RC circuit discharging or a radioactive sample decaying.

Worked example

Coffee at 90 °C in a 20 °C room, k = 0.05 per minute (the default)

Starting gap: 90 − 20 = 70 °C.
After 10 minutes: kt = 0.5, so T = 20 + 70 e^−0.5 = 20 + 70 × 0.6065 = 62.5 °C.
Time to cool to 50 °C: t = −ln(30/70)/0.05 = 16.9 minutes.
The time constant is 1/k = 20 minutes, and the gap halves every ln 2/k = 13.9 minutes.

Finding k from a measurement

k bundles together the convection coefficient, surface area, mass and specific heat: for a small object with uniform temperature, k = hA/(mc). Rather than estimating those, measure the temperature once after a known time and let the calculator back out k. A cup of coffee that drops from 90 °C to 70 °C in 5 minutes in a 20 °C room has k = −ln(50/70)/5 = 0.067 per minute.

Assumptions and limits

The surroundings stay at a constant temperature, large enough not to warm up.
k is constant. Natural convection and radiation both get weaker as the gap shrinks, so real objects often cool a little more slowly near the end than the model predicts.
The object’s temperature is uniform (lumped). This holds when the Biot number hL/k_solid is below about 0.1: small or highly conductive objects, or gentle cooling.
No phase change. Water freezing or boiling holds its temperature while it changes phase.

Frequently asked questions

Does the object ever reach room temperature?

Mathematically never, because the gap shrinks by the same fraction in each time step. In practice it is within 1% of the starting gap after about 4.6 time constants, which is usually “done”.

Can it be used for heating?

Yes. A cold drink warming in a warm room obeys the same law; the temperature rises toward the surroundings along the mirror-image curve.

How is this used to estimate time of death?

Investigators compare body temperature with the normal 37 °C and the room temperature, using a k estimated for body size and clothing. Real bodies don’t cool exactly exponentially at first, so forensic methods add corrections.

What units does k have?

Inverse time, such as 1/min or 1/s. The product kt must be dimensionless, so use the same time unit for both.