MotionLab

Newton's Cooling Law Calculator

Hot coffee, crime scene forensics, and industrial cooling — all governed by the same exponential decay.

Initial temperature (°C)

Ambient temperature (°C)

Cooling constant k (per min)

Time (minutes)

Final temperature (°C)

30.939

T(t) = T_env + (T₀ − T_env)·e^(−kt)

What you entered

Initial temperature (°C): 95Ambient temperature (°C): 22Cooling constant k (per min): 0.07Time (minutes): 30

1
Initial temperature difference
95 − 22= 73 °C
2
T(t) = T_env + (T₀ − T_env)·e^(−kt)
22 + 73 × e^(−0.07 × 30)= 30.9393 °C

Result

Final temperature (°C): 30.939

Starting at 95°C in a 22°C environment, after 30 minutes the object cools to 30.94°C.

Newton's law of cooling over time

The temperature difference from ambient decays exponentially at a rate set by the cooling constant.

Start (C)	Ambient (C)	Cooling constant	Time (min)	Temperature (C)
60	20	0.05	20	34.715
80	20	0.02	60	38.072
90	20	0.05	10	62.457
90	20	0.05	30	35.619
100	25	0.1	15	41.735

Rows three and four are the same coffee at 10 and 30 minutes: it loses 27.5 degrees in the first ten minutes and only another 26.8 across the next twenty, because cooling slows as the gap to ambient closes. That exponential shape is why a drink is scalding then merely warm for a long time, and why adding cold milk immediately rather than later leaves coffee hotter overall. The object can never fall below ambient. The cooling constant depends on surface area, material and airflow, so a wide shallow cup cools much faster than a tall narrow one.

Exponential approach to equilibrium

Newton's law of cooling says the rate of temperature change is proportional to the temperature difference — this produces an exponential decay curve where the object cools quickly at first (large difference) and progressively slower as it approaches ambient temperature, never quite reaching it.

Real-world applications beyond coffee

Forensic investigators use this law to estimate time of death from body temperature. Food safety guidelines use it to determine safe cooling times. HVAC engineers use it to size heating and cooling systems. The same simple equation appears across surprisingly diverse fields.

Frequently asked questions

A cup of coffee at 90°C is left in a 22°C room with cooling constant k = 0.05 /min. What temperature after 30 min?

T(t) = T_amb + (T₀ − T_amb)×e^(−kt) = 22 + (90 − 22)×e^(−0.05×30) = 22 + 68×e^(−1.5) = 22 + 68×0.223 = 37.2°C. The coffee lost most of its heat in the first 30 minutes — it will approach 22°C asymptotically from here.

How do forensic investigators estimate time of death?

A body cools from 37°C toward ambient temperature following Newton's cooling law. By measuring body temperature and ambient temperature, and using a known cooling constant for human bodies (~0.83°C/hour under standard conditions), investigators back-calculate the time of death — accurate to within a few hours.

Why does the first cup of coffee cool faster than the last degree?

The cooling rate is proportional to the temperature difference. At 90°C in a 22°C room, ΔT = 68°C drives fast cooling. By the time the coffee reaches 25°C, ΔT = 3°C — cooling is 23× slower. This exponential decay means 'almost there' takes much longer than 'most of the way there.'