Physics
How To Calculate Newton's Law Of Cooling
A hot object cools fastest when it is hottest, because the rate follows the temperature gap. That single idea produces an exponential curve.
Quick Answer
T(t) = Ta + (T0 - Ta) x e^(-kt)
- T(t)
- The temperature after time t
- Ta
- The ambient temperature the object approaches
- T0
- The starting temperature of the object
- k
- The cooling constant in units of one over time
Take the gap between the object and its surroundings, shrink it by e to the minus k times t, and add the ambient temperature back. Coffee at 90 degrees in a 20 degree room with k of 0.05 falls to 35.6191 degrees in 30 minutes.
What Is Newton's Law Of Cooling?
Newton's law of cooling says the rate at which an object loses heat is proportional to the difference between its temperature and the temperature of its surroundings.
That proportionality produces an exponential. The gap between the object and the room shrinks by the same fraction in every equal interval of time, rather than by the same number of degrees.
Written as a formula, the temperature at time t is the ambient temperature plus the starting gap times e to the minus k t. The ambient temperature is the asymptote the curve approaches.
The curve is steepest at the start. Coffee at 90 degrees in a 20 degree room loses 3.5 degrees in the first minute and less than a degree a minute by the time it has nearly arrived.
The cooling constant k carries everything about the object and the conditions. It depends on the surface area, the material, the container and how fast the air moves past it.
Because k is a rate, its units are one over time. A k of 0.05 means five percent of the gap disappears each minute, and the same number with minutes replaced by hours describes a much slower object.
The half difference time is a clean way to read k. It is the natural log of two divided by k, which is 13.8629 minutes for a k of 0.05, and it does not depend on the temperatures at all.
This is the same mathematics as radioactive decay and as a discharging capacitor. Any quantity whose rate of change is proportional to what is left produces this curve.
The law runs in reverse for warming. A cold drink in a warm room has a negative gap, and the same formula pulls its temperature up towards ambient.
Forensic science uses it to estimate time of death. Measure the body temperature, know the ambient, estimate k, and the formula gives the elapsed time.
Food safety depends on the same curve. Cooling a large pot of stock slowly in the fridge leaves it in the danger zone for hours, which is why the depth of the container matters.
The model assumes the object has one uniform temperature. That is the lumped capacitance assumption, and it holds when the inside conducts heat far faster than the surface loses it.
A big roast breaks that assumption. The surface cools quickly while the centre stays hot, so a single k cannot describe the whole joint, and the centre temperature lags the prediction.
Evaporation can dominate for an open hot liquid. A cup of coffee loses a surprising share of its heat to vapour rather than to conduction, which is why a lid changes the cooling rate so much.
For coffee at 90 degrees in a 20 degree room with k of 0.05, the temperature after 30 minutes is 35.6191 degrees, and it passes through 60 degrees at 11.1923 minutes.
Formula
T(t) = Ta + (T0 - Ta) x e^(-kt)
Formula page →The gap shrinks exponentially towards the ambient temperature.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| T(t) | Temperature | degrees | The temperature after time t. |
| Ta | Ambient | degrees | The temperature of the surroundings. |
| T0 | Initial | degrees | The starting temperature of the object. |
| k | Cooling constant | per unit time | The fractional rate the gap shrinks at. |
dT/dt = -k x (T - Ta)
Formula page →The differential statement, and the quickest way to read the initial rate.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| dT/dt | Cooling rate | degrees per unit time | How fast the temperature is falling. |
| k | Cooling constant | per unit time | The fractional rate of gap loss. |
| T - Ta | Temperature gap | degrees | How far the object is from ambient. |
t = -ln((Tt - Ta) / (T0 - Ta)) / k
Formula page →Inverts the curve, which is how time of death and cooling schedules are worked out.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| t | Time | unit of time | How long until the target is reached. |
| Tt | Target temperature | degrees | The temperature you are waiting for. |
| T0 | Initial | degrees | The starting temperature. |
| Ta | Ambient | degrees | The temperature of the surroundings. |
How To Calculate Newton's Law Of Cooling
- 1
Find the starting gap
Coffee at 90 degrees in a 20 degree room starts with a gap of 70 degrees. This gap is what decays, not the temperature itself.
- 2
Evaluate the decay factor
e to the minus k times t is e to the minus 0.05 times 30, which is e to the minus 1.5, giving 0.22313. Just over a fifth of the gap remains.
- 3
Apply it to the gap
70 times 0.22313 is 15.6191, so the coffee has fallen 54.3809 degrees and only 15.6191 of the original gap is left.
- 4
Add the ambient back
20 plus 15.6191 is 35.6191 degrees. Forgetting this last step is the most common error, because the formula decays the gap, not the temperature.
Examples
Example 1: Coffee at 90 degrees left in a 20 degree room for 30 minutes
- Starting temperature
- 90
- Ambient temperature
- 20
- Cooling constant
- 0.05
- Time elapsed
- 30
| Step | Calculation | Result |
|---|---|---|
| Starting temperature gap | 90 - 20 | 70 |
| Decay factor | e^(-0.05 x 30) | 0.22313 |
| Gap remaining after 30 minutes | 70 x 0.22313 | 15.6191 |
| Temperature of the coffee | 20 + 15.6191 | 35.6191 |
Result: The gap of 70 degrees decays to 15.6191 over 30 minutes, so the coffee sits at 35.6191 degrees. It started out losing 3.5 degrees a minute and the gap halves every 13.8629 minutes.
Example 2: How long until the same coffee reaches 60 degrees
- Starting temperature
- 90
- Ambient temperature
- 20
- Cooling constant
- 0.05
- Target temperature
- 60
| Step | Calculation | Result |
|---|---|---|
| Target gap as a fraction of the start | (60 - 20) / 70 | 0.5714286 |
| Natural log of that fraction | ln(0.5714286) | -0.5596158 |
| Time to reach 60 degrees | 0.5596158 / 0.05 | 11.1923 |
| Half difference time for comparison | 0.6931472 / 0.05 | 13.8629 |
Result: Reaching 60 degrees means shrinking the gap to 0.5714286 of its original size, which takes 11.1923 minutes. The gap halves completely at 13.8629 minutes, when the coffee passes 55 degrees.
Calculator
Temperature after time t
35.6191
- Time to reach the target
- 11.1923
- Half difference time
- 13.8629
- Initial cooling rate
- -3.5
- Share of the gap remaining
- 22.31%
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Newton's Law Of Cooling calculator page.
Common Mistakes
Forgetting to add the ambient temperature back
The exponential decays the gap, not the temperature. Reporting 15.6191 degrees instead of 35.6191 leaves the coffee colder than the room it is sitting in.
Assuming a constant cooling rate
The rate is proportional to the gap, so it slows as the object cools. Extrapolating the first minute's 3.5 degrees over 30 minutes predicts a temperature far below freezing.
Treating k as a universal constant
k belongs to the object and the conditions. The same coffee cools faster in a draught, in a metal cup, or with the lid off.
Mixing time units
k and t must share a unit. A k quoted per minute used with hours overstates the elapsed time by a factor of sixty.
Applying it to a large solid object
A big joint has a hot centre and a cooler surface, so one k cannot describe it. The centre lags the curve badly.
Asking for a temperature below ambient
The curve approaches ambient and never crosses it. A target below the room temperature has no solution, because the object warms instead.
FAQ
What is Newton's law of cooling?
The rate of temperature change is proportional to the difference between the object and its surroundings. That gives an exponential approach to the ambient temperature.
What does the cooling constant k mean?
The fraction of the temperature gap lost per unit time. A k of 0.05 per minute means about five percent of the gap disappears each minute.
What is the half difference time?
The natural log of two divided by k, which is 13.8629 minutes for a k of 0.05. It is the time for the gap to halve whatever the temperatures are.
Does the formula work for warming?
Yes. A cold object in a warm room has a negative gap, and the same curve pulls its temperature up towards ambient.
Can I use Fahrenheit?
Yes, as long as every temperature uses the same scale. The gaps are what matter, and a Fahrenheit degree is smaller, so the numbers change but the physics does not.
Why does my coffee cool faster than the formula predicts?
Evaporation. An open cup loses a large share of its heat as vapour, so the effective cooling constant is higher than a conduction only estimate suggests.
References
- [1]Wikipedia, Newton's law of cooling — https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling
- [2]Wikipedia, Heat transfer — https://en.wikipedia.org/wiki/Heat_transfer
- [3]Wikipedia, Exponential decay — https://en.wikipedia.org/wiki/Exponential_decay