Law of Cooling Calculator

Calculator Inputs

Solve for

Temperature unit

Time unit

Ambient temperature

Initial temperature

Observed or final temperature

Target temperature

Cooling constant k

Elapsed time

Mass, optional kg

Specific heat, optional J/kg K

Surface area, optional m²

Decimal places

Formula Used

The calculator uses Newton's law of cooling:

T(t) = Ta + (T0 - Ta)e^(-kt)

T0 is initial temperature. Ta is ambient temperature. k is the cooling constant. t is elapsed time.

For time solving, it uses t = -ln((T - Ta) / (T0 - Ta)) / k.

For constant solving, it uses k = -ln((T - Ta) / (T0 - Ta)) / t.

For heat transfer estimation, it uses k = hA / (mc), so h = kmc / A.

How to Use This Calculator

Select the unknown value from the solve menu.
Enter temperatures using one consistent temperature scale.
Enter the cooling constant per selected time unit.
Use observed final temperature when solving for the constant.
Use target temperature when solving for elapsed time.
Add mass, specific heat, and surface area only when needed.
Press Calculate to view results above the form.
Use CSV or PDF buttons to save the solved result.

Example Data Table

Ambient	Initial	k per minute	Time	Estimated temperature
22 °C	90 °C	0.04	10 minutes	67.58 °C
22 °C	90 °C	0.04	20 minutes	52.55 °C
22 °C	90 °C	0.04	30 minutes	42.47 °C
22 °C	90 °C	0.04	40 minutes	35.73 °C

Understanding Newton Cooling

Newton’s law of cooling describes how temperature changes when an object sits in a surrounding medium. The model is useful when the surrounding temperature stays nearly constant. It works well for warm drinks, hot metal parts, food cooling, and many classroom physics problems.

What the calculator solves

This calculator handles several unknowns. It can find the final temperature after a selected time. It can find the elapsed time needed to reach a target temperature. It can estimate the cooling constant from observed data. It can also work backward for the starting temperature or ambient temperature. Each option uses the same exponential model, so the results stay consistent.

Why the constant matters

The cooling constant is the main control value. A larger constant means faster change. A smaller constant means slower change. The constant depends on surface area, airflow, material, insulation, and contact conditions. A thin metal cup cools faster than a covered ceramic mug. Moving air also increases the constant because heat leaves the surface more quickly.

Reading the result

The result shows the solved value, remaining temperature difference, half approach time, and initial cooling rate. The half approach time is the time needed for the temperature difference from ambient to fall by one half. This is not radioactive half life. It is only an exponential comparison value. The cooling rate changes during the process. It is highest at the start, when the temperature difference is largest.

Using real measurements

For practical use, measure temperatures with the same unit. Keep the ambient condition steady. Record time carefully. When estimating the constant, choose two measurements that are not too close together. Small measurement errors can strongly affect the logarithm. If the target temperature crosses the ambient value, the model is not valid for positive cooling.

Physics limits

The law is an approximation. It assumes one uniform object temperature. Large objects may have internal temperature gradients. Boiling, freezing, evaporation, radiation dominance, or changing airflow can reduce accuracy. Still, the model is very helpful for quick estimates, lab checks, and educational planning. Use it as a clear guide, then compare with measured results. Document assumptions, units, and test conditions so later comparisons remain fair, repeatable, and easier to explain well.

FAQs

What is Newton's law of cooling?

It is an exponential model for temperature change. It says the rate of change is proportional to the difference between object temperature and ambient temperature.

Can this calculator handle heating?

Yes. If the ambient temperature is higher than the object, the same formula models heating toward ambient temperature.

What does the cooling constant mean?

The cooling constant shows how quickly the temperature difference shrinks. A larger value means the object approaches ambient temperature faster.

Do all temperatures need the same unit?

Yes. Enter every temperature in the same scale. Do not mix Celsius, Fahrenheit, or Kelvin values in one calculation.

Why is my time result invalid?

The target may be beyond ambient temperature or on the wrong side of ambient. A positive cooling constant cannot cross ambient in this model.

How do I find k from lab data?

Choose the cooling constant option. Enter ambient temperature, starting temperature, observed final temperature, and elapsed time between the two readings.

What is half approach time?

It is the time required for the temperature difference from ambient to reduce by half. It is a useful exponential comparison value.

When is this model less accurate?

It is less accurate when ambient temperature changes, airflow changes, the object has large internal gradients, or phase changes occur.