Newton Cooling Law Calculator

Calculator Inputs

Choose a mode to predict temperature, solve for time, or estimate the cooling constant from observed data.

Calculation Mode

Temperature Units

Initial Temperature

Ambient Temperature

Cooling Constant k (per minute)

Time Elapsed / Observation Time

Target Temperature

Observed Temperature

Graph Time Step

Graph Duration

Example Data Table

Case	Initial Temp	Ambient Temp	k	Time	Observed / Target Temp	Purpose
Hot Coffee	90 °C	25 °C	0.1200	10 min	46.1353 °C	Predict temperature after cooling
Metal Sample	150 °C	30 °C	0.0800	20 min	54.2278 °C	Check process cooling behavior
Soup Container	85 °C	22 °C	0.1000	—	40 °C	Find time needed to reach serving temperature
Lab Flask	100 °C	24 °C	Estimated	12 min	58 °C	Estimate cooling constant from measurement

Formula Used

Newton’s law of cooling equation:

T(t) = T_a + (T₀ - T_a)e^-kt

Where:

T(t) = temperature at time t
T₀ = initial object temperature
T_a = ambient temperature
k = cooling constant
t = elapsed time

Time to reach a target temperature:

t = -ln((T_target - T_a) / (T₀ - T_a)) / k

Cooling constant from one observation:

k = -ln((T_obs - T_a) / (T₀ - T_a)) / t

The model assumes heat transfer rate is proportional to the temperature difference between the object and its surroundings.

How to Use This Calculator

Select the calculation mode that matches your problem.
Enter the initial and ambient temperatures in the same unit system.
Provide the cooling constant, target value, or observation data as required.
Choose a graph duration and time step for the plotted curve.
Press Calculate to show results above the form.
Review the summary table and graph for interpretation.
Use the export buttons to save the result as CSV or PDF.

Frequently Asked Questions

1. What does Newton’s cooling law describe?

It describes how an object’s temperature approaches ambient temperature over time. The rate of change depends on the difference between the object and its surroundings.

2. When is this model accurate?

It works best when ambient temperature stays nearly constant, the object is reasonably uniform internally, and heat transfer conditions do not change significantly during cooling.

3. What is the cooling constant k?

The cooling constant measures how quickly temperature difference shrinks. Larger values mean faster cooling, while smaller values indicate slower thermal decay.

4. Can I use Fahrenheit or Kelvin?

Yes. The calculator supports multiple unit labels. Keep all entered temperatures in the same unit system for mathematically correct results.

5. Why must the target temperature be between initial and ambient values?

Because the model predicts exponential approach toward ambient temperature. A valid cooling target must lie between the starting value and the surrounding temperature.

6. How is the graph useful?

The graph reveals the full cooling path, not only one answer. It helps compare temperature decline, ambient reference, and the speed of approach across time.

7. Can this calculator estimate k from measured data?

Yes. Use the estimation mode with an initial temperature, ambient temperature, measured temperature, and observation time to compute the cooling constant.

8. What are common applications of this law?

Common uses include food cooling, industrial heat treatment, forensic temperature studies, chemical process monitoring, and laboratory thermal experiments.