Newton's Law of Cooling Calculator

Understanding Newton's Law of Cooling Calculator

Newton's Law of Cooling states that the rate of temperature change is proportional to the difference between the object and environment temperatures.

How to use this calculator

1. Enter the initial temperature of the object, \(T_0\).
2. Enter the surrounding temperature, \(T_{env}\).
3. Enter the time, \(t\), and the cooling constant, \(k\).
4. Compute \(T(t) = T_{env} + (T_0 - T_{env})e^{-kt}\) to find the temperature at that time.

The formula explained

The formula computes the object's temperature after time \(t\) as it moves toward the surrounding temperature. The exponential term \(e^{-kt}\) shows that the temperature change happens quickly at first and then slows down.

• T(t) = the temperature of the object after time \(t\)
• \(T_{env}\) = the surrounding or ambient temperature
• \(T_0\) = the initial temperature of the object at \(t=0\)
• k = the cooling constant, which controls how fast the temperature changes
• t = time elapsed

Step by step method

1. Start with the formula \(T(t) = T_{env} + (T_0 - T_{env})e^{-kt}\).
2. Substitute the values for \(T_0\), \(T_{env}\), \(k\), and \(t\).
3. Evaluate the exponential term \(e^{-kt}\).
4. Add the result to \(T_{env}\) to get the temperature.

Worked example

Problem. A cup of coffee starts at \(90^B0\mathrm{C}\). The room is \(20^B0\mathrm{C}\), the cooling constant is \(k=0.15\), and you want the temperature after \(10\) minutes. Find \(T(10)\).

1. Use the formula, \(T(10) = 20 + (90 - 20)e^{-0.15(10)}\).
2. Simplify, \(T(10) = 20 + 70e^{-1.5}\). Since \(e^{-1.5} \approx 0.2231\), \(70e^{-1.5} \approx 15.62\).
3. Add the room temperature, \(T(10) \approx 20 + 15.62 = 35.62\).

Tips and common mistakes

• Be sure \(T_{env}\) is the temperature of the surroundings, not the object.
• The constant \(k\) must be positive, and larger values mean faster cooling or warming toward the environment.

Frequently asked questions