Newton's Law of Cooling Calculator
Enter an initial temperature, ambient temperature, cooling constant and time to find the object's temperature at that moment. Switch to reverse mode to find how long cooling takes, or use the physical-properties panel to derive the cooling constant from heat-transfer coefficient, surface area, mass and specific heat. The cooling curve chart updates live.
What is Newton's Law of Cooling?
Newton's Law of Cooling states that the rate at which an object loses heat to its surroundings is proportional to the temperature difference between the object and those surroundings. Mathematically, dT/dt = -k(T - T_amb), where T is the object temperature, T_amb is the fixed ambient temperature, and k is the cooling constant (a positive number, in units of inverse time). Solving this differential equation yields the exponential decay formula T(t) = T_amb + (T0 - T_amb) × e^(-kt), which gives the temperature at any time t. The law is an excellent approximation when the temperature difference is not too large and radiative heat transfer is small compared to convective and conductive losses.
How to use this calculator
Three solve modes are available. In the default mode, enter the starting temperature, ambient temperature, cooling constant k and elapsed time to find the temperature at that moment. Switch to "Time to reach target T" to enter the same parameters plus a desired final temperature and find how long cooling takes. The "Cooling constant k" mode derives k from physical properties: enter the heat transfer coefficient h, exposed surface area A, mass and specific heat, and the calculator computes k = hA / (mc_p), which you can then carry over to the other modes. The cooling curve chart updates with each change so you can see the full trajectory.
The cooling constant k and what affects it
The cooling constant k = hA / (mc_p), where h is the convective heat transfer coefficient, A is the exposed surface area, m is mass and c_p is specific heat. A high h (forced air, moving liquid) increases k and speeds cooling. A large surface area also increases k, which is why fins on a heat sink or a wider coffee mug cool faster. A larger mass or high specific heat reduces k because more energy must be transferred for the same temperature drop. Typical k values for a cup of coffee in still air are around 0.01 to 0.05 per minute, while a small electronic component being cooled by a fan might have k around 0.5 to 2 per minute.
Half-life and cooling progress
The half-life of the temperature gap is the time for (T - T_amb) to drop by half, and equals ln(2) / k. After one half-life, 50% of the gap remains; after two, 25%; after ten, less than 0.1%. This makes it easy to estimate when an object is "essentially" at ambient. For example, if k = 0.05 min^-1, the half-life is ln(2) / 0.05 = 13.9 minutes. After 70 minutes (five half-lives) the temperature difference is 1/32 of the original, meaning a coffee that started at 90 C in a 20 C room would be within about 2 C of room temperature.
Typical heat transfer coefficients (h) and cooling constants
| Scenario | h (W/m²K) | Cooling rate |
|---|---|---|
| Still air, small object | 5-10 | Slow |
| Still air, large flat surface | 10-25 | Moderate |
| Light forced air (fan) | 25-100 | Fast |
| Strong forced air / wind | 100-300 | Very fast |
| Still water immersion | 200-1000 | Very fast |
| Flowing water | 1000-15000 | Extremely fast |
| Boiling water | 2500-35000 | Extremely fast |
Approximate values for natural convection in still air. Forced convection (fans, liquid cooling) can be 10-100x higher.
Frequently asked questions
What is Newton's Law of Cooling formula?
The formula is T(t) = T_amb + (T0 - T_amb) × e^(-kt), where T(t) is the temperature after time t, T_amb is the ambient (surrounding) temperature, T0 is the initial temperature, k is the cooling constant and e is Euler's number (~2.71828). The formula comes from solving the differential equation dT/dt = -k(T - T_amb).
How do I find the cooling constant k?
You can derive k from physical properties: k = hA / (mc_p), where h is the heat transfer coefficient in W/(m²K), A is surface area in m², m is mass in kg and c_p is specific heat in J/(kg K). Alternatively, if you have two measured temperatures at known times, you can solve k = -ln[(T2 - T_amb) / (T1 - T_amb)] / (t2 - t1). This calculator's "Cooling constant k" mode does the first approach automatically.
What units should k be in?
k has units of inverse time, such as 1/s, 1/min or 1/hr. The value you use in the formula must match the time unit you plug in. If k = 0.05 min^-1, you must measure time in minutes. This calculator keeps the time unit consistent throughout, so switching between seconds, minutes and hours automatically rescales k.
When does Newton's Law of Cooling break down?
The law is an approximation that works well when the temperature difference between the object and surroundings is modest (typically under a few hundred degrees Celsius). At very high temperature differences, thermal radiation (which follows the Stefan-Boltzmann law, proportional to T^4) becomes significant, and the simple exponential model underestimates cooling. The law also assumes the ambient temperature stays constant and the object has a uniform internal temperature (low Biot number).
Can I use this for warming as well as cooling?
Yes. If the initial temperature is below the ambient temperature (for example, a cold drink in a warm room), T0 - T_amb is negative and the formula still applies: the temperature rises exponentially toward ambient rather than falling. The cooling constant k and all formulas remain identical.