Newton's Law of Cooling
Calculate the final temperature of an object cooling over time using Newton's Law of Cooling. Enter initial temp, ambient temp, cooling coefficient, and time for instant results with interactive charts.
About This Calculator
Newton's Law of Cooling describes how an object's temperature changes over time when exposed to an environment with a different temperature. This calculator uses the formula T(t) = Tamb + (T0 − Tamb) × e−kt to compute the temperature of an object at any given time. The law applies to everyday scenarios like a cup of coffee cooling on your desk, food cooling in a refrigerator, or a hot engine cooling down after shutdown.
How It Works
The rate of cooling depends on two factors: the temperature difference between the object and its surroundings, and the cooling coefficient k. The larger the temperature difference, the faster the cooling rate. As the object approaches ambient temperature, the cooling rate slows exponentially — this is why a hot drink cools quickly at first but then takes much longer to reach room temperature.
Formula
T(t) = Tamb + (T0 − Tamb) × e−kt
Where:
- T(t) — Temperature of the object at time t
- Tamb — Ambient (surrounding) temperature
- T0 — Initial temperature of the object
- k — Cooling coefficient (1/s), related to surface area, heat transfer coefficient, and heat capacity
- t — Time elapsed (seconds)
When to Use
This calculator is ideal for students studying thermodynamics, engineers working on thermal management, food safety professionals monitoring cooling times, and anyone curious about how quickly objects heat up or cool down in different environments. It works best when the object is small and has high thermal conductivity (low Biot number).
Regional Notes
The calculator works with any temperature scale (°C, °F, or K) as long as all inputs use the same scale. Defaults are provided in °C. The formula itself is unit-agnostic — just ensure consistency across initial temperature, ambient temperature, and time.
Frequently Asked Questions
What is Newton's Law of Cooling?
Newton's Law of Cooling states that the rate of heat loss of an object is directly proportional to the temperature difference between the object and its surroundings. The formula is T(t) = T_amb + (T₀ − T_amb) × e⁻ᵏᵗ where T(t) is the temperature at time t, T_amb is the ambient temperature, T₀ is the initial temperature, and k is the cooling coefficient.
How do I calculate the final temperature using Newton's Law of Cooling?
Use the formula T(t) = T_amb + (T₀ − T_amb) × e⁻ᵏᵗ. Enter the initial temperature, ambient temperature, cooling coefficient k, and time t into the calculator. For example, a 100°C cup of coffee in a 22°C room with k = 0.015 cools to approximately 35°C after 2 minutes.
What is the cooling coefficient k?
The cooling coefficient k (measured in 1/s) represents how quickly heat is transferred between the object and its environment. It depends on factors like surface area, heat transfer coefficient, and heat capacity. A higher k means faster cooling. Typical values for a cup of liquid range from 0.01 to 0.05 1/s.
Can I use this calculator for both heating and cooling?
Yes. Newton's Law applies to both heating and cooling. If the initial temperature is lower than the ambient temperature, the formula predicts the object heating up toward ambient temperature. Simply enter a lower initial temperature than ambient temperature to model heating.
What units does this calculator use?
Temperatures are in degrees Celsius (°C), cooling coefficient is in reciprocal seconds (1/s), and time is in seconds (s). You can use the same formula with Fahrenheit or Kelvin as long as all temperature values use the same scale.
How long does it take for coffee to cool down?
A typical cup of coffee at 100°C in a 22°C room with a cooling coefficient of 0.015 1/s cools to about 35°C in 120 seconds (2 minutes). The exact time depends on the cup material, surface area, and air circulation.
Is Newton's Law of Cooling accurate for all situations?
Newton's Law of Cooling is most accurate when the Biot number is small (temperature is uniform throughout the object) and convection is the primary heat transfer mechanism. It works well for small objects like cups of liquid but is less accurate for large objects with significant internal temperature gradients.
What is the difference between Newton's Law of Cooling and the heat equation?
Newton's Law of Cooling is a simplified model that assumes the object's temperature is uniform and only changes at the surface. The heat equation (Fourier's law) is more general and accounts for temperature gradients inside the object. Newton's law is a special case that works well when internal conduction is much faster than surface convection.