Calculating Newton's law of cooling allows you to accurately model the effect of heat transfer in many processes. If you are searching for:

  • A simple explanation of Newton's law of cooling* equation;
  • A derivation of the formula for Newton's law of cooling;
  • The formula for the rate of cooling; or
  • A way to calculate the time to reach a temperature.

You are in the right place: our article and tool will answer all your questions!

Keep your cool: how to calculate the time to reach a temperature

Cooling and heating processes are at the core of thermodynamics. However, when studying variation in temperature due to heat transfer, we can forgo dealing with entropy, enthalpy, and all the rest.

Each body varies its temperature in specific ways, which depend on many factors. However, the fundamental mechanisms for heat transfer are just three:

  • Convection;
  • Conduction; and
  • Radiation.

Our Newton's law of cooling calculator will deal only with the first two, and it's good to remember that the law works better for small contributions due to convection.

The variation in temperature of a body depends on:

  • The difference between the body temperature and the environment; and
  • The physical properties of the body.

The cooling coefficient models the latter:

k=hACk =h\cdot \frac{A}{C}

Where the value of the coefficient kk depends on:

  • hh — the heat transfer coefficient (with units W/(m2K)\text{W}/\left(\text{m}^2\cdot\text{K}\right));
  • AA — The heat exchanging surface; and
  • CC — The heat capacity in J/K\text{J}/\text{K}.

Since we introduced the cooling coefficient, we can proceed with Newton's cooling formula.

Calculating Netwon's law of cooling: equation and derivation

Newton's law of cooling equation appeared first in differential form: the scientist found that the rate of variation of the temperature is directly proportional to the variation in temperature**.

Here's the formula for cooling in Newton's words:

T˙(t)=k(TRT(t))\dot{T}(t) = k\cdot \left(T_{\text{R}} - T(t)\right)

Where:

  • T˙(t)\dot{T}(t) and T(t)T(t) are, respectively, the rate of heat loss — which corresponds to a rate of variation of temperature — and the instantaneous temperature at time tt.
  • TRT_{\text{R}} is the temperature of the environment.

Even though rather pretty, this formula is unwieldy for many reasons. We can solve it as a differential equation by setting a known solution that T(0)=TinitT(0)=T_{\text{init}} and that for T(t)=TRT(t)=T_{\text{R}}, T˙(t)=0\dot{T}(t)=0.

The solution sees the appearance of an exponential function:

T(t)=TR(TinitTR)ektT(t) = T_{\text{R}}\cdot\left(T_{\text{init}} - T_{\text{R}}\right)\cdot e^{-k\cdot t}

This equation allows us to calculate the time to reach a temperature since both are explicit parameters.

🙋 Our Newton's law of cooling calculator implements both equations; the result of the differential form is available if you click on advanced mode.

What you can see from the equation is that cooling is an exponential process: it begins as fast as possible, and it slows down when the temperature of the hotter body approaches the one of the environment: it is the opposite of an exponential growth.

🙋 Use our temperature converter to switch seamlessly between various temperature measurement units.

Past Newton's law of cooling: is there a formula for Newton's law of heating?

Yup! Since physics is not scared by minus sign, we can apply Newton's law of cooling for negative differences in temperature without additional errors in the forecasted behavior.

Even if our daily experience makes cooling easier to observe than heating — for many reasons — worry not and plug your values in our Newton's law of cooling calculator! If you want to learn more about heating processes, our [water heating calculator(calc:4192) is here to help.

Wrong Newton's law? If you are looking for the uber-famous relationship between force and acceleration, head straight to our Newton's second law calculator!

Davide Borchia
Ambient temperature
°F
Initial temperature
°F
Cooling constant
sec⁻¹
What's the temperature after...
min
?
Temperature
°F
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