# Newton's Law of Cooling

## Key Questions

Set up an equation with all the knowns and solve for the unknown! Make sure to know your law of cooling too, shown in blue in the Explanation section.

#### Explanation:

Newton's Law of Cooling is given by the formula

color(blue)(T(t) = T_s + (T_0 - T_s)e^(-kt)

Where

$T \left(t\right)$ is the temperature of an object at a given time $t$
${T}_{s}$ is the surrounding temperature
${T}_{0}$ is the initial temperature of the object
$k$ is the constant

The constant will be the variable that changes depending on the other conditions. Let's take an example of a question where you would need to find $k$.

The average coffee temperature at a particular coffee shop is 75˚C. Marie purchases a coffee from the local coffee shop. After 10 minutes, the drink has cooled to 67˚ C. The temperature outside the coffee shop is steady at 16˚C. Assuming the coffee follows Newton's Law of Cooling, determine the value of the constant $k$

Let's identify our variables.

T_0 = 75˚C
T_s = 16˚C
$t = 10$
T(t) = 67˚C
k = ?

We have

$67 = 16 + \left(75 - 16\right) {e}^{- 10 k}$

$51 = 59 {e}^{- 10 k}$

$\frac{51}{59} = {e}^{- 10 k}$

$\ln \left(\frac{51}{59}\right) = \ln \left({e}^{- 10 k}\right)$

$\ln \left(\frac{51}{59}\right) = - 10 k$

$k = - \frac{1}{10} \ln \left(\frac{51}{59}\right)$

Use a calculator to get

$k \approx 0.01457$

Hopefully this helps!

$- \left(\mathrm{dT} / \mathrm{dt}\right) \propto \Delta T$
$\implies - \left(d \theta / \mathrm{dt}\right) \propto \Delta \theta$
Newton's Law of Cooling states that, if the temperature 'T' of the body is not very different from that of the surroundings '${T}_{0}$', then rate of Cooling '(-dT/dt)'or' $- \left(d \theta / \mathrm{dt}\right)$' is proportional to the temperature difference between them.