Question

How do I find `k` in Newton's Law of Cooling?

Answer 1

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(t)` 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 + (75 - 16)e^(-10k)`

`51 = 59e^(-10k)`

`51/59 = e^(-10k)`

`ln(51/59) = ln(e^(-10k))`

`ln(51/59) = -10k`

`k = -1/10ln(51/59)`

Use a calculator to get

`k~~ 0.01457`

Hopefully this helps!