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

1 Answer
May 2, 2017

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)T(t)=Ts+(T0Ts)ekt

Where

T(t)T(t) is the temperature of an object at a given time tt
T_sTs is the surrounding temperature
T_0T0 is the initial temperature of the object
kk 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 kk.

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!