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

1 Answer
May 2, 2017

Answer:

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!