How do I find #k# in Newton's Law of Cooling?
1 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
The average coffee temperature at a particular coffee shop is
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!